For Marston Conder’s 65th Birthday
I have said many times that everything you do is dependent on who you know, and who
you know is often a matter of luck. In my case, I must be very lucky to have had the chance
to work with Marston. Of course, having a Conder-number of one isn’t really a matter or
luck, since there are 100 other mathematicians with Conder-number one.
I probably have known Marston as long as any of his many co-authors. I still have a
handwritten letter I wrote to Graham Higman in 1982 asking about the work of one of his
students on Hurwitz groups 〈a, b : a2 = b3 = (ab)7 = 1, . . . 〉. I have it because Higman
returned it to me saying his student would write me. And I got a typed (!) response from
Marston summarizing his results, which I also still have. We arranged to meet at Tübingen
on my first mathematical trip to Europe for an Oberwolfach meeting. We missed connecting
on the first day, but we did get together the next day for a long walk and talk. It was then
16 years until we saw each other again in Flagstaff for the first SIGMAP workshop. But it
was not until Marston’s first New Zealand conference at Auckland in December 2000 that
we started working together on regular Cayley maps, mostly with Robert Jajcay.
Marston has made up for those 16 years of not traveling by becoming a presence at
combinatorics conferences throughout Europe, the US and the East (often bringing two or
three bottles of New Zealand wine). Last March I was stunned that he was going to make
the 24,000 mile round trip from New Zealand to Boston just for a 20-minute talk at a two-
day AMS Sectional conference at Tufts (cancelled of course). His frequent traveler miles
are near 2 million. When I ask him how he’ll use them, he talks about a round-the-world
trip with Jenny, but then I point out he actually needs to take ten such trips.
Working with Marston will always be one of the joys of my mathematical life. First, if
it comes to permutation groups, graphs, polytopes, maps, dessins, Riemann surfaces, he’s
been there. And if he hasn’t been there before, he picks it up immediately. Second, he
also brings with him all the mathematical machinery from those fields. I remember his
amusement when he managed to use most of the group theoretic techniques he had learned
in graduate school in one paper: the 2011 JEMS paper with Jozef, which to this day is the
paper I am proudest of (with no training in group theory, I was the bare-hands co-author).
Of course, when it comes to machinery, there is MAGMA. His files of all regular maps,
hypermaps, symmetric and semi-symmetric cubic graphs, up to some given order, are a
foundation for work in these fields. At one of Marston’s Queenstown conferences, there
was a short course on MAGMA. I got blown away in the first 10 minutes. Then I realized
I didn’t need MAGMA; I had Marston. So much easier to just email him. But there is
something much more important about Marston’s work with computers. In mathematics, it
is always examples first, theorems second. I remember him saying once: “All-but-finitely-
many results are nice, but I am interested in those finitely many." That is where the real
work and beauty are. And that is Marston.
Tom Tucker
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.01
https://doi.org/10.26493/2590-9770.1531.46a
(Also available at http://adam-journal.eu)
Hole operations on Hurwitz maps*
Gábor Gévay†
Bolyai Institute, University of Szeged
H-6720 Szeged, Hungary
Gareth A. Jones
School of Mathematics, University of Southampton
Southampton SO17 1BJ, UK
Dedicated to Marston Conder in celebration of his 65th birthday
Received 14 October 2021, accepted 19 March 2022, published online 27 June 2022
Abstract
For a given group G the orientably regular maps with orientation-preserving automor-
phism group G are used as the vertices of a graph O(G), with undirected and directed
edges showing the effect of duality and hole operations on these maps. Some examples of
these graphs are given, including several for small Hurwitz groups. For some G, such as
the affine groups AGL1(2e), the graph O(G) is connected, whereas for some other infinite
families, such as the alternating and symmetric groups, the number of connected compo-
nents is unbounded.
Keywords: Regular map, orientably regular, duality, hole operation, Hurwitz group.
Math. Subj. Class.: 05C10 (primary), 20B25 (secondary)
1 Introduction
For any group G let O(G) denote the set of (isomorphism classes of) orientably regular
maps M with orientation-preserving automorphism group Aut+M isomorphic to G. For
a given finite group G this set is finite since its elements correspond bijectively to the
orbits of AutG on certain generating sets for G. These maps are related to each other
*The authors thank the referees for a number of valuable comments.
†Corresponding author. Supported by the Hungarian National Research, Development and Innovation Office,
OTKA grant No. SNN 132625.
E-mail addresses: gevay@math.u-szeged.hu (Gábor Gévay), G.A.Jones@maths.soton.ac.uk (Gareth A.
Jones)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.01
by the application of certain operations, described by Wilson in [40], such as duality D
and the hole operations Hj for j coprime to the valency of the vertices, since all of these
preserve the group G. One way of understanding how the various maps associated with
G are related to each other is to regard this set O(G) as the vertex set of a graph, with
directed or undirected edges indicating the actions of these operations. For example, one
can consider whether or not this graph is connected, and if not, what invariants might be
used to distinguish maps in different components. We will describe the maps M and graphs
O(G) arising for some particularly interesting groups, including several of the smallest
Hurwitz groups. We will show that for the symmetric and alternating groups, the number
of connected components of O(G) is unbounded, building on work of Marston Conder [5]
on alternating groups as Hurwitz groups in the latter case.
In addition to topological graph theory and the theory of Riemann surfaces, Grothen-
dieck’s dessins d’enfants provide motivation for this study, as a preparation for an investi-
gation of the relationship between the actions of these operations and that of the absolute
Galois group GalQ/Q on finite oriented maps, regarded as algebraic curves defined over
number fields (see [26], for example). We also see this paper as a first step towards a study
of a wider class of operations, including Petrie duality, on the set R(G) of all regular maps
with a given automorphism group G. Constraints of space and time oblige us to defer these
extensions to a later date.
2 Orientably regular maps
If M is an orientably regular map then its arcs (directed edges) can be identified with the
elements of a group G, so that its orientation-preserving monodromy and automorphism
groups are identified with the right and left regular representations of G.
Let M have type {p, q}, meaning that its faces are p-gons and its vertices have valency
q. (The term Schläfli type, or Schläfli symbol, is also used [10]). We note that later on in
this paper, we shall use the same symbols p and q for denoting a prime and a prime power,
respectively; to avoid confusion, where it is not clear from the context, the actual meaning
of these symbols will be emphasized.)
Then G has generators x, y, z satisfying
xq = y2 = zp = xyz = 1,
where x and y act (as monodromy permutations) by rotating all arcs around their incident
vertices, following the orientation, and by reversing them, so that z rotates them around
faces. Conversely any such generating triple for G defines a map M of this type, with
isomorphic maps corresponding to triples equivalent under AutG. Of course the triple,
and hence the map, is uniquely determined by any two of its members, so we will often
restrict attention to the generating pair x, y.
Map duality D, transposing vertices and faces, replaces the triple (x, y, z) with (z, y,
(zy)−1 = xy), corresponding to the dual map D(M). Then D2 sends (x, y, z) to (xy, y, zy)
= (x, y, z)y , giving a map isomorphic to M.
If j is coprime to q then another orientably regular map Mj = Hj(M), of type {p′, q}
for some p′ and with the same orientation-preserving automorphism group G as M, can
be found by applying the hole operation Hj , described by Wilson in [40], to M. This map
embeds the same graph as M, but the rotation x of arcs around each vertex is replaced with
xj , so that the faces, and hence the underlying surface, may be changed. In terms of group
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 3
theory, Hj(M) corresponds to the generating triple (xj , y, zj) for G, where zj := (xjy)−1.
Thus the boundary of each face of Hj(M), given the orientation of the underlying surface,
follows a j-hole of M, at each vertex v taking the j-th incident edge, following the reverse
orientation around v, rather than the first; its length is the order of the element zj . (Note
that a j-hole, taken in the reverse direction, becomes an (q − j)-hole.)
Example 2.1. Figure 1 shows a 3-hole (16, 5, 6, 20, 16) of length 4 in Klein’s map K of
type {3, 7} and genus 3 with G = PSL2(7) (see Section 7 for details).
Figure 1: Klein’s map K of type {3, 7} and genus 3 represented in Poincaré’s disc model
of the hyperbolic plane. A 3-hole (16, 5, 6, 20, 16) is highlighted.
Clearly Hj = Hk whenever j ≡ k mod (q); since Hj ◦ Hk = Hjk for all j and k
coprime to q, these operations give a representation of the group Uq of units mod (q). The
operation H−1 sends each map to its mirror-image, so an orientably regular map M is
regular (has a flag-transitive full automorphism group AutM) if and only if H−1(M) ∼=
M; thus the action on regular maps gives a representation of the group Uq/{±1}.
One can also apply Hj to M when k := gcd(j, q) > 1. However, in this case the
operation is not invertible, and it does not preserve the embedded graph, since each vertex
is replaced with k vertices of valency q/k. Moreover, the orientation-preserving auto-
morphism group may also change: if G0 := ⟨xj , y⟩ is a proper subgroup of index n in
G, then Hj(M) is the disjoint union of n isomorphic connected orientably regular maps
M0, each with Aut+M0 ∼= G0, and Aut+Hj(M) is isomorphic to the wreath product
G0 ≀Sn. Since our aim is to study those maps with a given orientation-preserving automor-
phism group G, we will avoid such complications by concentrating on cases where k = 1.
4 Art Discrete Appl. Math. 5 (2022) #P3.01
A further reason for doing this is that we eventually hope to compare hole operations with
cyclotomic Galois operations, and these act by raising qth roots of unity to their jth powers
where gcd(j, q) = 1.
Another useful map operation is Petrie duality. A Petrie polygon in a map M is a closed
zig-zag path turning alternately first right and first left at successive vertices. The Petrie
dual P (M) of M embeds the same graph as M, but the faces of M are replaced with new
faces bounded by the Petrie polygons. If M is orientably regular their length r (the Petrie
length of M) is twice the order of the commutator [x, y] = x−1y−1xy = x−1yxy, so that
P (M) has type {r, q}. As in [11, 12], we will often refer to the extended type {p, q}r
of M. For instance, in Figure 1 the 3-holes (16, 5, 6, 20, 16) and (24, 17, 13, 19, 24) in K
enclose a Petrie polygon of length r = 8. (More generally, whenever p = 3 each Petrie
polygon is enclosed by two 3-holes of length r/2 in this way.)
Given any group G, one can regard O(G) as the vertex set of a graph (also denoted by
O(G)) by adding edges showing the actions of the operations D and Hj (for selected j);
they will be undirected for cycles of length 2, e.g. for involutions such as D, and directed in
the case of longer cycles for hole operations Hj . Loops, corresponding to invariant maps,
will be omitted. In particular, we will use dashed and dotted undirected edges for D and
H−1, and unbroken edges for other hole operations. Although the operation P preserves
the full automorphism group of a map, it has the disadvantage from our current point of
view of not always preserving orientability. For example, if M is the tetrahedral map on
the sphere, then P (M) is the antipodal quotient of the cube, on the real projective plane;
similarly, P (K) is a non-orientable regular map of genus 41. In this paper we will therefore
concentrate mainly on the operations D and Hj , which preserve automorphism groups and
orientability.
In general we will not draw edges labelled Hj for all j coprime to the relevant valencies
q, but merely enough to identify the connected components of O(G) by showing how
various maps can be tranformed into each other. For example, if Uq (or Uq/{±1} in the
case of regular maps) is cyclic it is sufficient to take j to be a generator.
Provided its genus g is neither too large nor less than 2, each map M ∈ O(G) can
be located in Conder’s computer-generated lists of regular or chiral orientable maps in [7].
Each entry there has the form Rg.n or Cg.n respectively, where g denotes the genus, and
n denotes the nth entry in the list of maps of that genus, ordered lexicographically by their
type {p, q}, with p ≤ q. Within each list, each entry refers either to a dual pair of maps of
types {p, q} and {q, p} or to a single self-dual map with p = q. If p < q we will denote
the maps of type {p, q} and {q, p} by Xg.na and Xg.nb respectively, where X is R or C; if
p = q and the entry denotes a non-isomorphic dual pair we will assign the labels Xg.na and
Xg.nb arbitrarily, whereas a single self-dual map will be denoted simply by Xg.n. Similar
conventions will apply to the list of non-orientable regular maps, denoted by Ng.n in [7].
2.1 Some simple examples of graphs O(G)
If G is the alternating group A4 then O(G) contains only the tetrahedral map {3, 3} of
genus 0.
If G is a dihedral group Dm for some m > 2 then O(G) consists of the dual pair of
maps {2,m} and {m, 2} of genus 0, whereas if m = 2, so that G = D2 ∼= V4, then O(G)
contains only the self-dual map {2, 2} of genus 0. In either case the operations Hj for j
coprime to m act trivially on O(G).
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 5
If G = S4 then the involution y must be a transposition: the double transpositions lie in
the normal subgroup K ∼= V4 with G/K ∼= S3 non-cyclic, so they cannot be members of
generating pairs. One can take x to be any 3-cycle or 4-cycle not inverted by y, giving the
cube {4, 3} or its dual, the octahedron {3, 4}. The operations Hj act trivially on O(G).
The graphs O(G) corresponding to these groups G are shown in Figure 2.
O(A4) O(Dn), n > 2 O(D2) O(S4)
{3, 3} {2, n} {n, 2} {2, 2} {3, 4} {4, 3}
Figure 2: The graphs O(A4), O(Dn) and O(S4)
If G is the alternating group A5 (also isomorphic to PSL2(4) and PSL2(5)) then O(G)
consists of three maps, namely the dodecahedron {5, 3}, the icosahedron {3, 5} and the
great dodecahedron {5, 5}6 of genus 4 (the self-dual map R4.6 in [7]); duality D transposes
the first two, while H2 transposes the last two, so O(G) is connected (see Figure 3).
{5, 5}6{3, 5}{5, 3}
Figure 3: The graph O(A5)
3 Isotactic polygons
Let M be an orientably regular map of type {p, q}. An isotactic polygon P of type
(d1, d2, . . . , dm)
n in M is a closed path in the underlying graph of M formed by suc-
cessively taking the di-th edge at the i-th vertex vi visited, following the local orientation
around vi, using the sequence d1, d2, . . . , dm ∈ Zq repeated n times. We call di the right
degree of P at vi, and l = mn the length of P . By orientable regularity, if such a poly-
gon P exists, then it does so starting at any directed edge in M, since it is equivalent to a
relation
(xd1yxd2y . . . xdmy)n = 1
in the monodromy group G = ⟨x, y⟩ of M.
This concept is a common generalisation of the classical notions of face, Petrie polygon
and hole. In fact,
• an isotactic polygon of type (1)p or (−1)p is the boundary of a p-valent face;
• more generally, an isotactic polygon of type (j)l with j coprime to q is a j-hole of
length l;
• an isotactic polygon of type (1,−1)r/2 is a Petrie polygon of length r (provided this
length is even);
• more generally, an isotactic polygon of type (j,−j)r/2 is a Petrie polygon of order
j, where r is its (even) length.
6 Art Discrete Appl. Math. 5 (2022) #P3.01
These well-known examples of isotactic polygons are distinguished by their roles in
the Petrie and hole operations, but other types also occur in various roles. For example,
hexagons of type (2, 4)3 occur in the polyhedral realisation of Klein’s map K of genus 3
(see Section 7) given by Schulte and Wills in [31]. On the other hand, the heuristic role of
hexagons of type (2, 4, 4)2 is emphasised in the investigation of 7-fold rotational symmetry
of the Fricke–Macbeath map F of genus 7 (see Section 8) in [3, 4]. Moreover, in this map
the 3-holes, together with three suitable triangular faces, form generalised Petersen graphs
of type GP(9, 3) (see Figure 4); the presence of such subgraphs in the underlying graph of
F confirms that this map has no polyhedral embedding in E3 with 9-fold symmetry [4].
Figure 4: A generalized Petersen graph of type GP(9,3) in the Fricke–Macbeath map F of
type {3, 7} and genus 7 (vertex labels correspond to those given in Figure 9 below).
4 Regularity
An orientably regular map M, corresponding to a generating pair x, y of G, is regular if
and only if there is an automorphism α of G inverting x and y, or equivalently inverting
x and centralising y. In this case the full automorphism group AutM is a semidirect
product G ⋊ ⟨α⟩ of G and C2, with α acting as a reflection of M. For example, the maps
in O(G) for G = A4, Dn and S4 are all regular, with AutM ∼= S4, Dn ×C2 and S4 ×C2
respectively.
We will say that a regular map M is inner or outer regular as α is or is not an inner
automorphism, that is, conjugation by some element c ∈ G. If α is inner then c2 is in
the centre Z(G) of G, so if Z(G) = 1 (as is the case with all the groups considered
here, apart from Dn for n even), then c2 = 1 and AutM ∼= G × C2 with αc = αc−1
generating the second direct factor. In this case M has a non-orientable regular quotient
map M = M/C2, of genus g + 1 where M has genus g, with automorphism group G;
then M is the canonical orientable double cover of M.
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 7
It is easy to check that regularity, inner regularity and AutM are preserved by the op-
erations D and Hj , so they are constant throughout each connected component of O(G). In
particular, maps in O(G) with different regularity properties must lie in different connected
components of the graph.
4.1 Non-orientable regular maps
Although the emphasis of this paper is on orientably regular maps M with a given group
Aut+M ∼= G, non-orientable regular maps N with AutN ∼= G will also be detected by
the construction of O(G). (Note that if G has no subgroup of index 2, then every map in
R(G) is non-orientable.) Each such map N has a canonical orientable double cover M,
an inner regular map with AutM ∼= G × Z where Z ∼= C2, and with Aut+M ∼= G, so
that N arises as the central quotient M = M/Z of a map M ∈ O(G).
Conversely, each inner regular map M ∈ O(G) has full automorphism group AutM =
G × Z, with Z ∼= C2 reversing orientation, so it has a non-orientable regular quotient
N = M = M/Z with automorphism group G. This gives a bijection M 7→ N between
the subsets of O×(G) ⊆ O(G) and R−(G) ⊆ R(G) consisting of their inner regular and
non-orientable maps; here N has the same type as M (but half its Petrie length), and has
genus g + 1 if M has genus g. Since this bijection commutes with the operations D and
Hj , it induces an isomorphism O×(G) → R−(G) of directed graphs.
4.2 O(G) for G = S5
The earlier examples were very straightforward, involving familiar maps. This case needs
rather more thought.
If G = S5 then the involution y can be a transposition or a double transposition. In
the first case, there are possible generators x of orders 4, 5 and 6, giving rise to two dual
pairs of regular maps, the dual pair R4.2 of types {4, 5}6 and {5, 4}6 on Bring’s curve of
genus 4 (see [38], for example), and the dual pair R9.16 of types {5, 6}4 and {6, 5}4, with
H2 transposing R4.2a and R9.16b; the component of O(G) containing these four maps is
a path graph, with three edges labelled D,H2, D.
In the second case x, which must be odd, can have order 4 or 6, giving the dual pair
R6.2 of types {4, 6}10 and {6, 4}10, together with the self-dual map R11.5 of type {6, 6}6
(for this last map one can take x = (1, 5, 3)(2, 4), y = (1, 2)(3, 4)); all operations Hj
act trivially on these maps, so the pair R6.2 and the map R11.5 form two more connected
components of O(G). The graph is shown in Figure 5: the vertices are labelled with the
extended types and our extension of the notation in [7] for the corresponding maps. This
example shows that the involution y (or more precisely its orbit under AutG) is insufficient
to characterise a component of O(G).
The maps M ∈ O(G) are all regular, and since Out S5 and Z(S5) are both trivial they
are inner regular, with automorphism group S5 × C2; their non-orientable quotients M,
reading Figure 5 from left to right, are the dual pairs N5.1, N10.4, N7.1 and the self-dual
map N12.3, all with automorphism group S5.
We note that R6.2b occurs first as one of Coxeter’s regular skew polyhedra [10]. It is re-
alised as a subcomplex of the boundary complex of the dual of the 4-polytope t1,2{3, 3, 3}.
Its faces form the faces of 10 equal Archimedean truncated tetrahedra (the facets of the
polytope). The non-identity element of C2 is realised in this case as a central inver-
sion (i.e. reflection in a point) in E4 interchanging two disjoint 10-tuples of hexagons.
8 Art Discrete Appl. Math. 5 (2022) #P3.01
R4.2b R4.2a R9.16b R9.16a R6.2a R6.2b R11.5
{5, 4}6 {4, 5}6 {6, 5}4 {5, 6}4 {4, 6}10 {6, 4}10 {6, 6}6
Figure 5: The graph O(S5).
Two triples of hexagons sharing a hole of length 3 belong to the same such 10-tuples; in
Figure 6 both the hexagonal faces and the triangular holes can clearly be seen.
Figure 6: The combinatorial structure of R6.2b [18], with one hole of length 3 highlighted.
R4.2a goes back even earlier, in fact to Gordan’s work [19]. Both the dual pairs R4.2a,
R4.2b and R6.2a, R6.2b have polyhedral realisations in Euclidean 3-space [32, 33].
5 Hurwitz surfaces, groups and maps
Many of our chosen examples of automorphism groups are Hurwitz groups. This is partly
because these groups exhibit interesting phenomena, and partly because they and their
associated maps and surfaces have been intensively studied. Here we summarise some of
their important properties.
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 9
Hurwitz [23] showed that the automorphism group G = AutS of a compact Riemann
surface S of genus g ≥ 2 has order at most 84(g − 1), attained if and only if S ∼= H/K,
where H is the hyperbolic plane and K is a normal subgroup of finite index in the triangle
group
∆ = ∆(7, 2, 3) = ⟨X,Y, Z | X7 = Y 2 = Z3 = XY Z = 1⟩
with G ∼= ∆/K. Such surfaces and groups attaining this upper bound are called Hurwitz
surfaces and Hurwitz groups. In each case, S carries an orientably regular map M of type
{3, 7}, called a Hurwitz map, with orientation-preserving automorphism group G. This
map is regular if and only if K is normal in the extended triangle group ∆[7, 2, 3] which
contains ∆ with index 2, in which case the full automorphism group of M is isomorphic to
∆[7, 2, 3]/K; this is equivalent to G having an automorphism inverting two of its standard
generators x, y and z. Conder has written some very useful surveys on Hurwitz groups
in [6, 8].
Every Hurwitz group is perfect, since ∆ is, so it is a covering of a non-abelian finite
simple group, which is itself a Hurwitz group. Two important infinite classes of simple
Hurwitz groups are given by the following theorems [30, 5]:
Theorem 5.1 (Macbeath, 1969). The group PSL2(q) is a Hurwitz group if and only if
(1) q = 7, with a unique Hurwitz surface and map, or
(2) q = p for some prime p ≡ ±1 mod (7), with three Hurwitz surfaces and maps for
each q, or
(3) q = p3 for some prime p ≡ ±2 or ±3 mod (7), with one Hurwitz surface and map
for each q.
(Note that here we use the standard group notation, so that q denotes a prime power.)
Theorem 5.2 (Conder, 1980). The alternating group An is a Hurwitz group for each
n ≥ 168.
In [5], Conder also determined which alternating groups of degree n < 168 are Hurwitz
groups; the smallest is A15.
6 Maps with G = PSL2(q)
6.1 Properties of PSL2(q)
Later in this paper we will construct the graphs O(G) for some specific groups G =
PSL2(q). Here we briefly summarise a few of the properties of these groups which we
will need; see [13, Ch. XII] or [22, II.8] for full details.
The group G = PSL2(q) = SL2(q)/{±I}, q = pe, has order q(q2 − 1) or q(q2 − 1)/2
as p = 2 or p > 2. It is simple for all q ≥ 4.
One useful property of G = PSL2(q) is that conjugacy classes of non-identity elements
g ∈ G are determined uniquely by their traces (more precisely, trace-pairs) ±tr(g) ∈ Fq .
For example, the elements of orders 2, 3 and p are the non-identity elements with traces 0,
±1 and ±2 respectively.
10 Art Discrete Appl. Math. 5 (2022) #P3.01
6.2 Consistent choice of y
For any group G, since the generator y is invariant (up to automorphisms) under the opera-
tions D and Hj , it makes sense to use the same element for each map in a given component
of O(G), or even for each map in O(G) when this is possible.
For example, if G = PSL2(q) then all involutions in G are conjugate to
y =
(
0 1
−1 0
)
.
(Here we identify each matrix A ∈ SL2(q) with −A.) With this as y, its zero entries make
it easier to see the traces of words such as xjy and [x, y], and hence to find their orders,
giving the extended types of various maps.
For instance, if y is as above and
x =
(
a b
c d
)
then xy =
(
−b a
−d c
)
, so z = (xy)−1 =
(
c −a
d −b
)
,
with trace ±(c − b). We will call τ = ±(a + d) and τ ′ = ±(c − b) the trace and cotrace
of the element x and of the triple (x, y, z). If this triple generates G then it corresponds to
a map M ∈ O(G) of Schläfli type {p, q}, where the pair consisting of the trace ±(a+ d)
and cotrace ±(c− b) (unique up to the action of GalFq) determine the orders q and p of x
and z and thus the type {p, q} of M, together with its genus
g = 1 +
|G|
2( 1
2
− 1p − 1q )
by the Riemann–Hurwitz formula. We also have
[x, y] = x−1y−1xy = x−1yxy =
(
−b2 − d2 ab+ cd
ab+ cd −a2 − c2
)
with trace σ = ±(a2 + b2 + c2 + d2), giving the Petrie length r (twice the order of [x, y])
and hence the extended type {p, q}r of M.
Once y is chosen, for example as above, all of this data for a map M is uniquely
determined by the generating element x, so instead of working with triples (x, y, z) it is
more efficient to work with the single elements x when applying operations such as D and
Hj .
For example, if τ = ±(a+ d) is the trace of x then
x2 =
(
a2 + bc b(a+ d)
c(a+ d) d2 + bc
)
has trace ±(a2 + 2bc+ d2) = ±(a2 + 2(ad− 1) + d2) = ±(τ2 − 2), and
(x2y)−1 =
(
c(a+ d) −a2 − bc
d2 + bc −b(a+ d)
)
has trace ±(a + d)(c − b) = ±ττ ′, giving the cotrace and hence the type and genus of
H2(M). Similarly, its Petrie length is determined by the trace
±((a2 + bc)2 + (b(a+ d))2 + (c(a+ d))2 + (d2 + bc)2)
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 11
of [x2, y]. Applying H−1 or D is achieved by replacing x with
x−1 =
(
d −b
−c a
)
or z =
(
c −a
d −b
)
.
Of course, these leave r and g invariant, while D transposes p and q.
By an observation of Singerman [35], if G = PSL2(q) then all maps M ∈ O(G) are
regular, so H−1(M) ∼= M and there is no need to apply H−1. However, this allows us, if
we wish, to replace x with
z−1 = xy =
(
−b a
−d c
)
instead of z when applying D.
Of course, many of the above expressions for matrices and traces are a little simpler if
q is a power of 2.
7 G = PSL2(7)
There is a unique Hurwitz surface of genus 3, Klein’s quartic curve (see [27, 28]) given in
projective coordinates by x30x1 + x
3
1x2 + x
3
2x0 = 0. The corresponding Hurwitz map K
of type {3, 7}, Klein’s map (see Figure 1), has orientation-preserving automorphism group
G ∼= PSL2(7) ∼= GL3(2), a simple group of order 168, and has full automorphism group
PGL2(7). This is the map R7.1 in [7], discussed as a polyhedron by Schulte and Wills
in [31].
The non-identity elements of PSL2(7) have trace 0, ±1, ±2 or ±3 as they have order
2, 3, 7 or 4 respectively. For a Hurwitz triple we require x and z to have orders 7 and 3,
that is, τ = ±(a+ d) = ±2, τ ′ = ±(c− b) = ±1 and of course ad− bc = 1, so we may
take a = 1, b = 1, c = 0, d = 1; the resulting generating triple for K is
x =
(
1 1
0 1
)
, y =
(
0 1
−1 0
)
, z =
(
0 −1
1 −1
)
.
Since σ = a2 + b2 + c2 + d2 = 3, [x, y] has order 4, so K has Petrie length 8 and extended
type {3, 7}8.
Squaring x to apply H2, we obtain
x2 =
(
1 2
0 1
)
,
corresponding to the map H2(K), which has extended type {7, 7}6 and genus 19. Squaring
again, we obtain
x4 =
(
1 4
0 1
)
,
corresponding to the map H4(K) = H−3(K) = H3(K) of type {4, 7}8 and genus 10.
Since 23 ≡ 1 mod (7), applying H2 again simply returns us to K.
For D(K), of type {7, 3}8 and genus 3, we can replace x with
z =
(
0 −1
1 −1
)
,
12 Art Discrete Appl. Math. 5 (2022) #P3.01
and for D(H4(K)), of type {7, 4}8 and genus 10, we can replace x4 with
(
0 −1
1 3
)
.
However, H2(K) is self-dual, that is, DH2(K) ∼= H2(K); this follows immediately from
the following result:
Proposition 7.1. There is a unique orientably regular map of type {7, 7} with orientation-
preserving automorphism group G ∼= PSL2(7).
Proof. Any such map M corresponds to a generating triple (x, y, z) for G of type (7, 2, 7).
We will use the Frobenius triple-counting formula [17], which states that if X , Y and Z
are conjugacy classes in any finite group G, then the number of triples x ∈ X , y ∈ Y and
z ∈ Z with xyz = 1 is equal to
|X | · |Y| · |Z|
|G|
∑
χ
χ(x)χ(y)χ(z)
χ(1)
, (7.1)
where the sum is over all irreducible complex characters χ of G.
In ATLAS notation [9], the conjugacy classes X , Y and Z of G = PSL2(7) containing
x, y and z are respectively 7A or 7B, 2A, and 7A or 7B. In all cases
|X | · |Y| · |Z|
|G| =
(23 · 3) · (3 · 7) · (23 · 3)
23 · 3 · 7 = 2
3 · 32 = 72.
If X ̸= Z then by using the character values in [9] we find that the character sum in (7.1)
is
1 + 2 · (−1) · (−1 + i
√
7)/2 · (−1− i
√
7)/2
3
+
(−1) · 2 · (−1)
6
= 0.
Thus there are no such triples in G and hence there are no corresponding maps. If X = Z =
7A or 7B then the character sum is
1 +
(−1) · ((−1 + i
√
7)/2)2
3
+
(−1) · ((−1− i
√
7)/2)2
3
+
(−1) · 2 · (−1)
6
=
7
3
,
so, adding these contributions, the total number of such triples is
2 · 72 · 7
3
= 336.
These triples all generate G, since no proper subgroup of G has order divisible by 14.
The triples are permuted by AutG = PGL2(7), semi-regularly since only the identity
automorphism can fix a generating set. Since |PGL2(7)| = 336, AutG is transitive on
the triples, so ∆(7, 2, 7) has one normal subgroup with quotient G, and hence there is
one orientably regular map of type {7, 7} with orientation-preserving automorphism group
G.
By the Riemann–Hurwitz formula, the above map M has genus 19. By its uniqueness,
it is regular and self-dual, and is isomorphic to H2(K) and to the map R19.23 of extended
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 13
Figure 7: The map H2(K) of type {7, 7}6 and genus 19. It is trisected, and each part is
represented in the disc model of the hyperbolic plane. A 2-hole is highlighted; this is the
same as the 3-hole in K highlighted in Figure 1.
type {7, 7}6, the only regular orientable map of this genus and type in [7]. This map is
shown in Figure 7.
Our other claims for uniqueness of maps, or enumerations of maps of a given genus and
type, can be proved in a similar way. However, in future we will not give such full details
unless there are special aspects of the calculation which need to be mentioned.
In fact the maps K, D(K), H2(K), H4(K) and DH4(K) are the only maps in O(G).
To see this, note that the face- and vertex-valencies p and q for any such map must be
orders of non-identity elements of G, so they must take the values 2, 3, 4 or 7. Since
p−1 + q−1 < 1/2 the only possible types (up to duality) are {3, 7}, {7, 3}, {4, 7}, {7, 4}
and {7, 7}; as in the case of {7, 7} it can be verified by using the Frobenius triple-counting
formula (or by inspection of [7]) that there is only one map of each type in O(G).
The graph O(G) is therefore as shown in Figure 8, where the actions of D and H2
are represented by undirected broken and directed unbroken edges; that of H3 is given by
reversing the directed edges for H2.
14 Art Discrete Appl. Math. 5 (2022) #P3.01
K D(K)
H2(K)
H4(K)DH4(K)
Figure 8: The graph O(PSL2(7))
In Table 1 we list the maps M ∈ O(PSL2(7)). The first column shows the corresponding
entry in [7], with letters ‘a’ and ‘b’ assigned as explained earlier. The second column gives
the extended type {p, q}r of M, with p, q and r denoting the face- and vertex-valencies and
the Petrie length. The third column shows how M may be obtained from Klein’s map K by
applying duality and hole operations. The final column gives the effect of applying the hole
operation H2 to M; this is left blank if M is unchanged (as when q = 3, for example) or q
is even. Of course, duality D transposes pairs Rg.na and Rg.nb, while leaving maps Rg.n
invariant. Since the graph O(G) is connected, and K is regular with full automorphism
group PGL2(7), the same applies to every map M in O(G). These maps are all outer
regular, so we obtain no non-orientable regular maps with automorphism group G.
Entry in [7] Type Relationship to K H2(M)
R3.1a {3, 7}8 K R19.23
R3.1b {7, 3}8 D(K)
R10.9a {4, 7}8 H4(K) R3.1a
R10.9b {7, 4}8 DH4(K)
R19.23 {7, 7}6 H2(K) R10.9a
Table 1: The maps M in O(PSL2(7)).
Example 7.2. Taking subscripts mod (7) and using regularity, we see that
H3(K) ∼= H−3(K) ∼= H4(K) ∼= H22 (K) ∼= R10.9a,
of type {4, 7}; Figure 1 confirms this, showing that the 3-holes of K, giving the faces of
H3(K), have length 4.
8 G = PSL2(8) = SL2(8)
There is a unique Hurwitz surface S of genus 7, described by Fricke [16] in 1899 and
rediscovered by Macbeath [29] in 1965; its automorphism group G, isomorphic to the
simple group PSL2(8) = SL2(8) of order 504, is the orientation-preserving automorphism
group of a regular map F of type {3, 7} on S , the Hurwitz map of genus 7, with full
automorphism group G× C2.
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 15
The combinatorial structure of this map is shown in Figure 9. We note that because of
the large size of the map, this representation is more suitable for studying its geometric
properties, in comparison with the examples in Figures 1 and 7 where Poincaré’s disc
model is used. For other representations, including various topological embeddings by
Carlo Sequin and by Jarke van Wijk, as well as for topological embeddings of regular maps
of large genus given by Polthier and Razafindrazaka, see [3] and the references therein.
For polyhedral realisations of the Hurwitz surface S of genus 7, see [2, 3, 4]. A detailed
summary of all the known polyhedral realizations of regular maps of genus g ≥ 2 can be
found in [4].
Figure 9: Combinatorial scheme of the Hurwitz map F of genus 7 (with vertex labels taken
from [4]). A 3-hole is highlighted.
8.1 The group G
In order to work with this group G, let us represent the field F8 of order 8 as F2[t]/(t3 +
t+ 1), with its elements represented as polynomials in F2[t] of degree at most 2. Then the
multiplicative group of F8 consists of
t, t2, t3 = t+ 1, t4 = t2 + t, t5 = t2 + t+ 1, t6 = t2 + 1, t7 = 1.
The non-identity elements of G form the following conjugacy classes:
• one class of elements of order 2, with trace 0;
• one class of elements of order 3, with trace 1;
• three classes of elements of order 7, with traces t+ 1, t2 + 1, t2 + t+ 1;
• three classes of elements of order 9, with traces t, t2, t2 + t.
Each class is inverse-closed. The outer automorphism group of G, isomorphic to C3, is
induced by the Galois group of F8 which is generated by the Frobenius automorphism
t 7→ t2. This group permutes the three classes of elements of order 7 in a single cycle, and
likewise for those of order 9.
16 Art Discrete Appl. Math. 5 (2022) #P3.01
Since OutG has odd order, and Z(G) is trivial, all maps in O(G) are inner regular,
with full automorphism group G× C2.
8.2 Construction of O(G)
Since all involutions in G are conjugate, in forming O(G) we can take
y =
(
0 1
−1 0
)
=
(
0 1
1 0
)
,
using the fact that F8 has characteristic 2 to eliminate minus signs. If
x =
(
a b
c d
)
then z =
(
c a
d b
)
,
with the trace τ = a + d and the cotrace τ ′ = b + c giving the type and hence the genus
of M. Also the trace σ = a2 + b2 + c2 + d2 of [x, y] gives its Petrie length and hence its
extended type. Then
x2 =
(
a2 + bc b(a+ d)
c(a+ d) d2 + bc
)
has trace a2 + d2 = τ2, and the trace of
(x2y)−1 =
(
c(a+ d) a2 + bc
d2 + bc b(a+ d)
)
gives the cotrace (a+ d)(b+ c) = ττ ′, giving the type and genus of H2(M). Similarly, its
Petrie length is determined by the trace
(a2 + bc)2 + (b(a+ d))2 + (c(a+ d))2 + (d2 + bc)2
of [x2, y]. Applying H−1 or D is achieved by replacing x with
x−1 =
(
d b
c a
)
or z =
(
c a
d b
)
.
Alternatively, one can replace x with
z−1 = xy =
(
b a
d c
)
instead of z when applying D.
The Frobenius triple-counting formula shows that F is the only map of type {3, 7} in
O(G). For x to correspond to such a map we require ad + bc = 1, a + d = t + 1, t2 + 1
or t2 + t+ 1 and b+ c = 1. Without loss of generality we can use the solution
a = t, b = 1, c = 0, d = t2 + 1,
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 17
so that F is represented by the matrix
x =
(
t 1
0 t2 + 1
)
with trace τ = t2 + t + 1 and cotrace τ ′ = 1, confirming that the type is {3, 7} and the
genus is 7. Moreover, since σ = t2+12+02+(t2+1)2 = t, the Petrie length is 18, so the
extended type is {3, 7}18. This is therefore the map R7.1a in [7]. The dual map D(F) =
R7.1b of extended type {7, 3}18 is represented by the matrix
z =
(
0 t
t2 + 1 1
)
.
The map H2(F) is represented by the matrix
(
t2 t2 + t+ 1
0 t2 + t+ 1
)
with trace τ2 = t+1, cotrace ττ ′ = t2+ t+1 and sum of squares t2+ t, so it has extended
type {7, 7}18 and genus 55. It must therefore be one of the dual pair R55.32 in [7]; we will
denote it by R55.32a, and its dual map DH2(F), corresponding to the matrix
(
0 t2
t2 + t+ 1 t2 + t+ 1
)
and also of type {7, 7}18, by R55.32b. (See Proposition 8.1 for a proof that there are just
two maps of type {7, 7} in O(G); the fact that the matrix x2 for H2(F) has distinct trace
and cotrace shows that they are not self-dual.)
Iterating this process, we find that H22 (F) = H4(F) = H3(F) is represented by the
matrix
(
t2 + t t
0 t+ 1
)
with trace t2 + 1, cotrace t and sum of squares t2 + 1, so it has extended type {9, 7}14 and
genus 63. It is therefore either R63.6b or R63.7b, since R63.6 and R63.7 are the only entries
of this genus and extended type in [7]. One can verify that it is R63.7b by checking that the
matrices x (given above) and z corresponding to this map satisfy the defining relations for
the full automorphism group of R63.7 given in [7], but not those for R63.6. Specifically,
the defining relations for R63.7, with generators R and S corresponding to our x and z,
and a third orientation-reversing generator T inverting them both, are given as
R−7 = S−9 = (RS)2 = (S−1R)3 = (RS−3R2)2 = 1,
T 2 = (RT )2 = (ST )2 = 1.
One can check that the matrices
x =
(
t2 + t t
0 t+ 1
)
and z =
(
0 t2 + t
t+ 1 t
)
corresponding to H3(F) satisfy these relations when substituted for R and S.
18 Art Discrete Appl. Math. 5 (2022) #P3.01
For example,
z−1x =
(
t t2 + t
t+ 1 0
)(
t2 + t t
0 t+ 1
)
=
(
t2 + t+ 1 t2 + 1
1 t2 + t
)
has trace 1, so (z−1x)3 = 1. On the other hand, the relations for R65.6 include (RS−2R)2 =
1, and since
xz−2x =
(
t2 + 1 t
t2 + t t2 + t+ 1
)
has trace t ̸= 0 we have (xz−2x)2 ̸= 1. Thus H3(F) is R63.7b.
Proposition 8.1. There are two orientably regular maps of type {7, 7} with orientation-
preserving automorphism group G = SL2(8). They form a dual pair.
Proof. We will use the Frobenius triple-counting formula (7.1), as in the proof of Propo-
sition 7.1, to count triples of type (7, 2, 7) in G. There are three choices for the conjugacy
classes X , Z of elements of order 7 in G, each containing 23 · 32 elements, and one class
Y of 32 · 7 involutions. Using the character values in [9], we find that for each of the six
choices of X ̸= Z the number of triples (x, y, z) of type (7, 2, 7) in G, with x ∈ X and
z ∈ Z , is
(23 · 32) · (32 · 7) · (23 · 32)
23 · 32 · 7

1 +
1
9
2
∑
j=0
(ζ2
j
+ ζ−2
j
)(ζ2
j+1
+ ζ−2
j+1
)


=
(23 · 32) · (32 · 7) · (23 · 32)
23 · 32 · 7 ·
7
9
= 23 · 32 · 7,
where ζ = exp(2πi/7), so the total number of such triples is
24 · 33 · 7 = 2|AutG|.
These triples all generate G, since no maximal subgroup of G contains such a triple with
non-conjugate x and z, so there are two corresponding maps.
There are also three choices of classes X = Z , and the total number of corresponding
triples in G is
3 · (2
3 · 32) · (32 · 7) · (23 · 32)
23 · 32 · 7

1 +
1
9
2
∑
j=0
(ζ2
j
+ ζ−2
j
)2


= 3 · (2
3 · 32) · (32 · 7) · (23 · 32)
23 · 32 · 7 ·
14
9
= 24 · 33 · 7.
Now G has nine Sylow 2-subgroups T , each with normaliser NG(T ) ∼= AGL1(8) ∼=
V8⋊C7 generated by 48·7 triples of type (7, 2, 7), all with conjugate x and z (see Section 10
for details). Since
9 · 48 · 7 = 24 · 33 · 7,
these account for all the triples of type (7, 2, 7) in G with x and z conjugate; thus no such
triples generate G, so they do not correspond to maps in O(G).
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 19
We have shown that there are two maps of type {7, 7} in O(G). One map corresponds
to the pair (t+1, t2+1) of traces for x and z, together with its Galois conjugates (t2+1, t2+
t+1) and (t2+ t+1, t+1), while the other map corresponds to the pairs (t+1, t2+ t+1),
(t2 + 1, t+ 1) and (t2 + t+ 1, t2 + 1). Since the first three pairs differ from the last three
by transposition of x and z, these two maps form a dual pair.
8.3 Summary of results for O(G)
Continuing in this way, using a similar criterion to identify the other maps of extended type
{9, 7}14 or {7, 9}14, one eventually finds that there are fourteen maps M ∈ O(G); they
are described in Table 2, where the first four columns are analogues of those in Table 1 for
PSL2(7), with the Fricke–Macbeath map F replacing Klein’s map K. The final column
gives the entries in [7] corresponding to the non-orientable quotients M/C2 by the centre
C2 of the full automorphism group G× C2 of M (see Section 8.4).
Entry in [7] Type Relationship to F H2(M) M/C2
R7.1a {3, 7}18 F R55.32a N8.1a
R7.1b {7, 3}18 D(F) N8.1b
R15.1a {3, 9}14 H2DH3(F) R71.15a N16.1a
R15.1b {9, 3}14 DH2DH3(F) N16.1b
R55.32a {7, 7}18 H2(F) R63.7b N56.5a
R55.32b {7, 7}18 DH2(F) R63.5b N56.5b
R63.5a {7, 9}6 DH2DH2(F) R71.15b N64.3a
R63.5b {9, 7}6 H2DH2(F) R63.6b N64.3b
R63.6a {7, 9}14 DH3DH2(F) R63.5a N64.4a
R63.6b {9, 7}14 H3DH2(F) R55.32b N64.4b
R63.7a {7, 9}14 DH3(F) R15.1a N64.5a
R63.7b {9, 7}14 H3(F) = H22 (F) R7.1a N64.5b
R71.15a {9, 9}18 H4DH3(F) R63.7a N72.9a
R71.15b {9, 9}18 DH4DH3(F) R63.6a N72.9b
Table 2: The maps in O(SL2(8)).
R7.1b R7.1a R63.7b R63.7a R15.1a R15.1b
R55.32a
R55.32b
R71.15a
R71.15b
R63.5b R63.6a
R63.6b R63.5a
Figure 10: The graph O(SL2(8)), with names of maps.
20 Art Discrete Appl. Math. 5 (2022) #P3.01
Figure 10 shows the graph O(G), with broken and unbroken edges representing the
actions of D and H2. The bilateral symmetry reveals an interesting duality between the
parameters 7 and 9 appearing in the types of the maps in O(G) (see Figure 11). However,
this does not extend consistently to the Petrie lengths: for example, R7.1b of extended type
{7, 3}18 is paired with R15.1b of extended type {9, 3}14, whereas R63.7b, of extended type
{9, 7}14, is paired with R63.7a, of extended type {7, 9}14.
{7, 3}18 {3, 7}18 {9, 7}14 {7, 9}14 {3, 9}14 {9, 3}14
{7, 7}18
{7, 7}18
{9, 9}18
{9, 9}18
{9, 7}6 {7, 9}14
{9, 7}14 {7, 9}6
Figure 11: The graph O(SL2(8)), with types of maps.
Example 8.2. H3(F) = H22 (F) ∼= R63.7b, of type {9, 7}; this is confirmed by Figure 9,
which shows that the 3-holes of F have length 4 + 5 = 9.
Note that for this group G an operation Hj with gcd(j, q) > 1 also transforms some
maps in O(G) into others in O(G), namely H3 for maps with q = 9, since one can check
that if x9 = 1 and ⟨x, y⟩ = G then ⟨x3, y⟩ = G also. However, since O(G) is already
connected, adding the resulting extra directed edges to the graph would not change its
connectivity properties.
8.4 Non-orientable quotients
Each map M ∈ O(G) is inner regular, with full automorphism group G× C2: this is true
for M = F since the corresponding generators x and y are both inverted by the matrix
(
t+ 1 t
t t+ 1
)
∈ G;
now the graph O(G) is connected, and AutM is preserved by D and Hj , so it is true for
all M ∈ O(G). Since G, being simple, has no subgroups of index 2, every map in R(G)
is non-orientable, so as explained in Section 4.1, we obtain an isomorphism O(G) →
R(G),M 7→ N = M/C2 of directed graphs. Using their extended types to identify the
maps N in the list of non-orientable regular maps in [7] gives the final column of Table 2.
9 Some other Hurwitz groups
We have seen that if G = PSL2(7) or SL2(8) then O(G) is connected, consisting of maps
which are respectively all outer or all inner regular, with automorphism groups isomorphic
to PGL2(7) or SL2(8)× C2.
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 21
In fact, a group G can have both inner and outer regular maps in O(G), necessarily in
different components. For instance, if G = PSL2(13) then O(G) contains three regular
Hurwitz maps R14.1, R14.2 and R14.3, of genus 14 and extended types {3, 7}12, {3, 7}26
and {3, 7}14. Calculations similar to those given earlier show that R14.2 is inner regu-
lar, with automorphism group G × C2 and non-orientable regular quotient N15.1 of type
{3, 7}13, while the other two maps are outer regular, with automorphism group PGL2(13).
This is interesting because, as shown by Streit [37], for each prime p ≡ ±1 mod (7) the
three Hurwitz dessins in O(PSL2(p)) are Galois conjugate, under the Galois group C3 of
the real cyclotomic field Q(cos 2πi/7), so although the orientation-preserving automor-
phism group of a dessin is a Galois invariant, the full automorphism group is not.
More generally, Wendy Hall [21] has shown that a Hurwitz map M with Aut+M ∼=
PSL2(q) is inner regular if and only if 3− τ2 is a square in Fq , where τ is the trace of the
canonical generator x of order 7. For primes q = p ≡ ±1 mod (7), where there are three
Hurwitz maps, her examples p = 167, 13, 43 and 181 show that none, one, two or all three
of them can be inner regular.
In order to realise alternating groups An as Hurwitz groups (Theorem 5.2), Conder [5]
constructed a sequence of planar maps Mn of type {7, 3} which have Hurwitz maps Hn
with Aut+Hn ∼= An as orientably regular covers. (Each Mn is presented as a coset
diagram for a subgroup An−1 < An = ⟨x, y⟩, but by shrinking the triangles for x to points
it can be interpreted as a cubic map with monodromy group An.) The maps Hn are all outer
regular: indeed, the bilateral symmetry in the construction of the maps Mn was designed
to show that each map Hn has full automorphism group Sn.
Example 9.1. Among the alternating groups, the smallest Hurwitz group is A15. There
are three Hurwitz maps H with Aut+H ∼= A15. Instead of drawing them (they have genus
7 783 776 001), we show their planar quotients M = H/A14 in Figure 12. On the left is
the map M15, corresponding to Conder’s diagram B from his set of basic coset diagrams
A, . . . , N in [5]; it is covered by the outer regular Hurwitz map H15. The other two maps
are covered by a chiral pair of Hurwitz maps H.
Figure 12: Three maps of type {7, 3} with monodromy group A15.
By contrast, the simple Ree groups Re(q) = 2G2(q) (q = 3e for odd e ≥ 3) are also
Hurwitz groups, but as shown in [25] the Hurwitz maps associated with them are all chiral
(the generator x of order 3 is not inverted by any automorphism). In the next section we
will consider a more straightforward example of this last phenomenon, but this time not
involving Hurwitz maps.
22 Art Discrete Appl. Math. 5 (2022) #P3.01
10 AGL1(q), q = 2
e
In this section, instead of considering individual groups such as PSL2(7) and SL2(8), we
will consider an infinite family of groups G for which O(G) exhibits uniform behaviour.
Let G be the 1-dimensional affine group AGL1(q) for q = 2e, consisting of the affine
transformations
t 7→ at+ b, (a, b ∈ Fq, a ̸= 0)
of the field Fq . This is a semidirect product T⋊S of an elementary abelian normal subgroup
T ∼= (Fq,+) ∼= Vq = (C2)e
consisting of the translations t 7→ t+ b, by a complement
S ∼= (F∗q ,×) ∼= Cq−1,
consisting of the transformations t 7→ at, a ̸= 0. To avoid trivial cases, we will assume
from now on that e ≥ 2. (Note that these groups G, being solvable, are not perfect, so they
are not Hurwitz groups.)
Theorem 10.1. (a) If e > 2 there are φ(q − 1)/e maps in M ∈ O(G), all of extended
type {q − 1, q − 1}4 and genus (q − 1)(q − 4)/4; they are all chiral, and satisfy
D(M) ∼= H−1(M) and H2(M) ∼= M.
(b) If e = 2 there is a single map M ∈ O(G), the tetrahedral map {3, 3} of genus 0,
which is outer regular with AutM ∼= AΓL1(4) ∼= S4.
(c) For each e ≥ 2 the graph O(G) is connected.
Proof. The q − 1 involutions in T are all conjugate in G, and the remaining non-identity
elements, all of order dividing q−1, form q−2 conjugacy classes of size q (the non-identity
cosets of T in G). These fuse into (q − 2)/e orbits of size qe under the action of
AutG = AΓL1(q) ∼= G⋊GalFq ∼= G⋊ Ce.
Any map M ∈ O(G) corresponds to a generating triple (x, y, z) for G, where y has
order 2 and xT generates G/T . There are φ(q − 1)q choices for x and q − 1 choices for
y, giving φ(q − 1)q(q − 1) = φ(q − 1)|AutG|/e triples, so there are φ(q − 1)/e maps
M in O(G). Since x and z have order q − 1 these maps have type {q − 1, q − 1} and
hence have genus (q− 1)(q− 4)/4. Since [x, y] is a non-identity element of T it has order
2, so the Petrie length is 4. In particular, if e = 2 there is a single map M ∈ O(G); this
is the tetrahedral map {3, 3}, which is outer regular with AutM ∼= AΓL1(4) ∼= S4. If
e > 2 then no element of GalFq inverts F∗q , so the maps M ∈ O(G) are all chiral; since
z is conjugate to x−1 they satisfy D(M) ∼= H−1(M). Since x 7→ x2 is an automorphism
of Fq they are all invariant under H2. Since any two generators of S are powers of each
other, any two maps M,M′ ∈ O(G) satisfy M′ = Hj(M) for some j coprime to q − 1,
so O(G) is connected. (It is, in fact, a quotient of a Cayley graph for the group of units
Uq−1.)
The involutions in G all lie in the proper subgroup T , so R(G) is empty. These maps
M are instances of the orientably regular embeddings of complete graphs Kq constructed
by Biggs in [1], where he showed that Kq has such an embedding if and only if q is a prime
power; see [24] for the classification of such maps.
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 23
10.1 Examples
The maps M arising for e = 3 are the chiral pair of Edmonds maps of type {7, 7}4,
corresponding to the entry C7.2 in [7]; these both lie on the Fricke–Macbeath surface S of
genus 7 realising the Hurwitz group SL2(8). The affine group AutM ∼= AGL1(8) is a
subgroup of index 9 in AutS = Aut+F ∼= SL2(8): it is the stabiliser of ∞ in the natural
representation of SL2(8) on the projective line P1(F8), and also the normaliser of a Sylow
2-subgroup. This inclusion lifts to an index 9 inclusion ∆(7, 2, 7) < ∆ = ∆(7, 2, 3) of
triangle groups, namely item (B) in Singerman’s list of triangle group inclusions [34].
For e = 4 we have the chiral pair C45.2 of type {15, 15}2, and for e = 5 we have the
three chiral pairs C217.45–47 of type {31, 31}4. The index 9 inclusions of automorphism
groups and triangle groups mentioned above for e = 3 do not generalise to higher powers
of 2.
When e = 5, if we take a to be a generator of F∗32 = ⟨a | a31 = 1⟩ ∼= C31, then the
chiral pairs of maps M±i (i = 1, 3, 5) correspond to the following mutually inverse pairs
of orbits Ω±i of GalF32 = ⟨t 7→ t2⟩ ∼= C5 on F∗32:
Ω1 = {ai | i = 1, 2, 4, 8, 16}, Ω−1 = {ai | i = 15, 30 ≡ −1, 29, 27, 23};
Ω3 = {ai | i = 3, 6, 12, 24, 17}, Ω−3 = {ai | i = 7, 14, 28 ≡ −3, 25, 19};
Ω5 = {ai | i = 5, 10, 20, 9, 18}, Ω−5 = {ai | i = 11, 22, 13, 26 ≡ −5, 21}.
Then D(Mi) = H−1(Mi) = M−i and H2(Mi) = Mi for all i, while H3 induces
a 6-cycle (1, 3, 5,−1,−3,−5) on the subscripts i. The graph O(AGL1(32)) is shown
in Figure 13, with directed edges representing the action of H3, and undirected dashed
and dotted edges the actions of D and H−1. The identification of these maps with the
entries C217.45–47 in [7] depends on the choice of a, or more precisely that of its minimal
polynomial in the action on the additive group of F32, one of the six irreducible factors of
the cyclotomic polynomial Φ31(t) = t30 + t29 + · · ·+ t+ 1 in F2[t] (see [24]).
M1
M3
M5M−1
M−3
M−5
Figure 13: The graph O(AGL1(32))
11 Groups G for which O(G) has many components
In contrast with the groups G = AGL1(2e), for which O(G) is connected for all e, we
will now consider some families of groups G for which O(G) has an unbounded number
of connected components.
24 Art Discrete Appl. Math. 5 (2022) #P3.01
The map operations D and Hj on orientably regular maps preserve the involution y ∈ G
in generating triples, up to the action of AutG, so if AutG has i = i(G) orbits on useful
involutions y ∈ G, where ‘useful’ means ‘member of a generating pair for G’, then the
number c = c(G) of connected components of O(G) satisfies c ≥ i. The example G = S5,
with i = 2 but c = 3 (see Section 4.2), shows that c can exceed i; the following example
shows that both can be arbitrarily large.
Lemma 11.1. If G = Sn with n ≥ 5 then each involution y ∈ G is useful.
Proof. It is sufficient to show that at least one involution in each conjugacy class of G is
useful. The result is straightforward if y is a transposition (i, j), since one can then take
x to be an n-cycle with ix = j, so we may assume that y consists of t transpositions and
n− 2t fixed points for some t ≥ 2.
First we will deal with the case n ≥ 8. Let m = ⌊n/2⌋, so that m > 3. By the
Bertrand–Chebyshev Theorem (see [36], for example) there is a prime p such that m <
p < 2m − 2, so n/2 < p ≤ n − 3. Let us take x to have cycles of length p and n − p if
n is odd, and p, n − p − 1 and 1 if n is even, so that x is odd in either case. Now x has at
most three cycles, so given any t ≥ 2 we can choose the involution y, which moves 2t ≥ 4
points, so that H := ⟨x, y⟩ is transitive. It follows that H must be primitive, for otherwise
H is contained in a wreath product Sa ≀Sb for some proper factorisation n = ab of n, which
is impossible since x has order divisible by p whereas Sa ≀ Sb has order (a!)bb! coprime to
p. Thus H is a primitive group containing a p-cycle (a suitable power of x) with n− p ≥ 3
fixed points, so by a classic theorem of Jordan (see [39, Theorem 13.9]) H ≥ An. Since H
contains the odd permutation x we must have H = Sn, as required.
The case n = 5 was considered in Section 4.2, while the cases n = 6 and n = 7 can
easily be dealt with by hand or by using GAP.
The restriction n ≥ 5 is required in this lemma: see Section 2.1.
Corollary 11.2. For each m ∈ N there exists Nm ∈ N such that if n ≥ Nm then O(Sn)
has at least m connected components.
Proof. By Lemma 11.1 the involutions in G = Sn are all useful for n ≥ 5. They have
⌊n/2⌋ possible cycle-structures, so if n ̸= 6 then AutG (= G) has ⌊n/2⌋ orbits on them.
Thus O(G) has c(G) ≥ i(G) = ⌊n/2⌋ connected components for all n ≥ 7. We may
therefore take Nm = max{2m, 7}.
By adapting Conder’s proof in [5] of Theorem 5.2 one can also prove:
Theorem 11.3. For each m ∈ N there exists N ′m ∈ N such that if n ≥ N ′m then O(An)
has at least m connected components containing Hurwitz maps.
Outline of proof. In [5] Conder proved that An is a Hurwitz group for each n ≥ 168 (and
also for some smaller n) by constructing coset diagrams for subgroups of index n in ∆,
and showing that the induced permutation group on the cosets is An. These diagrams are
constructed by joining copies of 14 basic cosets diagrams A,B, . . . , N of degrees between
14 and 108. An i-handle in a coset diagram for ∆ is a pair (a, b) of fixed points of y with
a = bzi, where i = 1, 2 or 3. (In this paper we have transposed Conder’s notation in [5] for
the generators x and y of ∆.) Two coset diagrams of degrees d and d′ with i-handles (a, b)
and (a′, b′) can be joined by an i-join to create a diagram of degree d + d′ by replacing a
and a′ with a transposition (a, a′) for y, and similarly for b and b′.
G. Gévay and G. A. Jones: Hole operations on Hurwitz maps 25
Conder’s construction (simplified a little here) is as follows. His diagram G of degree
42 has three 1-handles. First use (1)-joins to form a chain of k copies of G for some k ≥ 1.
For each of the 42 congruence classes [c] = [0], . . . [41] mod (42), join two specified
combinations of basic diagrams, of total degree dc ≡ c mod (42), to the ends of the chain,
to give a coset diagram of degree n = 42k+ dc ≡ c mod (42). By taking k = 1, 2, . . . this
realises An for each sufficiently large n ∈ [c] as a quotient of ∆ and hence as a Hurwitz
group.
Each of the k copies of G in the chain has an unused (1)-handle, giving two fixed
points of y. By using these to make i further joins, for any i ≤ ⌊k/2⌋, one can reduce
the total number of fixed points of y by 4i; this does not change the fact that the resulting
permutation group is An since Conder’s proof of this still applies. We thus obtain ⌊k/2⌋+1
Hurwitz maps in O(An), with mutually distinct cycle-structures for y and hence in distinct
components of this graph. Taking k sufficiently large proves the result. □
12 The order of O(G)
One obvious problem which we have not yet addressed is to determine |O(G)| for any
finite group G; however, a method for solving this is already known. The maps in O(G)
correspond bijectively to the orbits of AutG on generating pairs x, y for G satisfying
y2 = 1. Since AutG acts semiregularly (i.e. fixed-point-freely) on all generating sets
for G, we have
|O(G)| = φ(G)/|AutG|,
where φ(G) is the number of such generating pairs x, y for G. If σ(G) denotes the total
number of pairs x, y ∈ G with y2 = 1, then by applying Philip Hall’s technique of Möbius
inversion in groups [20] to the obvious equation σ(G) =
∑
H≤G φ(H) we obtain
φ(G) =
∑
H≤G
µ(H)σ(H), (12.1)
where µ is the Möbius function for the subgroup lattice of G, defined by
µ(G) = 1 and
∑
K≥H
µ(K) = 0 for all H < G.
This applies to generating sets satisfying any given set of relations, but in our case, with
y2 = 1 the only relation, we have
σ(H) = |H|(|H|2 + 1),
where |H|2 is the number of involutions in H .
In [20] Hall determined the function µ for many groups, including PSL2(p) for primes
p. Downs [14] extended this to PSL2(q) for prime powers q = pe. Their results for
p > 2 are complicated but |O(PSL2(13))| = 33 is a simple example. If q = 2e then
|AutG| = e|G| and as in [15] we have
|O(G)| = 1
e
∑
f |e
µ
(
e
f
)
(2f − 1)(2f − 2),
26 Art Discrete Appl. Math. 5 (2022) #P3.01
where µ now denotes the Möbius function of elementary number theory (see Sections 2.1
and 8 for the cases e = 2 and e = 3). For any q the sum in (12.1) is dominated by the
summand with H = G, so that for G = PSL2(q) we have
|O(G)| ∼ |G|(|G|2 + 1)|AutG| ∼
q2
e
or
q2
4e
as q → ∞,
where q is respectively even or odd.
We close with a question: if G = PSL2(q), how does the number c(G) of connected
components of O(G) behave as q → ∞?
ORCID iDs
Gábor Gévay https://orcid.org/0000-0002-5469-5165
Gareth A. Jones https://orcid.org/0000-0002-7082-7025
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.02
https://doi.org/10.26493/2590-9770.1436.0f3
(Also available at http://adam-journal.eu)
Cyclotomic association schemes of broad classes
and applications to the construction of
combinatorial structures*
Luis Martínez† , Maria Asunción García ,
Leire Legarreta‡ , Iker Malaina§
Department of Mathematics,
University of the Basque Country UPV/EHU, Bilbao, Spain
Received 12 August 2021, accepted 31 October 2021, published online 27 June 2022
Abstract
In 2010, G. Fernández, R. Kwashira and L. Martínez gave a new cyclotomy on any
product R =
∏n
i=1 Fqi , where Fqi is a finite field with qi elements. They defined a certain
subgroup H of the group of units of this product ring R for which the quotient is cyclic.
The orbits of the corresponding multiplicative action of the subgroup on the additive group
of R are of two types:
(1) The cyclotomic cosets of the quotient of the group of units of R over the subgroup
H .
(2) The n-tuples with arbitrary non-zero elements in positions indicated by a proper
subset S of {1, . . . , n} and zeroes elsewhere.
In this paper, we introduce and study a fusion of a class of association schemes derived
from the mentioned cyclotomy. The association schemes that we are proposing correspond
with a fusion of orbits associated to subsets S of {1, . . . , n} of the same cardinality. We
call standard cyclotomic association schemes of broad classes to these association schemes.
The fusion corresponds to the operation of adding to the permutation group that determines
the original association scheme the permutations of R induced by the permutations of the
symmetric group Sn.
*The authors were supported by the UPV/EHU and Basque Center of Applied Mathematics grant US18/21.
The authors would like to thank G.A. Fernández-Alcober and the anonymous referee for helpful suggestions.
†Corresponding author. The author was supported by the Basque Government, grant IT974-16.
‡The author was supported by the Basque Government, grant IT974-16 and by the Spanish Government, grant
MTM2017-86802-P.
§The author was supported by the Basque Government, grant IT974-16.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.02
We use these association schemes to obtain sporadic examples and infinite families of
difference sets and partial difference sets.
Keywords: Cyclotomy, difference sets, partial difference sets, association schemes.
Math. Subj. Class.: 05C25, 05E18, 05B10
1 Standard cyclotomy
Cyclotomy of finite rings is based on considering the action of a subgroup G of the multi-
plicative group of a finite ring on the additive group of the same ring through multiplication
by the elements of G. Cyclotomy is a fundamental tool to build many types of combina-
torial structures, in which the objects are formed by taking a union of orbits of the action,
so that the group G becomes a group of automorphisms of the combinatorial structure
considered (not necessarily equal to the complete group of automorphisms).
Fernández, Kwashira and Martínez introduced in [8] a new type of cyclotomy, which
they called standard cyclotomy, in finite Cartesian products of finite fields, that general-
ized the two most widely used in the literature (namely, the classic one used by Gauss
and the one introduced by Whiteman in [19], and they used it to find certain combinato-
rial structures, including divisible difference sets, relative difference sets, which give rise
to groop divisible designs, partial difference sets, which originate strongly regular (undi-
rected) graphs, and also three class association schemes.
Araluze, Kutnar, Martínez and Marusic used this standard cyclotomy in [1] to construct
an infinite series of partial sum families that originate strongly regular digraphs with new
parameters.
Subsequently, Momihara found in [12] an infinite series of strongly regular graphs
with parameters like those of Proposition 4.3 of the article by Fernández, Kwashira and
Martínez, as well as another infinite family with new parameters.
Ott used in [13] and [14] the standard cyclotomy to obtain partial difference sets that
originate new strongly regular graphs.
Later, Michel and Wang used the standard cyclotomy in [11] to construct partial geo-
metric designs from subgroups of F⋆q × Fq .
Let us recall now the basic definitions and results from the standard cyclotomy. Let
R = Fq1 × · · · × Fqn be a product of finite fields (thus q1, . . . , qn are prime powers) and
let X = {1, . . . , n}. For every k ∈ X we choose a primitive root θk in Fqk . Let e be a
common divisor of all the qk − 1 for k ∈ X , and let us write qk − 1 = efk. Then the
following subgroup of the multiplicative group R× is well-defined:
H = {(θr11 , . . . , θ
rn
n ) |
∑n
k=1 rk ≡ 0 mod e}.
The multiplicative subgroup H acts by multiplication on the additive group of R, and
our first result describes the corresponding orbits.
Proposition 1.1 ([8, Proposition 2.1]). The orbits of R under the action of H are the
following subsets:
E-mail addresses: luis.martinez@ehu.eus (Luis Martínez), mariasun.garcia@ehu.eus (Maria Asunción
García), leire.legarreta@ehu.eus (Leire Legarreta), iker.malaina@ehu.eus (Iker Malaina)
L. Martínez et al.: Cyclotomic association schemes of broad classes and applications . . . 3
(1) Ci = {(θ
r1
1 , . . . , θ
rn
n ) |
∑n
k=1 rk ≡ i mod e}, for all 0 ≤ i ≤ e− 1.
(2) FS = {(x1, . . . , xn) ∈ R | xk ̸= 0 for k ∈ S and xk = 0 for k ̸∈ S}, for all
S ⊊ X . In other words, FS = U1 × · · · × Un, where Uk = F
×
qk
if k ∈ S and
Uk = {0} otherwise.
Moreover,
|Ci| =
∏
k∈X (qk − 1)
e
and |FS | =
∏
k∈S (qk − 1).
We call the orbits of the form Ci cyclotomic cosets.
We define the standard cyclotomy of order e with respect to the primitive roots θ1, . . . , θn
as the partition of R into orbits of the action of H .
Definition 1.2 ([8, Definition 2.2]). Given a standard cyclotomy of order e over a Galois
domain corresponding to primitive roots of unity θ1, . . . , θn, its inverse cyclotomy is the
cyclotomy of order e defined by θ−11 , . . . , θ
−1
n .
Definition 1.3 ([8, Definition 2.3]). Let two standard cyclotomies of the same order e
be given over the rings R and R′, defined by primitive roots of unity θ1, . . . , θn and
θ′1, . . . , θ
′
m, respectively. Then the product cyclotomy of the two is the cyclotomy over
R × R′ of order e and primitive roots of unity θ1, . . . , θn, θ
′
1, . . . , θ
′
m. This generalizes
obviously to the product of more than two cyclotomies.
Proposition 1.4 ([8, Proposition 2.5]). Let A, B and C be three orbits of R. Then the
number of solutions a ∈ A and b ∈ B to the equation a + b = c, with c ∈ C fixed, is
independent of the choice of c. The same result holds if we fix a ∈ A or b ∈ B.
Thus, if we we consider the additive group (R,+) and the group ring ZR, and if ∗
denotes the product in this group ring, we can write
A ∗B =
∑
C orbit of R
α(A,B,C)C (1.1)
for some non-negative integers α(A,B,C).
Proposition 1.5 ([8, Proposition 2.6]). For any three orbits A, B and C of R, the following
properties hold:
(1) α(B,A,C) = α(A,B,C).
(2) α(B,C,A) =
|C|
|A|
α(A,−B,C).
(3) α(uA, uB, uC) = α(A,B,C) for all u ∈ R×.
(4) Let a be any element of A, and denote by N the number of b ∈ B such that a+b ∈ C.
(Note that N is independent of the choice of a by the previous proposition.) Then
α(A,B,C) =
N |A|
|C|
.
4 Art Discrete Appl. Math. 5 (2022) #P3.02
As a consequence of these simple properties, in order to determine
α(A,B,C) for all orbits A, B and C, it suffices to know the values α(i, j, k), α(i, j, S),
α(S, T, i) and α(S, T, U), for 0 ≤ i, j, k ≤ e− 1 and S, T, U ⊊ X .
Proposition 1.6 ([8, Proposition 2.7]). Let S, T and U be proper subsets of X and let
i ∈ {0, . . . , e− 1}. Then
α(S, T, i) =
{
∏
k∈S∩T (qk − 2), if S ∪ T = X ,
0, otherwise,
and
α(S, T, U) =





∏
k∈S∩T∩U (qk − 2) ·
∏
k∈(S∩T )∖U (qk − 1),
if (S ∪ T )∖ (S ∩ T ) ⊆ U ⊆ S ∪ T ,
0, otherwise.
The δ symbol in the following proposition and in the rest of the paper is defined by
δi =
{
1, if i = r
0, in other case
,
where r is defined modulo e by the condition (−1, . . . ,−1) ∈ Cr. It was proved in Propo-
sition 2.4 in [8] that, if Y denotes the set of indices k ∈ X for which qk is odd, then
r =
{
0, if
∑
k∈Y fk is even
e/2, in other case
Proposition 1.7 ([8, Proposition 2.10]). Let S be a proper subset of X of cardinality s.
Then
α(i, j, S) =
∏
k ̸∈S (qk − 1)
e
{
∏
k∈S (qk − 2)− (−1)
s
e
+ (−1)sδj−i
}
.
with respect to the integers α(i, j, k), which satisfy α(i, j, k) = α(0, j − i, k − i) for
every i, j, k and are denoted by (j − i, k − i) and called cyclotomic numbers, there is not
a closed algebraic form to express all of them (except for certain especial cases, since for
instance the regular cyclotomies and other special cyclotomies similar to the regular one),
and they are calculated (with pain and effort!) for each particular value of e.
We will calculate next the values for e = 2 and e = 3.
We begin by a couple of simple results.
Proposition 1.8 ([8, Proposition 3.1]).
(1) (i, j) = (−i, j − i).
(2) (i, j) = (j + r, i+ r).
Proposition 1.9 ([8, Proposition 3.2]).
∑e−1
j=0 (i, j) = (−1)
n(δi + Ṁ), where
Ṁ =
∏n
k=1 (2− qk)− 1
e
.
L. Martínez et al.: Cyclotomic association schemes of broad classes and applications . . . 5
By using the previous two propositions, we can already calculate the cyclotomic num-
bers for e = 2. We only need to give the values of (0, 0) and (0, 1), since the other
cyclotomic numbers can be deduced from these according to Proposition 1.8. Note also
that all the qk are odd for e = 2.
Proposition 1.10. For e = 2, we have that:
(1) If f1 + · · ·+ fn is odd, then
(0, 0) = (−1)n
Ṁ + 1
2
and (0, 1) = (−1)n
Ṁ − 1
2
.
(2) If f1 + · · ·+ fn is even, then
(0, 0) = (−1)n
(Ṁ
2
+ 1
)
and (0, 1) = (−1)n
Ṁ
2
.
As in the classical case, for the calculation of the cyclotomic numbers of order greater
than 2, it is convenient to introduce periods and Stickelberger-Gauss sums. For each k ∈
{1, . . . , n}, let pk be the characteristic of the field Fqk and choose a complex root of unity
ξk of order pk. Then we define the periods ηi to be
ηi =
∑
(x1,...,xn)∈Ci
ξ
T1(x1)
1 . . . ξ
Tn(xn)
n , for i = 0, . . . , e− 1,
where Tk is the trace map corresponding to the extension Fqk/Fpk . Now, for any complex
e-th root of unity β, we define the Stickelberger-Gauss sum H(β) by means of
H(β) =
q1−2
∑
i1=0
· · ·
qn−2
∑
in=0
ξ
T1(θ
i1
1 )
1 . . . ξ
Tn(θ
in
n )
n β
i1+···+in =
e−1
∑
i=0
ηiβ
i.
Since for any d > 1 the sum of all d-th roots of unity is zero, we have the following
inversion formula expressing the periods ηi in terms of the Stickelberger-Gauss sums:
eηi =
∑
β
H(β)β−i, (1.2)
where the sum runs over all complex e-th roots of unity.
It is clear that these generalized Stickelberger-Gauss sums satisfy the following multi-
plicative property: if we decompose the ring R as a cartesian product, say R = R1 × R2,
and we view the cyclotomy over R as a product of cyclotomies over R1 and R2, then
H(β) = H1(β)H2(β). (Here H , H1 and H2 are the Stickelberger-Gauss sums over R,
R1 and R2, respectively.) In particular, if Hk denotes the classical Stickelberger-Gauss
sums over Fqk for the primitive root of unity θk, then H(β) = H1(β) . . . Hn(β). As a
consequence, H(1) = (−1)n.
Lemma 1.11. For every i and j,
ηiηj =
∑
S⊊X
α(i, j, S)(−1)|S| +
e−1
∑
k=0
(j − i, k − i)ηk.
6 Art Discrete Appl. Math. 5 (2022) #P3.02
Proof. We have that
ηiηj =
∑
(x1,...,xn)∈Ci
∑
(y1,...,yn)∈Cj
ξ
T1(x1+y1)
1 . . . ξ
Tn(xn+yn)
n
=
∑
A
{
α(i, j, A)
∑
(z1,...,zn)∈A
ξ
T1(z1)
1 . . . ξ
Tn(zn)
n
}
,
where the sum runs over all orbits A of R. Since the sum corresponding to an orbit of the
type FS is clearly equal to (−1)
|S|, the result follows.
Proposition 1.12. Let β ̸= 1 be a complex e-th root of unity. Then
H(β)H(β−1) = q1 · · · qnβ
r.
Proof. By using the multiplicative property of H , we have that this a direct consequence
of the case n = 1, which is well known (see, for instance, Equation (5) in [4]).
Proposition 1.13. Let β be a complex primitive e-th root of unity, and let u, v be integers
such that u ̸≡ 0 (mod e), v ̸≡ 0 (mod e) and u+ v ̸≡ 0 (mod e). Then:
(1) H(βu) H(βv) = R(u, v) H(βu+v), where
R(u, v) =
e−1
∑
i=0
e−1
∑
j=0
(i, j)βviβ−(u+v)j .
(2) R(u, v) R(−u,−v) = q1 · · · qn.
Proof. The product H(βu)H(βv) can be developed with the help of Lemma 1.11. By
using the expression given in Proposition 1.7 for the α(i, j, S) and the identity
∏
k∈X
(xk + yk) =
∑
T⊆X
∏
k∈T
xk
∏
k ̸∈T
yk,
which is valid in any commutative ring and is a direct consequence of the distributive law,
we obtain easily the result in (1).
In order to prove (2), multiply together the expression in (1) corresponding to βu, βv
with the one corresponding to β−u, β−v , and then use Proposition 1.12.
By combining these two propositions we get the following result.
Corollary 1.14. In the conditions of the previous proposition,
R(u, v) = R(v, u) = βurR(−v − u, u).
Now, by using Propositions 1.8, 1.9, and 1.13, we can derive the cyclotomic numbers
for e = 3. All cyclotomic numbers can be easily obtained from the ones in the following
proposition.
L. Martínez et al.: Cyclotomic association schemes of broad classes and applications . . . 7
Proposition 1.15. For e = 3, we have that
9(0, 0) = (−1)n(7− x+ 3Ṁ),
18(0, 1) = (−1)n(2 + x+ 9y + 6Ṁ),
18(0, 2) = (−1)n(2 + x− 9y + 6Ṁ),
9(1, 2) = (−1)n(−2− x+ 3Ṁ),
for some integers x, y satisfying the Diophantine equation x2 + 27y2 = 4q1 . . . qn, and
where x ≡ 1 (mod 3).
Proof. Take u = v = 1 in Proposition 1.13.
2 Broad classes
It is evident from (1.1) that, given an standard cyclotomy, if we define for each orbit A the
relation
RA = {(x, y) ∈ R×R | x− y ∈ A}
then these relations form an association scheme. When n = 1, that is, when we have a
single field, we obtain the classical cyclotomic association scheme (see, for instance, [3]).
This is why we will call to this association scheme the standard cyclotomic association
scheme.
Now we will introduce the main object in this work. We will assume that all the fields
have the same cardinality (therefore we can assume, without losing generality that they are
the same field. Nonetheless, we are not necessarily taking the same primitive root in all of
them).
Definition 2.1. Let us suppose that qi = q ∀i ∈ {1, . . . , n}. For i ∈ {0, . . . , n − 1}, we
define
Bi =
⋃
|S|=i
FS .
We will call broad orbits to these sets Bi. It is obvious that the cyclotomic cosets Ci
and the broad orbits Bj determine an association scheme with the relations Ci = {(x, y) ∈
R × R | x − y ∈ Ci} and Bj = {(x, y) ∈ R × R | x − y ∈ Bj}. It is a fusion of
the standard cyclotomic association scheme defined before. We will call it the standard
cyclotomic association scheme of broad classes, and the associated cyclotomy, standard
cyclotomy of broad classes. Besides of the already mentioned connection with the classical
cyclotomic association schemes, when e = 1 we obtain the classical Hamming association
schemes (see [2]).
When we identify the orbits Ci and Bj with the corresponding sum of its elements in
the group ring ZR, we have a decomposition for the product of orbits similar to the one in
(1.1). To simplify the notation, when we consider an orbit Ci we will put simply i in the
α(A,B,C), and when we take an orbit Bj we will put j. Thus, for instance, α(Bi, Cj , Bk)
will be denoted simply by α(i, j, k).
An analysis similar to that done in the first section shows that to calculate the coef-
ficients α(A,B,C) it is sufficient to know the values α(i, j, k), α(i, j, k), α(i, j, k), and
α(i, j, k). Obviously, the numbers α(i, j, k) are the α(i, j, k) introduced in the previous
section. With respect to the other three types of coefficients we have:
8 Art Discrete Appl. Math. 5 (2022) #P3.02
Proposition 2.2.
(1) The α(i, j, k) are the corresponding structure constants pki,j of the Hamming associ-
ation scheme.
(2) The α(i, j, k) are the corresponding structure constants pni,j of the Hamming associ-
ation scheme.
(3)
α(i, j, k) =
(q − 1)n−k
e
{
(q − 2)k − (−1)k
e
+ (−1)kδj−i
}
.
Proof. The first two parts of the proposition are evident.
The third part is a consequence of Proposition 1.7, because the number in that proposi-
tion depends on the cardinality of S, but not on the specific set S.
As a consequence of what we told in the first section, the standard cyclotomic associ-
ation scheme admits the group H (when viewed as a group of permutations) as a group of
automorphisms. When we consider the broad cyclotomy we have a richer group of auto-
morphisms, because any permutation of the symbols in an n-tuple of R is an automorphism
of the association scheme, and these permutations form a group K isomorphic to the sym-
metric group Sn, and therefore the permutation group generated by H ∪ K is a group of
automorphisms.
3 Combinatorial applications
The computational search of combinatorial structures with the standard cyclotomy of broad
classes is easier than with the standard cyclotomy, because the number of orbits is much
less, and hence we have to consider unions of fewer orbits.
The following constructions given in [8] can be interpreted in terms of broad classes:
Proposition 3.1 ([8, Proposition 4.1]). Let q be an odd prime power such that 3q − 2
is also a prime power, and consider a standard cyclotomy of order e = q − 1 over
R = F3q−2 × F3q−2 × Fq where the cyclotomies in the first two factors are inverse of
each other. Then D = B1 ∪ C0 is a divisible difference set in R with respect to the sub-
group {0} ∪ F{3}, with λ = 9q − 10 and µ = q − 2.
Proposition 3.2 ([8, Proposition 4.5]). Let q = 2n, and consider a cyclotomy over
R = Fq × Fq × Fq of order e = q − 1, where the cyclotomies in the first two factors
are inverse of each other. Then D = B1 ∪ C0 is a partial difference set with v = 2
3n,
k = (2n − 1)(2n + 2), λ = 2n − 2 and µ = 2n + 2.
Proposition 3.3 ([8, Proposition 4.6]). Let q be a prime power. Then, for any standard
cyclotomy of order e = q − 1 over R = Fq2 × Fq2 × Fq2 , the union D = B1 ∪ C0 is a
partial difference set if and only if q = 2 or if q ≥ 3 and the following conditions hold:
(0, 0) = q4+q3−7q−2, (0, i) = q4+2q3−2q2−9q−4, for all i = 1, . . . , e− 1.
In this case, v = q6, k = (q + 2)(q2 − q + 1)(q2 − 1), λ = q4 + q3 − q − 2 and
µ = (q − 1)(q + 2)(q2 + q − 1).
In the following two propositions we consider the product of a cyclotomy over Fq ×Fq
and its inverse cyclotomy.
L. Martínez et al.: Cyclotomic association schemes of broad classes and applications . . . 9
Proposition 3.4 ([8, Proposition 4.10]). If f = 1, then B2 ∪ C0 ∪ C1 ∪ C2 is a
(q4, 3(q + 1)(q − 1)2, (5q − 6)(2q − 3), 3(q − 1)(3q − 4))-partial difference set.
Proposition 3.5 ([8, Proposition 4.11]). If f = 3, then B2 ∪C0 is a (q
4, 3(q+1)(q− 1)2,
(5q − 6)(2q − 3), 3(q − 1)(3q − 4))-partial difference set.
Proposition 3.6 ([8, Proposition 4.15]). Consider a cyclotomy over R = F4 × F4 × F4
of order 3, where the cyclotomies of the first two factors are inverse of each other. Then
D1 = B2, D2 = B1 ∪ C0 is a basic blueprint of a three-class association scheme with
intersection matrices
L1 =




10 8 8
12 9 6
12 6 9




, L2 =




8 6 4
9 2 6
6 6 6




, and L3 =




8 4 6
6 6 6
9 6 2




.
Here we will present some new constructions not appearing in [8]:
Lemma 3.7. Let U be a non-empty finite set. Then:
(1) For any z ∈ C,
∑
V⊆U
z|V | = (z + 1)|U |.
(2) For any z ∈ C,
∑
V⊆U
|V |≡|U | mod 2
z|V | =
(z + 1)|U | + (z − 1)|U |
2
and
∑
V⊆U
|V |̸≡|U | mod 2
z|V | =
(z + 1)|U | − (z − 1)|U |
2
(3) #{V ⊆ U | |V | odd} = #{V ⊆ U | |V | even} = 2|U |−1.
Proof. Note first that (1) is simply the binomial theorem. In order to get (2) it suffices to
sum and subtract the two equalities
∑
V⊆U z
|V | = (z + 1)|U | and
∑
V⊆U
(−1)|U |+|V | z|V | = (−1)|U |(−z + 1)|U | = (z − 1)|U |.
Finally, (3) follows from (2) by setting z = 1.
Theorem 3.8. For all n ≥ 1, the set D of tuples in Fn4 with an odd number of components
equal to zero is a difference set.
Proof. Let us write R = Fn4 and X = {1, . . . , n}, and consider the standard cyclotomy
over R of order 1. Then
D = ∪{FS | |S| ̸≡ |X| mod 2}.
10 Art Discrete Appl. Math. 5 (2022) #P3.02
We will write FX to denote R
×. In this case FX = C0 is an orbit of the cyclotomy, and
the formula for α(S, T, 0) coincides with the formula for α(S, T, U) if it is allowed to take
U = X .
We want to prove that for any x ∈ R, x ̸= (0, . . . , 0), the number σ of different ways
of writing x = a − b with a, b ∈ D is independent of x. There exists a non-empty subset
U of X such that x ∈ FU and then
σ =
∑
S,T⊆X
|S|,|T |̸≡|X| mod 2
α(S, T, U).
Here we have used that −FT = FT .
By Proposition 1.6, we know that α(S, T, U) = 0 unless (S∪T )∖(S∩T ) ⊆ U ⊆ S∪T ,
in which case
α(S, T, U) = 2|S∩T∩U | 3|(S∩T )∖U |.
Thus σ is a sum of a number of products of the type 2|V |3|W |, with V ⊆ U and W ⊆ U c.
More precisely,
σ =
∑
V⊆U
∑
W⊆Uc
τ(V,W ) 2|V | 3|W |, (3.1)
where τ(V,W ) is the number of subsets S, T of X satisfying the following conditions:
(1) |S|, |T | ̸≡ |X| mod 2.
(2) (S ∪ T )∖ (S ∩ T ) ⊆ U ⊆ S ∪ T .
(3) S ∩ T ∩ U = V and (S ∩ T )∖ U = W .
Given U ⊆ X , V ⊆ U and W ⊆ U c, it is easy to determine all the subsets S and T
fulfilling these conditions: it suffices to choose P ⊆ U ∖ V and put S = P ∪ V ∪W and
T = (U ∖ P ) ∪W , as shown in Figure 1 below.
Figure 1: Relative positions of S and T .
Since all the unions in the definition of S and T are disjoint, it follows that
|S| = |P | + |V | + |W | and |T | = |U | − |P | + |W |. Hence |S| ̸≡ |X| mod 2 if and
L. Martínez et al.: Cyclotomic association schemes of broad classes and applications . . . 11
only if |P | ̸≡ |X| + |V | + |W | mod 2, and once this holds, we get that |T | ̸≡ |X| mod 2
if and only if |V | ≡ |U | mod 2.
In particular, in the sum (3.1) we only need to consider the subsets V of U satisfying
|V | ≡ |U | mod 2. In that case, Lemma 3.7 yields that
τ(V,W ) = #{P ⊆ U ∖ V | |P | ̸≡ |X|+ |V |+ |W | mod 2}
=





2|U |−|V |−1, if V ⊊ U ,
0, if V = U and |W | ≡ |X|+ |U | mod 2,
1, if V = U and |W | ̸≡ |X|+ |U | mod 2.
Hence, we split the sum in (3.1) according as V = U or V ⊊ U , and by using repeatedly
Lemma 3.7 we get that
σ = 2|U |
∑
W⊆Uc
|W |̸≡|X|+|U | mod 2
3|W | + 2|U |−1
∑
V⊊U
|V |≡|U | mod 2
∑
W⊆Uc
3|W |
= 22|X|−|U |−1 − 2|X|−1 + 2|U |−1
∑
V⊊U
|V |≡|U | mod 2
4|X|−|U |
= 22|X|−|U |−1 − 2|X|−1 + 22|X|−|U |−1 (#{V ⊆ U | |V | ≡ |U | mod 2} − 1)
= 22|X|−|U |−1 − 2|X|−1 + 22|X|−|U |−1(2|U |−1 − 1)
= 22|X|−2 − 2|X|−1.
Thus σ is independent of x, as desired.
The underlying additive groups are 2-groups, and it is well known that the parameters
of these difference sets must be of the form v = 4u2, k = 2u2 − u and λ = u2 − u, in
other words the parameters of a Hadamard difference set. Hadamard difference sets are
known to exist over any elementary abelian 2-group whose order is a square (see [5]). In
fact, our construction provides an alternative proof of this result, since it applies to arbitrary
elementary abelian 2-groups of square order.
An sporadic Hadamard difference set with v = 256, k = 120, λ = 56 can be obtained
for e = 3 and n = 4, and is
D = B1 ∪B2 ∪ C0 ∪ C1
Although the parameters for this sporadic difference set are the same that the ones in
Theorem 3.8 with n = 4, they are not isomorphic. In fact, calculations in GAP show that
the order of the automorphisms group of the associated graph in this case is 331776, and
hence has the minimum possible size, that is, the group is the sum G = G1 + G2 + G3,
where G1 is the group derived from the regular action of the additive group of F
4
4, G2 is the
group derived from the group H associated to the cyclotomy, and G3 is a group of order
8 associated to the ways to choose a primitive root in each one of the four factors of F44.
On the other hand, the automorphisms group for the graph associated to the difference set
constructed in Theorem 3.8 contains strictly to G and has order 89181388800.
By using a similar way of reasoning to the one in Theorem 3.8, the following result can
be obtained:
12 Art Discrete Appl. Math. 5 (2022) #P3.02
Theorem 3.9. For all n ≥ 1, the set D of tuples in Fn2 with an odd number of components
equal to zero is a partial difference set.
By using the fact that, if D is a partial difference set over a group G (and, in particular,
if D is a difference set with 0 ̸∈ D and D = −D), then G − {0} − D is also a partial
difference set, we obtain the following two corollaries from the two previous theorems,
respectively:
Corollary 3.10. For all n ≥ 1, the set D of non-zero tuples in Fn4 with an even number of
components equal to zero is a partial difference set.
Corollary 3.11. For all n ≥ 1, the set D of non-zero tuples in Fn2 with an even number of
components equal to zero is a partial difference set.
Partial difference sets with parameters as in Corollary 3.10 are known by constructions
using projective 4-ary codes, affine polar graphs and 2-graphs.
Now we will give some sporadic constructions of partial difference sets obtained for e
from 1 to 3 and n from 3 to 5. They will be described in the form (v, k, λ, µ, S, T ), where
v, k, λ, µ are the parameters of the strongly regular graph and S ⊆ {0, . . . , n − 1} and
T ⊆ {0, . . . , e− 1}, and where the corresponding partial difference set is
D =
⋃
i∈S
Bi ∪
⋃
j∈T
Cj .
Thus, for instance, ({1, 3}, {0, 1}) would represent the union of the broad orbits B1 and
B3 and the cyclotomic cosets C0 and C1. When searching the partial difference sets we
can assume without losing generality by taking the complement if necessary that 0 ∈ T
whenever T ̸= ∅.
The proof that the sets D are in fact partial difference set is easy to do and will be
omitted. It is only necessary to check that 0 ̸∈ D and D = −D and that the identity
D ⋆ D = λD + µ(R − D − {0}) holds in the group ring ZR, by using
Propositions 1.10, 1.15 and 2.2.
For e = 1, n = 4
(16, 5, 0, 2, {1}, {0}), (81, 24, 9, 6, {1}, {0}), (81, 24, 9, 6, {2}, {}),
(16, 5, 0, 2, {3}, {0}), (81, 32, 13, 12, {3}, {})
For e = 2, n = 4
(625, 240, 95, 90, {1, 2}, {0}), (2401, 864, 319, 306, {2}, {0})
For e = 3, n = 3
(64, 18, 2, 6, {}, {0, 1})
For e = 3, n = 4
(2401, 672, 203, 182, {1, 2}, {0}), (10000, 2673, 748, 702, {2}, {0})
For e = 3, n = 5
(1024, 186, 50, 30, {1, 2}, {0})
L. Martínez et al.: Cyclotomic association schemes of broad classes and applications . . . 13
For the parameters of the associated strongly regular graphs with v up to 1024, several
constructions appear for each one of the parameters set in the table of
Andries E. Brouwer [6].
With respect to the parameters set (2401, 864, 319, 306), it is known that strongly reg-
ular graphs with these parameters exist coming from two-weight codes over F7 of length
144 and dimension 4, with weights 119 and 126 [7], and that they belong to the families of
SU2, CY 1 and CY 4 codes (and one that can be constructed as a complement of a [256, 4;
217:1536, 224:864] 7-code), and with respect to the parameters set (2401, 672, 203, 182),
also several strongly regular graphs with these parameters exist coming from two-weight
codes over F7 of length 112 and dimension 4, with weights 91 and 98 [7], and they be-
long to the families of SU2, CY 1 and CY 4 codes (and one that can be constructed as a
complement of a [288, 4; 245:1728, 252:672] 7-code).
To finish, we would like to note that when considering cyclotomic classes of order e in
Fq1 × · · · × Fqn such that e|qi − 1 ∀i, if each cyclotomic class of order e in each field is a
partial difference set then it can be viewed as a product construction. For instance, the dif-
ference set in Theorem 3.8 (note that difference sets are special cases of partial difference
sets) is a union of products of trivial partial difference sets consisting of a single element
or all the elements in F22 . Some works in which product constructions of partial difference
sets appear in the mathematical literature are [9, 10, 15, 16, 17, 18]. For instance, the pa-
rameters v, k, λ, µ of the 5 previously shown sporadic partial difference sets corresponding
to e = 1, n = 4 can be obtained with a product construction in the following two results,
that can be found in [10] and also in [16]:
Result 1 Let p be a prime, and let u and s2, s4, . . . , s2v be nonnegative integers. Let
G = Z2up ×
∏v
i=1 Z
2s2i
p2i
and let n =
√
|G|. Then there is a partition of G − 1G into p
regular partial difference sets in G, of which p− 1 are of (n, n
p
) Latin square type and one
is of (n, n
p
+ 1) Latin square type.
Result 2
(1) Let u,w and s4, s6, . . . , s2v be nonnegative integers satisfying u + w ≥ 1. Let
G = Z4u2 ×Z
2w
4 ×
∏v
i=2 Z
4s2i
22i and let n =
√
|G|. Then there is a partition of G−1G
into 4 regular partial difference sets in G, of which three are of (n, n4 ) negative Latin
square type and one is of (n, n4 − 1) negative Latin square type.
(2) Let u and s2, s4, . . . , s2v be nonnegative integers. Let G = Z
2u+2
3 ×
∏v
i=1 Z
2s2i
32i and
let n =
√
|G|. Then there is a partition of G−1G into 3 regular partial difference sets
in G, of which two are of (n, n3 ) negative Latin square type and one is of (n,
n
3 − 1)
negative Latin square type.
ORCID iDs
Luis Martínez https://orcid.org/0000-0002-6297-1449
Maria Asunción García https://orcid.org/0000-0002-3884-356X
Leire Legarreta https://orcid.org/0000-0002-2058-0466
Iker Malaina https://orcid.org/0000-0002-1305-6623
14 Art Discrete Appl. Math. 5 (2022) #P3.02
References
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Mathematics and its Applications, Cambridge University Press, Cambridge, 2nd edition, 1999,
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[6] A. E. Brouwer, Parameters of Strongly Regular Graphs, Eindhoven University of Technology,
https://www.win.tue.nl/~aeb/graphs/srg/srgtab.html.
[7] E. Chen, Online database of two-weight codes, Kristianstad University, http://www.tec.
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2009.08.005.
[9] K. Hayashida and K. Momihara, Note on a construction of partial difference sets, Kumamoto
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[10] J. Jedwab and S. Li, Packings of partial difference sets, Comb. Theory 1 (2021), Paper No. 18,
41, doi:10.5070/c61055384.
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(2019), 2655–2670, doi:10.1007/s10623-019-00644-7.
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from cyclotomy, Finite
Fields Appl. 25 (2014), 280–292, doi:10.1016/j.ffa.2013.10.006.
[13] U. Ott, A generalization of a cyclotomic family of partial difference sets given by Fernández-
Alcober, Kwashira and Martínez, Discrete Math. 339 (2016), 2153–2156, doi:10.1016/j.disc.
2016.03.002.
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[17] J. Polhill, Paley partial difference sets in groups of order n4 and 9n4 for any odd n > 1, J.
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1255631810.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.03
https://doi.org/10.26493/2590-9770.1464.7cd
(Also available at http://adam-journal.eu)
The graphs with a symmetrical Euler cycle*
Jiyong Chen
School of Mathematical Sciences, Xiamen University, Xiamen, P. R. China
Cai Heng Li
Department of Mathematics and SUSTech International Center for Mathematics, Southern
University of Science and Technology, Shenzhen, P. R. China
Cheryl E. Praeger†
Department of Mathematics and Statistics, The University of Western Australia,
Crawley 6009, WA, Australia
Shu Jiao Song
School of mathematics and information science, Yantai University, Yantai, P. R. China
Dedicated to our friend and colleague Marston Conder
on the occasion of his 65th birthday.
Received 17 September 2021, accepted 1 December 2021, published online 27 June 2022
Abstract
The graphs in this paper are finite, undirected, and without loops, but may have more
than one edge between a pair of vertices. If such a graph has ℓ edges, then an Euler cycle is
a sequence (e1, e2, . . . , eℓ) of these ℓ edges, each occurring exactly once, such that ei, ei+1
are incident with a common vertex for each i (reading subscripts modulo ℓ). An Euler cycle
is symmetrical if there exists an automorphism of the graph such that ei → ei+2 for each i.
The cyclic group generated by this automorphism has one orbit on edges if ℓ is odd, or two
orbits of length ℓ/2 if ℓ is even: that is to say, the group is regular or bi-regular on edges,
respectively. Symmetrical Euler cycles arise naturally from arc-transitive embeddings of
graphs in surfaces since, for each face of the embedded graph, the sequence of edges on
the boundary of the face forms a symmetrical Euler cycle for the induced subgraph on this
edge-set.
We first classify all finite connected graphs which admit a cyclic subgroup of auto-
morphisms that is regular or bi-regular on edges, and identify more than a dozen infinite
*This work was partially supported by Australian Research Council Discovery project DP160102323, and
NSFC projects nos. 11771200, 11931005, 61771019, 12101518, and NSFS no. ZR2020MA044. The authors
sincerely thank the reviewer for carefully reading their manuscript and for several very helpful comments.
†Corresponding author.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.03
families of examples. We then prove that exactly six of these families consist of graphs
with symmetrical Euler cycles. These are the (only) candidates for the induced subgraphs
of the boundary cycles of the faces of arc-transitive maps.
Keywords: edge-transitive graphs, graphs with multiple edges, graph embeddings, arc-transitive
maps.
Math. Subj. Class.: 20B25, 05C25, 05C35
1 Introduction
The graphs studied in this paper are finite, undirected, without loops, but may have multiple
edges. Thus a graph Γ = (V,E, I) consists of finite sets V of vertices, and E of edges,
together with an incidence relation I ⊆ V ×E such that each edge e is incident with exactly
two distinct vertices. We often suppress I in the notation and write simply Γ = (V,E). An
edge e of Γ incident with the two vertices α and β is sometimes denoted by [α, e, β]. Many
graphs of this type admit natural embeddings as maps into closed surfaces: perhaps the
simplest being the graph K
(λ)
2 with exactly two vertices V = {α, β} and λ edges forming
E = {[α, ei, β] | 1 ≤ i ≤ λ}, and embedded into a sphere with α, β at the poles, and the
edges ei arranged as λ lines of ‘longitude’ joining the two poles.
Throughout the paper all graphs will be of the kind just described. Motivating our in-
vestigation was our wish to study embeddings of graphs as arc-transitive maps in surfaces.
If such a map has at least two faces then the edge-sequence C obtained by moving around
the boundary of a face forms a ‘cycle’, as defined in (1.1). We emphasise that we are avoid-
ing certain degeneracies by making the assumption that each edge in a map is incident with
exactly two faces, and as a consequence each ‘boundary cycle’ of a face has pairwise dis-
tinct edges. We show in Lemma 3.2 that this cycle C is a ‘symmetrical Euler cycle’ for the
induced subgraph [C]: this is a cycle which admits a large subgroup of the corresponding
dihedral group acting with at most two orbits on edges, see Subsection 1.2 and Section 3
for more details. The most natural example for the cycle C is the sequence obtained by
traversing the edges around a simple n-cycle several times, say λ times, and we call the
induced subgraph [C] in this example C
(λ)
n , see Section 3. Maps for which all boundary
cycles are of this form with λ = 1 (simple cycles) have been studied in [10, 11, 14]. On the
other hand, quite different subgraphs [C] have been identified for maps with a single face in
[2, 12]. The problem, which we address in the paper, is to determine the kinds of subgraphs
[C] that arise, induced by symmetrical Euler cycles C, and to describe the possible groups
induced on these cycles by the subgroup of automorphisms leaving [C] invariant.
Note that the subgraph induced by the edges of a boundary cycle of a map is con-
nected, and for an arc-transitive map, the group induced on the cycle contains a cyclic
subgroup having at most two edge-orbits which acts faithfully on each of these edge-orbits
(Lemma 3.2). In our first main result, Theorem 1.1, we broaden the scope of this study
slightly, and classify all connected graphs admitting a cyclic group with at most two edge-
orbits and acting faithfully on each of its edge-orbits. We find a dozen infinite families of
examples. Then, in Theorem 1.2, we show that only six of these families contain graphs for
which the edge set can be sequenced into a cycle preserved by a cyclic group with at most
E-mail addresses: chenjy1988@xmu.edu.cn (Jiyong Chen), lich@sustech.edu.cn (Cai Heng Li),
cheryl.praeger@uwa.edu.au (Cheryl E. Praeger), shujiao.song@hotmail.com (Shu Jiao Song)
J. Chen et al.: The graphs with a symmetrical Euler cycle 3
two edge-orbits (of equal size). To assist with our analysis we develop, in Section 5, the
theory of coset graphs which may have multiple edges and which admit an edge-transitive
group with two vertex-orbits.
1.1 Graphs admitting cyclic edge regular or bi-regular groups
A graph Γ = (V,E) is a simple graph if, for all distinct α, β ∈ V , the number of edges
incident with both α and β is 0 or 1. Given a graph Γ = (V,E) and a positive integer λ,
the λ-extender of Γ is the graph Γ(λ) with vertex set V such that each edge [α, e, β] of Γ
is replaced by λ edges [α, ei, β] of Γ
(λ), for 1 ≤ i ≤ λ. If Γ is simple, then Γ is said to
be the base graph of Γ(λ); in this case if α, β are adjacent in Γ, that is, if there exists an
edge [α, e, β] in Γ, then there are exactly λ edges of Γ(λ) incident with α and β, and we
say that Γ has edge-multiplicity λ. For each edge [α, e, β], we have [α, e, β] = [β, e, α] and
the edge e corresponds to two arcs (α, e, β) and (β, e, α).
Let a group G act on a set Ω. Then G is called transitive or bi-transitive if G has a
single orbit or exactly two orbits in Ω, respectively. Further, if G is finite, then G is said to
be regular or bi-regular on Ω if the permutation group GΩ induced by G on Ω is transitive
and |GΩ| = |Ω|, or GΩ is bi-transitive and |GΩ| = 12 |Ω|, respectively; in other words, GΩ
has at most two orbits in Ω and is faithful and regular on each.
An automorphism of a graph Γ = (V,E, I) is a permutation of V ∪E which preserves
V , E and the incidence relation I. The set of automorphisms forms the automorphism
group AutΓ. Usually AutΓ acts faithfully on E, see Lemma 2.1 for details: the unique
exceptions among connected graphs are the graphs K
(λ)
2 mentioned above. Our first re-
sult Theorem 1.1 presents a classification of connected graphs admitting a cyclic subgroup
of automorphisms that is regular or bi-regular on edges. The exceptional graphs K
(λ)
2
are treated separately in detail in Proposition 2.2. The families of examples, apart from
K
(λ)
2 , are defined in Section 4. Note that for graphs Γ = (V,E) with an edge partition
E = E1 ∪ E2, we sometimes write Γ = [E1] + [E2] to give a rough description of the
graph structure, even though this notation does not uniquely define the graph in general,
see Subsection 2.1. We give a precise description in Section 4 of all the graphs in the tables
for Theorem 1.1.
Theorem 1.1. Let Γ = (V,E) be a connected graph with |V | ≥ 3 such that a cyclic sub-
group G ≤ AutΓ is regular or bi-regular on E, and has NV orbits on V .
Then Γ = Γ
(λ)
0 and |G| = λN , for some λ,N , and either
(i) G is regular on E and Γ0, N,NV are as in one of the lines of Table 1, or
(ii) G is bi-regular on E and Γ0, N are as in one of the lines of Table 2 if NV = 1, and
Γ0, N,NV are as in one of the lines of Table 3 if NV ≥ 2. In particular NV ≤ 3.
1.2 Cycles in graphs and boundary cycles of maps
A cycle of length ℓ in a graph Γ = (V,E), sometimes called an ℓ-cycle, is a sequence
C = (e1, e2, . . . , eℓ) (1.1)
of ℓ pairwise distinct edges each of the form [αi−1, ei, αi], for 1 ≤ i ≤ ℓ, and we read the
subscripts modulo ℓ so that, in particular, αℓ = α0. The edge induced subgraph [C] of a
4 Art Discrete Appl. Math. 5 (2022) #P3.03
Γ0 N NV Conditions Reference
Cn n 1 n ≥ 3 Definition 4.1, Lemma 4.2(a)
Ks,t st 2 gcd(s, t) = 1, st > 1 Definition 4.8, Lemma 4.9
Table 1: Table for Theorem 1.1 with G regular on E.
Γ0 N Conditions Reference
C
(2)
n n n ≥ 3 Definition 4.1, Lemma 4.2(b)
C2n + nK
(2)
2 2n n ≥ 2 Definition 4.3, Corollary 4.5(a)
2Cn + nK
(2)
2 2n n ≥ 3 odd Definition 4.3, Corollary 4.5(b)
Circ(n, S) n n ≥ 5, |S| = 4, Definition 4.3, Lemma 4.4
S = {a,−a, b,−b},
gcd(n, a, b) = 1
Table 2: Table for Theorem 1.1 with G bi-regular on E and transitive on V .
Γ0 N NV Conditions Reference
Cn =
1
2n 2 n ≥ 4 even Def. 4.1, Lemma 4.2(c)
n
2K2 +
n
2K2
K
(2)
s,t st 2 gcd(s, t) = 1, st > 1 Def. 4.8, Lemma 4.9
C2r[sK1, tK1] rst 2 r ≥ 2, gcd(s, t) = 1, st > 1 Def. 4.6, Lemma 4.7
rK
(2t)
2 +K2r,t 2rt 2 rt ≥ 1, gcd(2r, t) = 1 Def. 4.10, Lemma 4.11
rK
(2t)
2 + 2Kr,t 2rt 2 r odd, rt ≥ 1, gcd(r, t) = 1 Def. 4.12, Lemma 4.13
rC
(t)
n +Knr,t nrt 2 n ≥ 3, rt ≥ 1, gcd(nr, t) = 1 Def. 4.10, Lemma 4.11
rC
(t)
su + uKsr,t srut 2 su ≥ 3, u ≥ 2, gcd(r, u) = 1 Def. 4.12, Lemma 4.13
and gcd(sr, t) = 1
rK
(t)
sr′,ut′+ rr
′stt′u 3 gcd(r, r′) = gcd(t, t′) = 1, Def. 4.15, Lemma 4.16
r′K
(t′)
sr,ut gcd(sr, ut) = gcd(sr
′, ut′) = 1
Table 3: Table for Theorem 1.1 with G bi-regular on E and intransitive on V .
J. Chen et al.: The graphs with a symmetrical Euler cycle 5
cycle C is the graph with vertex set V (C) := {αi | 1 ⩽ i ⩽ ℓ}, edge set E(C) := {ei |
1 ⩽ i ⩽ ℓ}, and incidence as in Γ. We call C an Euler cycle of Γ if E(C) = E, that is, the
cycle ‘passes through each edge of Γ exactly once’. If Γ possesses an Euler cycle then, in
particular, Γ is connected. Further, if C as in (1.1) is an Euler cycle of Γ, then since each
edge of E occurs exactly once in C, and since whenever a vertex α = αi then both ei and
ei+1 are incident with α, it follows that for each vertex α ∈ V , the number of edges (of C,
and hence of Γ) incident with α is even.
For any cycle C, the following bijections on E(C) form a dihedral group of order 2ℓ:
φ : ei → ei+1 and τ : ei → eℓ+1−i (for each i, reading subscripts modulo ℓ). (1.2)
The reflection Cτ = (eℓ, eℓ−1, . . . , e1), and each shift Cφ
i = (ei+1, . . . , eℓ, e1, . . . , ei)
are also ℓ-cycles of Γ, and
D(C) := ⟨φ, τ⟩ = ⟨φ, τ | φℓ = τ2 = 1, φτ = φ−1⟩ ∼= D2ℓ (1.3)
is the subgroup of all permutations of E(C) which preserve the edge-sequencing of C,
up to rotations and reflections. Thus the subgroup of AutΓ leaving the cycle C invariant
induces on [C] a subgroup H(C) of Aut[C] contained in D(C). If this subgroup contains
φ2, that is to say, if there is an element σ ∈ Aut(Γ) such that σ|[C] = φ2, then we say
that C is symmetrical in Γ. In particular, for an Euler cycle C of a graph Γ = (V,E), C
is symmetrical if and only if AutΓ contains a cyclic subgroup which preserves C and is
regular or bi-regular on E. For future reference, we note the definitions of the elements φ2
and φτ of D(C):
φ2 : ei → ei+2 and φτ : ei → eℓ−i (for each i, reading subscripts modulo ℓ). (1.4)
For the exceptional graphs K
(λ)
2 , we show in Proposition 2.2 that there is a symmetrical
Euler cycle if and only if λ is even. For all other (connected) graphs we apply Theorem 1.1
to determine whether or not they have a symmetrical Euler cycle.
Theorem 1.2. Let Γ = (V,E) be a graph with |V | ≥ 3 which has a symmetrical Euler
cycle C, and suppose that the subgroup of AutΓ which preserves C induces the subgroup
H(C) of D(C), as in (1.3).
(a) Then (Γ, H(C)) are as in one of the lines of Table 4.
(b) Conversely, if Γ is one of the graphs in lines 1−5 of Table 4, then Γ has a symmetrical
Euler cycle C with the given H(C); and also if Γ is as in line 6 of Table 4 with at
least one of gcd(n, a+ b) = 1 or gcd(n, a− b) = 1, then Γ has a symmetrical Euler
cycle.
As we discuss in Remark 7.5, if Γ = Γ
(λ)
0 with Γ0 = Circ(n, S) as in line 6 of
Table 4, and if Γ0 is a normal Cayley graph (that is the translation subgroup Zn is normal
in AutΓ0), then the the condition ‘gcd(n, a+b) = 1 or gcd(n, a−b) = 1’ is necessary and
sufficient for existence of a symmetrical Euler cycleC. A complete analysis of these graphs
would need a better understanding of any non-normal, edge-transitive, Cayley graphs in this
family.
The principle motivation for our work was the study of boundary cycles of faces of
arc-transitive maps. The link between arc-transitive maps and symmetrical Euler cycles is
6 Art Discrete Appl. Math. 5 (2022) #P3.03
Γ Conditions for C to exist H(C) Reference
C
(λ)
n all n ≥ 3, λ ≥ 1 H(C) = D(C) Lemma 7.1(a)
K
(λ)
s,t st > 1, gcd(s, t) = 1, λ even H(C) = ⟨φ2, φτ⟩ Lemma 7.1(b)
(C2n + nK
(2)
2 )
(λ) n ≥ 2, n even, λ ≥ 1 H(C) = ⟨φ2, φτ⟩ Lemma 7.3(a)
(2Cn + nK
(2)
2 )
(λ) n ≥ 3, n odd, λ ≥ 1 H(C) = ⟨φ2, φτ⟩ Lemma 7.3(b)
(C2r[sK1, tK1])
(λ) r ≥ 2, st > 1, gcd(s, t) = 1, λ ≥ 1 H(C) = ⟨φ2, τ⟩ Lemma 7.6
(Circ(n, S))(λ) n ≥ 5, S = {a,−a, b,−b}, H(C) ≥ ⟨φ2⟩ Lemma 7.4
|S| = 4, gcd(n, a, b) = 1, λ ≥ 1 see Remark 7.5
Table 4: Table for Theorem 1.2 with C a symmetrical Euler cycle for Γ.
made explicit in Lemma 3.2. A natural problem which we plan to explore in further work
is to understand which of these graphs from Theorem 1.2 actually arise in arc-transitive
maps.
Problem 1.3. Determine which of the graphs in Table 4 arise as the induced subgraph of
the boundary cycle of a face in an arc-transitive map.
We were able to find a partial answer in our study of vertex-rotary maps, that is, arc-
transitive maps for which a vertex-stabiliser is cyclic and regular on the edges incident
with it. We prove in [8, Theorem 1.9] that each face boundary cycle C for a vertex-rotary
map has induced subgraph [C] = C
(λ)
n as in line 1 of Table 4, and moreover the cycle C
is of the form C
(λ)
0 , with C0 a simple n-cycle, as in Proposition 3.4(b). In addition, for
n ≡ 2 (mod 4), we construct in [8, Section 6] infinitely many examples of such maps
from extenders of complete bipartite graphs, namely, if n = 2m with m odd, and if λ
is such that gcd(n, λ) = 2, we construct an embedding of K
(2λ)
m,m with all face boundary
cycles of the form C
(λ)
0 for a simple n-cycle C0. In particular, it would be interesting to
know precisely which values of n, λ are possible for vertex-rotary maps.
2 Preliminaries and the exceptional graph K
(λ)
2
2.1 Notation
For a graph Γ = (V,E, I), and any subset of edges E′ ⊆ E, the edge-induced subgraph is
the graph [E′] := (V (E′), E′, I′), where
V (E′) = {α | (α, e) ∈ I, for some e ∈ E′}, and I′ = I ∩ (V (E′)× E′.
An isolated vertex of a graph Γ, is a vertex which is not incident to any edge of Γ. Thus Γ
has no isolated vertices if and only if V (E) = V . The graph Γ is connected if, for each pair
of vertices α, β there exists a sequence of edges e1, . . . , er such that ei = [αi−1, ei, αi] for
each i, and α0 = α and αr = β. A connected component of Γ is either (i) a one-vertex
graph ({α}, ∅) where α is an isolated vertex, or (ii) a connected edge-induced subgraph
[E′] for some non-empty E′ ⊆ E such that, for each edge e ∈ E, either e ∈ E′, or e is
incident with no vertex of V (E′).
For some of the graphs in Theorem 1.1, the group considered has two edge-orbits,
and we view the graph as an edge-disjoint union of two smaller graphs. We use the
J. Chen et al.: The graphs with a symmetrical Euler cycle 7
following notation: given graphs Γi = (Vi, Ei, Ii), for i = 1, 2, where the vertex sets
V1, V2 may overlap but E1 ∩ E2 = ∅, the edge-disjoint union of Γ1 and Γ2 is the graph
Γ1 + Γ2 = (V,E, I), such that
V = V1 ∪ V2, E = E1 ∪ E2, and I = I1 ∪ I2 ⊂ V × E.
2.2 The graph K
(λ)
2
First we prove the assertion, mentioned in the introduction, that essentially the only graphs
for which the automorphism group is not faithful on edges are those with at least one
component K
(λ)
2 .
Lemma 2.1. Let Γ = (V,E, I) be a graph with no isolated vertices and G ≤ AutΓ. Then
either G acts faithfully on E, or some connected component of Γ is isomorphic to K
(λ)
2 ,
for some λ.
Proof. Let K = G(E) be the kernel of the action of G on E. By the definition of an
automorphism, K acts faithfully on V . If K is trivial on V then K = 1 and there is
nothing to prove, so assume that x ∈ K and α ∈ V such that αx ̸= α. Since Γ has no
isolated vertices there exists an edge e = [α, e, β], and since x fixes e it follows that x
interchanges α and β. Since x fixes all edges, it follows that each edge incident with α
is also incident with β, and conversely, each edge incident with β is also incident with α.
Hence Γ has a connected component with vertex set {α, β}, and it is isomorphic to K(λ)2 ,
for some λ. ✷
The graph Γ = K
(λ)
2 = (V,E) has automorphism group AutΓ = Sym(E)×Sym(V ) ∼=
Sym(λ)×Z2, with Sym(E) = Sym(λ) acting naturally on E and fixing V pointwise, and
Sym(V ) acting naturally on V and fixing E pointwise. The additional automorphisms,
since Aut(Γ) is not faithful on edges, give rise to extra examples of edge-regular and bi-
regular actions, and it is instructive to consider these graphs separately here. Of the six
edge regular or bi-regular groups G identified in Table 5, for precisely two of them (lines 1
and 4) the induced regular or bi-regular action on the edge set is unfaithful.
Proposition 2.2. Let Γ = K
(λ)
2 = (V,E) for some λ ≥ 1, and let G = ⟨g⟩ ≤ AutΓ be
regular or bi-regular on E. Then
(a) G ≤ L × ⟨y⟩, where ⟨y⟩ = Sym(V ) ∼= Z2, L = ⟨x⟩ ∼= Zλ is a cyclic edge-regular
subgroup of Sym(E), and G, λ are as in one of the lines of Table 5.
(b) Moreover, Γ has a symmetrical Euler cycle C if and only if λ is even, and in this case
H(C) = ⟨φ, τ⟩ = D(C), with φ, τ,D(C) as in (1.2) and (1.3).
Proof. (a) The projection π : G→ Sym(E) has image a cyclic edge-regular or bi-regular
subgroup, and in either case there exists a cyclic edge-regular subgroup L ≤ Sym(E)
containing π(G). Thus G ≤ L × ⟨y⟩ with L = ⟨x⟩ ∼= Zλ. Suppose first that G is edge-
regular, so π(G) = L and |G| is a multiple of λ. One possibility is that G = L× ⟨y⟩, and
since G is cyclic this implies that λ is odd and line 1 of Table 5 holds. So suppose that G is
a proper subgroup of L× ⟨y⟩. Since |G| is a multiple of λ, it follows that |G| = λ. If G is
vertex-transitive then G = ⟨xy⟩ ∼= Zλ and this implies that λ is even, and line 2 of Table 5
8 Art Discrete Appl. Math. 5 (2022) #P3.03
G |G| Action on V Action on E Conditions/comments
⟨x⟩ × ⟨y⟩ 2λ transitive regular λ odd, G(E) ̸= 1
⟨xy⟩ λ transitive regular λ even
⟨x⟩ λ trivial regular λ arbitrary
⟨x2⟩ × ⟨y⟩ λ transitive bi-regular λ = 2λ0, λ0 odd, G(E) ̸= 1
⟨x2y⟩ λ0 transitive bi-regular λ = 2λ0, λ0 even
⟨x2⟩ λ0 trivial bi-regular λ = 2λ0, λ0 arbitrary
Table 5: Edge regular and bi-regular actions on K
(λ)
2 .
holds. On the other hand, if G is intransitive, then G = ⟨x⟩ ∼= Zλ and line 3 of Table 5
holds (and here λ is arbitrary).
Suppose now that G is edge-bi-regular, so λ = 2λ0 is even and π(G) = ⟨x2⟩ ∼= Zλ0 ,
and G ≤ ⟨x2⟩ × ⟨y⟩. If G = ⟨x2⟩ × ⟨y⟩, then λ0 = |x2| is odd since G is cyclic, and line
4 of Table 5 holds. Suppose now that G is a proper subgroup of ⟨x2⟩ × ⟨y⟩, so |G| = λ0.
If G is vertex-transitive then G = ⟨x2y⟩ ∼= Zλ0 and this implies that λ0 is even, and line 5
of Table 5 holds. On the other hand, if G is vertex-intransitive, then G = ⟨x2⟩ ∼= Zλ0 and
line 6 of Table 5 holds (and here λ0 is arbitrary).
(b) By our comments in Subsection 1.2, in an Euler cycle each vertex is incident with
an even number of edges. Hence there is no Euler cycle if λ is odd. Suppose that λ is even.
Then it is easy to sequence the edges into a symmetrical Euler cycle using, say, the edge-
regular automorphism x: choose e ∈ E and define e0 = e and ei = ex
i
for 1 ≤ i ≤ λ− 1.
Then C = (e0, . . . , eλ−1) is an Euler cycle and x induces φ ∈ D(C) and fixes each of the
vertices. Also there is an involution z ∈ NSym(E)(⟨x⟩) given by z : ei → eλ−i for each i,
and hence z induces φτ in H(C), and it follows that H(C) = D(C). ✷
3 Cycles, symmetrical Euler cycles, and maps
Let C be an ℓ-cycle in a graph Γ = (V,E, I), as in (1.1), with edges [αi−1, ei, αi], reading
the subscripts modulo ℓ. If the vertices αi are pairwise distinct, then [C] is a simple graph
of valency 2, and hence [C] = Cℓ. In this case we callC a simple ℓ-cycle, and since the ver-
tices αi are pairwise distinct, we can identify C with the vertex sequence (α1, α2, . . . , αℓ).
Other examples of ℓ-cycles C arise naturally with [C] = C
(λ)
n where ℓ = nλ, with the
edges sequenced in any way so that αi = αj whenever i ≡ j (mod n). The set of all
ℓ-cycles with this property forms a single orbit under the automorphism group of C
(λ)
n , and
we call each such ℓ-cycle standard, or a standard edge-sequencing. However not all cycles
in graphs are of this kind. Even for [C] = C
(λ)
n with λ > 1, there are many non-standard
ways to seqence the edges to form a cycle. For example, if e1ij , e
2
ij denote the two edges of
C
(2)
3 incident with vertices i, j, then (e
1
12, e
2
12, e
1
13, e
1
33, e
2
23, e
2
13) is a non-standard 6-cycle.
Moreover the induced subgraph [C] of a cycle C may be quite different from C
(λ)
n , as the
next examples show (some of which appear again in our main theorems).
Example 3.1. Let Γ = Σ(λ), with λ = 2 and with Σ one of the following simple graphs.
Then Γ has constant edge-multiplicity 2, and its edge set can be sequenced to form an Euler
cycle. (Finding these sequencings is left as an easy exercise.)
J. Chen et al.: The graphs with a symmetrical Euler cycle 9
(a) Σ is the complete bipartite graph K2,3, so Γ has ℓ = 12 edges and n = 5 vertices;
(b) Σ is the cartesian product Cr ✷K2, so Γ has ℓ = 6r edges and n = 2r vertices, and
Γ is regular of valency 3 (each vertex is adjacent to 3 vertices);
(c) Σ = (V,E, I) with V = Zr × Z2 and E = {ei, fi | i ∈ Zr} such that, for each
i ∈ Zr, ei is incident with (i, 0) and (i + 1, 0), and fi is incident with (i, 0) and
(i, 1). Here Γ = Σ(2) has ℓ = 4r edges, n = 2r vertices, and we note that ℓ = nλ.
These examples illustrate that, for an ℓ-cycle C, the induced graph [C] need not be
regular (examples (a) and (c)), or [C] may be regular of valency greater than 2 (example
(b)), and even if [C] has constant edge-multiplicity λ and nλ edges, where n is the number
of vertices, [C] need not be C
(λ)
n (example (c) again). These observations hold in particular
for the Euler cycles of the graphs in Example 3.1.
3.1 Symmetrical cycles and arc-transitive maps
Let C = (e1, e2, . . . , eℓ) be an ℓ-cycle in a graph Γ. As we noted in Section 1.2, each
rotation (shift) Cφi and reflection (reversal followed by a shift) Cτφi of C is again a
cycle involving the same edge-set E(C) as C. This collection of cycles is called the se-
quence class of cycles containing C. We say that an automorphism g ∈ AutΓ preserves
C if g leaves invariant the sequence class of C. Thus the subgroup (AutΓ)C of AutΓ
preserving C induces a subgroup H(C) of automorphisms of the induced subgraph [C]
such that H(C) ≤ D(C) with D(C) as in (1.3). As discussed above, C is called sym-
metrical if H(C) contains ⟨φ2⟩, and for such cycles, Theorem 1.2 determines all possible
induced subgraphs [C]. Apart from ⟨φ2⟩ itself, the subgroups H(C) of D(C) contain-
ing ⟨φ2⟩ are listed in Table 6. These subgroups are relevant for studying edge-transitive
or face-transitive embeddings of graphs in Riemann surfaces, see for example, [12] or
[4, Chapter 3, especially Theorem 3.5].
A map M = (V,E, F ) is a 2-cell embedding of a graph Γ = (V,E) in a closed surface
with face set F , and the graph Γ is called the underlying graph of M. The automorphism
group of the map is the subgroup of Aut(Γ) consisting of those graph automorphisms
which extend to self-homeomorphisms of the supporting surface [7, Section 6]. It is also
sometimes defined as the group of permutations of the flags (mutually incident vertex-edge-
face triples) that ‘preserve vertex-vertex and face-face adjacencies’ [12, Section 2]. An arc
of a map, or a graph, is an incident vertex–edge pair, and a map M is called arc-transitive if
its automorphism group AutM is transitive on the arc-set of M. The next result describes
the the kinds of groups H(C) which may arise for boundary cycles C of arc-transitive
maps. In [9] we discuss finite maps M = (V,E, F ) with arc-transitive automorphism
groups, where the underlying graph Γ = (V,E) may have multiple edges, as in this paper.
The only facts about maps we use in the proof of Lemma 3.2 are that each edge in E is
incident with exactly two faces in F , together with the arc-transitivity of the map group
AutM ≤ AutΓ.
Lemma 3.2. Let M = (V,E, F ) and G ⩽ AutM such that G is arc-transitive on M,
and each edge in E is incident with exactly two faces in F . Let f ∈ F be a face, and
C = (e1, . . . , eℓ) the boundary cycle of f . Then the stabiliser Gf induces a subgroup
H(C) of D(C) as in one of the lines of Table 6, which gives also the H(C)-orbits on
edges E(C) and vertices V (C) of C. In each case, H(C) contains the cyclic subgroup
10 Art Discrete Appl. Math. 5 (2022) #P3.03
H(C) orbits in E(C) orbits in V (C) Comments
D(C) E(C) V (C)
⟨φ⟩ E(C) V (C) equals ⟨φ2⟩ if ℓ odd
⟨φ2, φτ⟩ two orbits V (C) ℓ even
⟨φ2, τ⟩ E(C) two orbits if λ = 1 ℓ even
Table 6: Groups H(C) for a boundary cycle C of an arc-transitive map M.
⟨φ2⟩ which is regular (if ℓ is odd) or bi-regular (if ℓ is even) on E(C), and ⟨φ2⟩ has at
most two orbits in V (C). In particular C is a symmetrical Euler cycle of [C].
Proof. For C = (e1, . . . , eℓ) we have ei = [αi−1, ei, αi] for each i, reading subscripts
modulo ℓ. Let fi be the second face incident with ei, for each i. The setwise stabiliser
Gf preserves C and induces a subgroup H(C) of D(C). As M is G-arc-transitive, there
exist g, h ∈ G such that (α0, e1, α1)h = (α1, e1, α0) and (α0, e1, α1)g = (α1, e2, α2).
Suppose first that fh = f . Then also fh1 = f1. For each i there exists gi ∈ AutM such
that egi1 = ei and so gi maps the pair {f, f1} to the pair of faces {f, fi} incident with ei.
In particular fg
−1
i is one of f, f1 and so is fixed by h and hence f
g−1
i
hgi = fg
−1
i
gi = f .
It follows that H(C) contains each edge-reflection, and hence contains φ2 as well as the
edge-reflection induced by h. Thus in this case H(C) is either ⟨φ2, φτ⟩ with ℓ even and
has two orbits in E(C), or is D(C), and in either case H(C) is transitive on V (C). Thus
we may assume that each element h such that (α0, e1, α1)
h = (α1, e1, α0) interchanges
f and f1, and hence that g
−1
i hgi interchanges f and fi, for each i. If g leaves the face f
invariant then H(C) contains φ so H(C) = ⟨φ⟩ (by our assumption on the elements h)
and H(C) is transitive on both E(C) and V (C).
So assume also that g does not fix f for any such g mapping (α0, e1, α1) to (α1, e2, α2).
Then fg1 = f, f
g = f2, and hence hg leaves f invariant. Now hg induces a reflection of C
in the vertex α1. Similar arguments, with the edges e1, e2 replaced by e2, e3, either yield
one of the groups above for H(C), or show that H(C) contains also the reflection of C
in the vertex α2. In the latter case the product of the reflection in α1 and the reflection
in α2 is a generator of the group ⟨φ2⟩, and hence these two reflections generate the group
H(C) = ⟨φ2, τ⟩, which is regular on E(C) and has two vertex-orbits when ℓ is even. (If ℓ
is odd the group generated is D(C).) ✷
Remark 3.3. We note that the subgroup ⟨φ2, τ⟩ contains all reflections of C ‘through a
vertex’: τ reflectsC through the vertex α0 (between the edges eℓ and e1), while τφ
2 reflects
C through the vertex α1 (between the edges e1 and e2), etc. Thus, for a symmetrical cycle
C as in (1.1), to prove that H(C) = ⟨φ2, τ⟩ we need only prove that H(C) contains one
of these reflections. Similar comments apply to the subgroup ⟨φ2, φτ⟩, which contains all
reflections of C ‘through an edge’, where φτ reflects C through the edge eℓ, etc. To prove
that H(C) = ⟨φ2, φτ⟩, for a symmetrical cycle C, we need only prove that H(C) contains
one reflection through an edge. We note also that it is sometimes more convenient in our
analysis to label the edges of an ℓ-cycle as (e0, e1, . . . , eℓ−1) for working with the action
of the cyclic group Zℓ.
J. Chen et al.: The graphs with a symmetrical Euler cycle 11
ψ induces in H(C) ψ(λ) on edges, for i mod ℓ, j mod λ
φ eji →
{
eji+1 if 1 ≤ i ≤ ℓ− 1
ej+1i+1 if i = ℓ
φ2 eji →
{
eji+2 if 1 ≤ i ≤ ℓ− 2
ej+1i+2 if i ∈ {ℓ− 1, ℓ}
τ eji →
{
eλ+1−jℓ+1−i if 1 ≤ i ≤ ℓ− 1
eλ−jℓ+1−i if i = ℓ
φτ eji →





eλ+1−jℓ−i if 1 ≤ i ≤ ℓ− 2
eλ−jℓ−i if i = ℓ− 1
eλ−jℓ if i = ℓ
Table 7: Elements in H(C(λ)) for an Euler cycle C of Γ; in all cases ψ(λ)|V = ψ|V .
3.2 Symmetrical Euler cycles and λ-extenders
It turns out that if a graph Γ (not necessarily a simple graph) has a symmetrical Euler cycle,
then so does its λ-extender for each λ. We prove this as a consequence of a more general
result about cyclic edge regular or bi-regular subgroups of automorphisms. It is useful to
use the following notation: for Γ = (V,E), we have Γ(λ) = (V,E(λ)), and we label the λ
edges in E(λ) corresponding to the edge e = [β, e, α] ∈ E by
ej = [β, ej , α], for 1 ⩽ j ⩽ λ. (3.1)
Proposition 3.4. Suppose that, for a graph Γ = (V,E), AutΓ has a cyclic subgroup that
is regular or bi-regular on E with t orbits on V , and let λ be a positive integer. Then, for
the λ-extender Γ(λ),
(a) AutΓ(λ) has a cyclic regular or bi-regular subgroup, respectively, with t orbits on
V ;
(b) if C = (e1, . . . , eℓ) is an Euler cycle of Γ, and if ψ ∈ AutΓ induces one of
φ,φ2, τ, φτ in H(C), as in (1.2) and (1.4), then
C(λ) = (e11, . . . , e
1
ℓ , e
2
1, . . . , e
2
ℓ , . . . , e
λ
1 , . . . , e
λ
ℓ )
is an Euler cycle of Γ(λ), and the map ψ(λ) as in Table 7 lies in AutΓ(λ) and in-
duces the element φ,φ2, τ, φτ in H(C(λ)), respectively. In particular, if Γ has a
symmetrical Euler cycle, then also Γ(λ) has a symmetrical Euler cycle.
Proof. (a) By assumption, there is an element ψ ∈ AutΓ such that G = ⟨ψ⟩ is reg-
ular or bi-regular on E. Thus we may label the edge-set E = {e1, . . . , eℓ}, such that
ei = [βi, ei, αi] for each i, and for 1 ⩽ i ⩽ ℓ, reading subscripts modulo ℓ,
ψ : ei → ei+1, if ψ is regular on E, or ψ : ei → ei+2, if ψ is bi-regular on E,
and hence ψ : {βi, αi} → {βi+1, αi+1} or {βi, αi} → {βi+2, αi+2}, respectively, for
each i. For each i, we label the λ edges of Γ(λ) = (V,E(λ)) which correspond to the edge
12 Art Discrete Appl. Math. 5 (2022) #P3.03
ei = [βi, ei, αi] by e
j
i , for 1 ⩽ j ⩽ λ, as in (3.1), and we define the map ψ
(λ) : V ∪E(λ) →
V ∪ E(λ) (reading subscripts modulo ℓ, and superscripts modulo λ) by ψ(λ)|V = ψ|V and
ψ(λ) : eji →
{
eji+1 if 1 ≤ i ≤ ℓ− 1
ej+1i+1 if i = ℓ
if ⟨ψ⟩ is regular, or
ψ(λ) : eji →
{
eji+2 if 1 ≤ i ≤ ℓ− 2
ej+1i+2 if i ∈ {ℓ− 1, ℓ}
if ⟨ψ⟩ is bi-regular. Then ψ(λ) is a bijection and preserves incidence in Γ(λ), and hence
ψ(λ) ∈ AutΓ(λ). Further ⟨ψ(λ)⟩ is regular or bi-regular on E(λ), according as G is regular
or bi-regular on E, respectively, and as ψ(λ)|V = ψ|V , the groups ⟨ψ⟩, ⟨ψ(λ)⟩ have the
same number of vertex-orbits. This proves part (a).
(b) Suppose now that Γ has an Euler cycle C = (e1, . . . , eℓ) with each
ei = [αi−1, ei, αi], and that ψ induces one of the maps φ,φ
2, τ, φτ in H(C). Let C(λ) and
ψ(λ) be as in the statement and Table 7. From the definition of E(λ) it is clear that C(λ) is
a cycle, and hence an Euler cycle of Γ(λ). Clearly ψ(λ) is a bijection on V ∪E(λ) and pre-
serves incidence and hence lies in AutΓ(λ). A somewhat tedious checking shows that the
map ψ(λ) is equal to the element φ,φ2, τ, φτ in H(C(λ)), respectively. In particular if C
is a symmetrical Euler cycle of Γ, so there is an element ψ ∈ AutΓ inducing φ2 in H(C),
then we have just shown that ψ(λ) induces φ2 in H(C(λ)). Hence C(λ) is a symmetrical
Euler cycle in Γ(λ). ✷
4 Examples of graphs
Here we introduce the families of graphs occurring in the tables for Theorem 1.1. In the
light of Proposition 3.4, to show that a λ-extender Γ = Γ
(λ)
0 satisfies the hypotheses of
Theorem 1.1, it is sufficient to show that the graph Γ0 satisfies them. Thus we will in
general consider graphs which are not λ-extenders.
4.1 Circulants
View Zn = {0, 1, . . . , n − 1} as a cyclic group of order n under addition. The circulants
we construct may have multiple edges. Let S be a multiset of elements from Zn \ {0}
such that, if an element z ∈ Zn appears exactly λ times in S, we write z(λ) ∈ S. Assume
further that S is self-inverse, that is, if z(λ) ∈ S then (−z)(λ) ∈ S. For such a self-inverse
multiset S, Circ(n, S) denotes the (Cayley) graph with vertex set Zn such that, for each
v ∈ Zn and each z(λ) ∈ S, there are exactly λ edges between v and v + z. Graphs
of this form are called circulants of order n. If each element of a multiset has the same
multiplicity λ, then we sometimes denote the multiset by S(λ) for some subset S ⊆ Zn,
that is, S(λ) = {s(λ) | s ∈ S}. First we construct the graphs occurring in Theorem 1.1
which are essentially cycles.
Definition 4.1. Let n be an integer, n ≥ 3, and let S = {1,−1}, regarded as a multiset
in Zn. Let Γ = Circ(n, S) = (V,E) with V = Zn and E = {ei | i ∈ Zn}, where
ei = [i, ei, i+ 1], for i ∈ Zn. Define the map g : V ∪ E → V ∪ E by
g : i→ i+ 1, and g : ei → ei+1 for i ∈ Zn.
J. Chen et al.: The graphs with a symmetrical Euler cycle 13
Define the edge sequence C = (e0, . . . , en−1) of length n.
Lemma 4.2. With the notation of Definition 4.1, the graph Γ ∼= Cn, Γ is connected,
g ∈ AutΓ, and
(a) the group ⟨g⟩ is transitive on V and regular on E and line 1 of Table 1 holds.
Moreover, C is a symmetrical Euler cycle for Γ, g induces the map φ of (1.2), and
H(C) = D(C) ∼= D2n.
(b) For the 2-extender C
(2)
n , the group ⟨g⟩ is transitive on V and bi-regular on E(2) and
line 1 of Table 2 holds.
(c) If n is even, n ≥ 4, then ⟨g2⟩ ∼= Zn/2 is bi-transitive on V , and bi-regular on E with
orbits E0 and E1 such that [E0] ∼= [E1] ∼= n2K2, and line 1 of Table 3 holds.
Proof. Clearly Γ ∼= Cn, Γ is connected, and g ∈ AutΓ, and most of part (a) follows
immediately from the definitions of g and a symmetrical Euler cycle for Γ, with g inducing
the map φ in H(C). Further AutΓ contains an automorphism τ as in (1.2), and ⟨g, τ⟩ ∼=
D2n, so H(C) = D(C) = D2n. For part (b), we note that ⟨g⟩ has two orbits in E(2),
namely, for j ∈ {1, 2}, Ej := {eji | i ∈ Zn}, where eji = [i, eji , i+ 1] for i ∈ Zn. Finally
for part (c), for n even, ⟨g2⟩ ∼= Zn/2 is bi-transitive on V , and bi-regular on E with edge-
orbits E0 := {e2i | 0 ≤ i < n/2} and E1 := {e2i+1 | 0 ≤ i < n/2}, and the induced
subgraphs [E0] ∼= [E1] ∼= n2K2. ✷
Next we consider the remaining circulants occurring in Table 2 for Theorem 1.1. For
a ∈ Zn, by |a| we mean the additive order of a, that is, the least positive integer m such
that ma ≡ 1 (mod n).
Definition 4.3. Let n be a positive integer with n ≥ 3, and let a, b ∈ Zn \ {0} such that
gcd(n, a, b) = 1. Let Γ = Γ(n, a, b) = Circ(n, S) where
S =
{
{a,−a, b,−b} if |a| ≥ 3, |b| ≥ 3, a ̸= ±b
{a,−a, b(2)} if |a| ≥ 3, 2b = n
so Γ has 2n edges, namely ei,a = [i, ei,a, i+ a], and ei,b = [i, ei,b, i+ b], for i ∈ Zn, and
we note that, if 2b = n then ei,b and ei+b,b are both incident with the same pair of vertices
{i, i+ b}. Let Ea = {ei,a | i ∈ Zn}, Eb = {ei,b | i ∈ Zn}. Thus Γ is not always simple,
but in all cases the natural generator g of Zn acts as follows:
g : i→ i+ 1, ei,a → ei+1,a, ei,b → ei+1,b, for i ∈ Zn.
Lemma 4.4. With the notation of Definition 4.3, the graph Γ = Γ(n, a, b) is connected,
and the group ⟨g⟩ = Zn induces a cyclic subgroup of AutΓ which is regular on vertices
and bi-regular on edges, with edge-orbits Ea and Eb. Moreover, if 2b ̸= n then Γ is simple
of valency 4 as in line 4 of Table 2.
Proof. The vertices reached by sequences of edges beginning at 0 are precisely those of
the form ia + jb for some integers i, j, and these vertices are the multiples of gcd(a, b),
modulo n. Since gcd(n, a, b) = 1 it follows that all vertices occur so Γ is connected and
Zn = ⟨a, b⟩. By definition of circulants, Zn, acting naturally by ‘right multiplication’
14 Art Discrete Appl. Math. 5 (2022) #P3.03
induces a cyclic subgroup of AutΓ which is regular on vertices, and it has Ea, Eb as its
edge-orbits. If 2b ̸= n then Γ is simple, has valency |S| = 4, and is a graph in line 4 of
Table 2. ✷
Note that if |b| = 2 in Definition 4.3 then n is even and b = n/2. The following
corollary describes two special cases of the construction in Definition 4.3 where |b| = 2.
We note that in case (b) below, the graph Γ(2r, 2, r) is isomorphic to the cartesian product
Cr✷K
(2)
2 .
Corollary 4.5. Let n = 2r ≥ 4. Then the following hold, with the notation of Defini-
tion 4.3,
(a) Γ = Γ(2r, 1, r) = Circ(2r, {1,−1}) + Circ(2r, {r(2)}) = C2r + rK(2)2 ; and
(b) if r is odd, then Γ = Γ(2r, 2, r) = Circ(2r, {2,−2}) + Circ(2r, {r(2)}) =
2Cr + rK
(2)
2 .
In either case, Γ is connected, and AutΓ has a cyclic subgroup that is transitive on vertices
and bi-regular on edges, as in line 2 or 3 of Table 2.
Proof. This follows immediately from Lemma 4.4, noting that gcd(2r, 1, r) = 1, and
when r is odd also gcd(2r, 2, r) = 1. ✷
4.2 Modified Kronecker product graphs
Our next construction is a modification of the Kronecker product construction for graphs,
so we use similar notation. It produces the graphs in line 3 of Table 3.
Definition 4.6. Let r, s, t be positive integers such that r ≥ 2, st ≥ 2 and gcd(s, t) = 1,
and let S = Zs, T = Zt. Then Γ = C2r[sK1, tK1] = (V,E) is the graph defined as
follows. The vertex set V = VS ∪ VT , where
VS = {(2k, i) | 0 ≤ k ≤ r− 1, i ∈ S}, and VT = {(2k+ 1, j) | 0 ≤ k ≤ r− 1, j ∈ T},
and edge set E = ES ∪ ET where
ES = {[(2k, i), e2ki,j , (2k + 1, j)] | 0 ≤ k ≤ r − 1, i ∈ S, j ∈ T}, and
ET = {[(2k + 1, j), e2k+1j,i , (2k + 2, i)] | 0 ≤ k ≤ r − 1, i ∈ S, j ∈ T},
and we read the entry 2k+2 modulo 2r. Define the maps g, y : V ∪E → V ∪E as follows,
for 0 ≤ k ≤ r − 1, i ∈ S, j ∈ T :
g : (2k, i) → (2k + 2, i), (2k + 1, j) → (2k + 3, j) if k ≤ r − 2,
(2r − 2, i) → (0, i+ 1), (2r − 1, j) → (1, j + 1)
e2ki,j → e2k+2i,j , e2k+1j,i → e2k+3j,i if k ≤ r − 3,
e2r−4i,j → e2r−2i,j , e2r−3j,i → e2r−1j,i+1
e2r−2i,j → e0i+1,j+1, e2r−1j,i → e1j+1,i
y : (0, i) → (0,−i), (1, j) → (2r − 1,−j − 1)
(2k, i) → (2(r − k),−i− 1), (2k + 1, j) → (2(r − k − 1) + 1,−j − 1)
e0i,j → e2r−1−j−1,−i, e2r−1j,i → e0−i,−j−1
e2ki,j → e
2(r−k−1)+1
−j−1,−i−1 , e
2k−1
j,i → e
2(r−k)
−i−1,−j−1
where i ∈ S, j ∈ T , and in lines 2 and 4 in the definition of y we have 1 ≤ k ≤ r − 1.
J. Chen et al.: The graphs with a symmetrical Euler cycle 15
e00,0
e20,0e
4
0,0
e01,1
e41,2
e10,0
e30,0
e50,1
e52,0
e11,1
(0, 0)
(0, 1)
(1, 0)
(1, 1)
(1, 2)
(2, 0)(2, 1)
(3, 0)
(3, 2)
(4, 0)
(4, 1)
(5, 0) (5, 2)
Figure 1: Graph Γ = C2r[sK1, tK1] from Definition 4.6, with r = 3, s = 2, t = 3.
We could have defined the graph C2r[sK1, tK1] with s = t = 1, but then it is just the
cycle C2r so we avoid this degenerate case. It is not difficult to see that the automorphism
group of C2r[sK1, tK1] is isomorphic to (Sym(s)×Sym(t)) ≀D2r. We show that the map
g generates a vertex-bi-transitive, edge-bi-regular group of automorphisms, and moreover
that ⟨g, y⟩ is an edge-regular dihedral group.
Lemma 4.7. With the notation of Definition 4.6, Γ = C2r[sK1, tK1] is connected and is
the graph in line 3 of Table 3, the maps g, y are automorphisms of Γ, and
(a) ⟨g⟩ is cyclic of order rst, bi-transitive on V with orbits VS , VT , and bi-regular on E
with orbits ES , ET ; also ⟨g, y⟩ ∼= D2rst and is regular on E;
(b) further, for each positive integer λ, the graph Γ(λ), admits a cyclic subgroup Zrstλ,
and a dihedral subgroup D2rstλ of automorphisms that are both bi-transitive on
vertices, and are bi-regular or regular on edges, respectively.
Proof. It is straightforward to check that Γ is connected, and that Γ is the graph in line 3 of
Table 3. Also g, y are bijections and it is straightforward, if tedious, to check that each of
g, y preserves incidence. Hence g, y ∈ AutΓ. First we consider repeated applications of g
on the edge e = e00,0 = [(0, 0), e
0
0,0, (1, 0)] ∈ ES . Suppose that eg
m
= e. Now eg
m
= e2mi,j
for some i ∈ S, j ∈ T , reading the superscript 2m modulo 2r, and this means, since
eg
m
= e, that m = rℓ for some integer ℓ. Now eg
r
= [(0, 1), e01,1, (1, 1)], and repeating
this we find that eg
rℓ
= [(0, ℓ), e0ℓ,ℓ, (1, ℓ)], where in the first entry we read ℓ modulo s, and
in the third entry we read ℓ modulo t. Thus, again since eg
rℓ
= eg
m
= e, we conclude that
ℓ is divisible by both s and t, and hence by st since gcd(s, t) = 1. Thus m is a multiple of
rst, and since |ES | = rst and g leaves ES invariant it follows that the ⟨g⟩-orbit containing
16 Art Discrete Appl. Math. 5 (2022) #P3.03
e is ES . A similar proof shows that the ⟨g⟩-orbit containing e10,0 is ET , and it follows that
⟨g⟩ is bi-regular on E. Also the vertex-orbits of ⟨g⟩ are clearly VS and VT and we conclude
that ⟨g⟩ has order rst, and is bi-transitive on V . This proves the first assertion.
Now we consider y. It follows from the previous paragraph that y2, (yg)2 ∈ AutΓ.
Again it is tedious, but not too difficult, to verify that both y2 and (yg)2 act trivially on the
vertex set V , and hence each of them leaves invariant the edge subsets
ES,k = {e2ki,j | i ∈ S, j ∈ T} and ET,k = {e2k+1j,i | i ∈ S, j ∈ T}, for each
k = 0, . . . , r − 1. For a fixed k, since each vertex pair {(2k, i), (2k + 1, j)} (for i ∈
S, j ∈ T ) is incident with a unique edge e2ki,j ∈ ES,k, it follows that y2 and (yg)2 act triv-
ially on ES,k. Similarly y
2 and (yg)2 act trivially on ET,k, and since this holds for each k,
it follows that y2 = (yg)2 = 1. Thus ygy = g−1 and so ⟨g, y⟩ ∼= D2rst. Moreover, since y
interchanges ES and ET it follows that ⟨g, y⟩ is regular on the edge set E. This completes
the proof of part (a). Part (b) now follows immediately from Proposition 3.4(b). ✷
4.3 Complete bipartite graphs
Next we study complete bipartite graphs, their λ-extenders, and constructions which com-
bine complete bipartite graphs with circulants. First we establish the properties of these
graphs occurring in Tables 1 and 3.
Definition 4.8. Let s, t be positive integers such that st ≥ 2 and gcd(s, t) = 1, and let
S = Zs, T = Zt. Then Γ = Ks,t = (V,E) with V = S ∪ T and
E = {[i, ei,j , j] | i ∈ S, j ∈ T}.
Define the map g : V ∪ E → V ∪ E as follows, for i ∈ S, j ∈ T :
g : i→ i+ 1 (mod s), j → j + 1 (mod t), ei,j → ei+1,j+1.
Lemma 4.9. With the notation of Definition 4.8 so in particular gcd(s, t) = 1, the map g
is an automorphism of Γ = Ks,t, and ⟨g⟩ ∼= Zst is cyclic, bi-transitive on V with orbits
S, T , and regular on E, and Γ is as in line 2 of Table 1. Further, the graph K
(2)
s,t admits a
cyclic subgroup of automorphisms that is bi-transitive on vertices, and bi-regular on edges,
as in line 2 of Table 3.
Proof. By definition both g|V and g|E are bijections, and g preserves incidence, so
g ∈ AutΓ. Also, since gcd(s, t) = 1, |g| = st, and it follows that the cyclic group
⟨g⟩ is bi-transitive on V with orbits S, T , and regular on E. The fact that AutK(2)s,t has a
cyclic subgroup bi-transitive on vertices, and bi-regular on edges follows on considering
K
(2)
s,t as an edge-disjoint union of two copies of Ks,t on the same vertex set. ✷
The next construction starts with a complete bipartite graph and adds a second set of
edges incident with the vertices of one bipart which form a perfect matching or an edge-
disjoint union of cycles (with multiple edges). These are the graphs in lines 4 and 6 of
Table 3.
Definition 4.10. Let r, n, t be positive integers such that rt ≥ 1, n ≥ 2, and
gcd(nr, t) = 1, and let V1 = Zrn, V2 = Zt. Define a graph Γ = CK(r, n, t) = (V,E)
J. Chen et al.: The graphs with a symmetrical Euler cycle 17
with vertex-set V = V1 ∪ V2, and edge-set E = E0 ∪ E1 where
E0 = {[k, ek,j , j] | k ∈ V1, j ∈ V2}, and
E1 = {[k, ejk, k + r] | k ∈ V1, j ∈ V2}.
Define the map g : V ∪ E → V ∪ E as follows, for k ∈ V1, j ∈ V2:
g : k → k + 1, j → j + 1
ek,j → ek+1,j+1, ejk → e
j+1
k+1
Lemma 4.11. With the notation of Definition 4.10 so in particular gcd(rn, t) = 1, the
graph Γ = CK(r, n, t) is connected and the map g is an automorphism of Γ, and ⟨g⟩ ∼=
Zrnt is cyclic, bi-transitive on V with orbits V1, V2, and bi-regular onE with orbitsE0, E1.
Further Γ = [E0] + [E1], where [E0] = Krn,t and, if n = 2 then [E1] = rK
(2t)
2 as in line
4 of Table 3, while if n ≥ 3 then [E1] = rC(t)n as in line 6 of Table 3.
Proof. It is clear that Γ = [E0] + [E1] is connected and [E0] = Krn,t. Further, if n = 2
then for each k ∈ V1, j ∈ V2, both ejk and e
j
k+r are incident with k and k + r, and hence
[E1] = rK
(2t)
2 . On the other hand if n ≥ 3 then [E1] = rC
(t)
n , so the last assertions hold.
By definition, both g|V and g|E are bijections, and g preserves incidence, so g ∈ AutΓ.
Consider the action of g onE. First let e = e00 ∈ E1, and suppose thatm is the least positive
integer such that eg
m
= e. Now, by definition of g, eg
m
= [m, emm,m], where the first entry
m ∈ Zrn and the last entry m ∈ Zt. Thus, since eg
m
= e the integer m is divisible by rn
and t, and hence by rnt since gcd(rn, t) = 1. On the other hand grnt fixes e and hence
m = rnt and as |E1| = rnt we conclude that ⟨g⟩ acts regularly on E1. A similar argument
shows that ⟨g⟩ acts regularly on E0, and so |g| = rnt and ⟨g⟩ is bi-transitive on V and
bi-regular on E. ✷
In a more general but similar construction, we start with an edge-disjoint union of
complete bipartite graphs uKsr,t, and add a second set of edges incident with the vertices
of the bipart of size usr, which form either a perfect matching of this bipart, or an edge-
disjoint union of cycles. In this case no edge of the second edge-set is incident with two
vertices of the same component Ksr,t. These are the graphs in lines 5 and 7 of Table 3, and
are depicted in Figure 2.
Definition 4.12. Let r, s, t, u be positive integers such that u ≥ 2, and gcd(r, u) =
gcd(sr, t) = 1, and let V1 = Zsru, V2 = Zut. Define a graph Γ = CK(2)(r, s, t, u) =
(V,E) with vertex-set V = V1 ∪ V2, and edge-set E = E0 ∪ E1 where
E0 = {[i+ uk, ei,k,j , i+ uj] | 0 ≤ i ≤ u− 1, 0 ≤ k ≤ sr − 1, 0 ≤ j ≤ t− 1}, and
E1 = {[ℓ+ rk, ejℓ,k, ℓ+ r(k + 1)] | 0 ≤ ℓ ≤ r − 1, 0 ≤ k ≤ su− 1, 0 ≤ j ≤ t− 1}.
Note that for edges in E0, we read the first entry as an element of Zsru and the third entry
as an element of Zut, while for edges in E1, the first and third entries are elements of Zsru.
Define the map g : V ∪ E → V ∪ E as follows, for k ∈ V1, j ∈ V2:
g : k → k + 1, and j → j + 1,
ei,k,j → ei+1,k,j+1,
ejℓ,k → e
j+1
ℓ+1,k, if 0 ≤ ℓ ≤ r − 2, and e
j
r−1,k → e
j+1
0,k+1
18 Art Discrete Appl. Math. 5 (2022) #P3.03
where in the edge-image ei+1,k,j+1 we read i + 1 modulo u and j + 1 modulo t, and for
the edge-image ej+1ℓ+1,k we read ℓ+ 1 modulo r and j + 1 modulo t.
Lemma 4.13. With the notation of Definition 4.12, the graph Γ = CK(2)(r, s, t, u) is
connected and the map g is an automorphism of Γ, and ⟨g⟩ ∼= Zsrut is cyclic, bi-transitive
on V with orbits V1, V2, and bi-regular on E with orbits E0, E1. Further Γ = [E0] + [E1],
where [E0] = uKsr,t and, if (s, u) = (1, 2) then [E1] = rK
(2t)
2 as in line 5 of Table 3,
and otherwise su ≥ 3 and [E1] = rC(t)su as in line 7 of Table 3.
Proof. By definition Γ = [E0] + [E1] and [E0] = uKsr,t. Further, if (s, u) = (1, 2), then
for each ℓ, j, both ejℓ,0 and e
j
ℓ,1 are incident with ℓ and ℓ+r in V1 and hence [E1] = rK
(2t)
2 .
On the other hand if (s, u) ̸= (1, 2), then su ≥ 3 and [E1] = rC(t)su , so the last assertions
hold.
By definition, both g|V and g|E are bijections, and g preserves incidence, so g ∈ AutΓ.
Consider the action of g on E. First let e = e0,0,0 ∈ E0, and suppose that m is the least
positive integer such that eg
m
= e. By definition of g, eg
m
= [m, em,0,m,m], where the
first entry m ∈ Zsru and the last entry m ∈ Zut. Thus, since eg
m
= e the integer m is
divisible by sru and ut, and hence by lcm{sru, ut} = srut since gcd(sr, t) = 1. On
the other hand gsrut fixes e and hence m = srut and as |E0| = srut we conclude that
⟨g⟩ acts regularly on E0. Now let e = [0, e00,0, 0] ∈ E1, and suppose that eg
m
= e with
m ≥ 1 minimal. By definition of g, egm = [m, emm,k,m], for some k, where the first entry
m ∈ Zsru and the last entry m ∈ Zut, and as before we deduce that m = srut = |E1| and
that ⟨g⟩ acts regularly on E1. Thus |g| = srut and ⟨g⟩ is bi-transitive on V and bi-regular
on E. ✷
Remark 4.14. We give some details about Figure 2, for the graph Γ = CK(2)(r, s, t, u) =
rC
(t)
su + uKsr,t with su ≥ 3, from Definition 4.12. The subset V1 of vertices admits
two ⟨g⟩-invariant partitions: first {B1, . . . , Br} where the Bi are the components of the
edge-induced subgraph [E1] = rC
(t)
su , and second {C1, . . . , Cu} where the Ci are the
intersections with V1 of the components of [E0] = uKsr,t. The vertex-subset V2 admits
the ⟨g⟩-invariant partition {D1, . . . , Du}, where the Di are the intersections with V2 of the
components of [E0] = uKsr,t.
Each of the final family of graphs admitting a cyclic edge-bi-regular action involves
three vertex-orbits. They are the graphs in the last line of Table 3, and Figure 3 gives a
broad description of their structure, see Remark 4.17.
Definition 4.15. Let r, r′, s, t, t′, u be positive integers such that
gcd(r, r′) = gcd(t, t′) = gcd(sr, ut) = gcd(sr′, ut′) = 1,
and let V1 = Zrut′ , V2 = Zsrr′ and V3 = Zr′ut. Define a graph Γ = KK(r, r
′, s, t, t′, u) =
(V,E) with vertex-set V = V1 ∪ V2 ∪ V3, and edge-set E = E0 ∪ E1 where
E0 = {[i+ rℓ, eji,ℓ,k, i+ rk] | 0 ≤ i ≤ r − 1, 0 ≤ ℓ ≤ ut′ − 1, 0 ≤ k ≤ sr′ − 1,
0 ≤ j ≤ t− 1},
E1 = {[i′ + r′ℓ′, ej
′
i′,ℓ′,k′ , i
′ + r′k′] | 0 ≤ i′ ≤ r′ − 1, 0 ≤ ℓ′ ≤ ut− 1, 0 ≤ k′ ≤ sr − 1,
0 ≤ j′ ≤ t′ − 1}.
J. Chen et al.: The graphs with a symmetrical Euler cycle 19
uKsr,t
rC
(t)
su
...
...
...
B1B2· · ·Br
C1
C2
...
Cu
D1
D2
...
Du
. . .
. . .
. . .
. . .
. . .
. . .
Figure 2: Γ = rC
(t)
su + uKsr,t, where [Bi] = C
(t)
su and [Cj ∪Dj ] = Ksr,t.
Note that the last entry in each edge lies in V2 = Zsrr′ , while the first edge-entries lie in
V1 = Zrut′ for edges in E0, and in V3 = Zr′ut for edges in E1. For c = 1, 2, 3, each
vc ∈ Vc can be written uniquely as follows:
v1 = i+ rℓ with 0 ≤ i ≤ r − 1, 0 ≤ ℓ ≤ ut′ − 1
v2 = i+ rk with 0 ≤ i ≤ r − 1, 0 ≤ k ≤ sr′ − 1
= i′ + r′k′ with 0 ≤ i′ ≤ r′ − 1, 0 ≤ k′ ≤ sr − 1
v3 = i
′ + r′ℓ′ with 0 ≤ i′ ≤ r′ − 1, 0 ≤ ℓ′ ≤ ut− 1
We will define a map g : V ∪E → V ∪E so that the vertex action is given by vc → vc +1
where we evaluate the vertex image modulo rut′, srr′, r′ut for c = 1, 2, 3 respectively. To
define the g-action on edges, set x := rr′sut′ and x′ := rr′sut, so that |E0| = xt and
|E1| = x′t′. Then for eji,ℓ,k ∈ E0 with i, ℓ, k, j in the ranges above, we have
1 ≤ (i+ rℓ+ 1)(k + 1)(j + 1) ≤ (rut′)(sr′)t = xt = |E0|,
and similarly, for ej
′
i′,ℓ′,k′ ∈ E1, we have
1 ≤ (i′ + r′ℓ′ + 1)(k′ + 1)(j′ + 1) ≤ (r′ut)(sr)t′ = x′t′ = |E1|.
Then, writing (i+rℓ+1)(k+1)(j+1) = ax+b and (i′+r′ℓ′+1)(k′+1)(j′+1) = a′x′+b′,
with 0 ≤ a ≤ t− 1, 1 ≤ b ≤ x, 0 ≤ a′ ≤ t′− 1 and 1 ≤ b′ ≤ x′, we define the g-action by
g : eji,ℓ,k → eJI,L,K and e
j′
i′,ℓ′,k′ → eJ
′
I′,L′,K′
where I, J,K,L and I ′, J ′,K ′, L′ are as in Table 8.
20 Art Discrete Appl. Math. 5 (2022) #P3.03
a i J I L K
≤ x− 2 ≤ r − 2 j i+ 1 ℓ k
≤ x− 2 r − 1 j i+ 1 ℓ+ 1 k + 1
x− 1 ≤ r − 2 j + 1 i+ 1 ℓ k
x− 1 r − 1 j + 1 i+ 1 ℓ+ 1 k + 1
a′ i′ J ′ I ′ L′ K ′
≤ x′ − 2 ≤ r′ − 2 j′ i′ + 1 ℓ′ k′
≤ x′ − 2 r′ − 1 j′ i′ + 1 ℓ′ + 1 k′ + 1
x′ − 1 ≤ r′ − 2 j′ + 1 i′ + 1 ℓ′ k′
x′ − 1 r′ − 1 j′ + 1 i′ + 1 ℓ′ + 1 k′ + 1
Table 8: Edge action for g in Definition 4.15.
Lemma 4.16. With the notation of Definition 4.15, the graph Γ = KK(r, r′, s, t, t′, u) is
connected, and Γ = [E0] + [E1], where [E0] = rK
(t)
sr′,ut′ and [E1] = r
′
K
(t′)
sr,ut as in the
last line of Table 3. Further the map g ∈ AutΓ, and ⟨g⟩ is cyclic of order srr′utt′, with
three vertex-orbits V1, V2, V3, and is bi-regular on E with orbits E0, E1.
Proof. By definition Γ = [E0] + [E1] and [E0] = rK
(t)
sr′,ut′ and [E1] = r
′
K
(t′)
sr,ut, so the
first assertion holds. Further, it follows from the definition of g that both g|V and g|E are
bijections, and g preserves incidence, so g ∈ AutΓ. Consider the action of g on E. First
let e = e00,0,0 ∈ E0, and suppose that m is the least positive integer such that eg
m
= e. It
follows from Table 8, that after xt repeated applications of g, we have eg
xt
= etxt,y,y where
y = sr′ut′, and as y is divisible by both ut′ and sr′, this edge-image is equal to e. Thus gxt
fixes e and so m divides xt. Suppose for a contradiction that 0 < m < xt. Then m is of
the form m = cx+ d where 0 ≤ c ≤ t− 1 and 0 ≤ d ≤ x− 1, and e = egm = ecm,v,w, for
some v, w. This implies that c is a multiple of t and hence c = 0, so 0 < m < x. Now, with
y = sr′ut′ so x = ry, we writem = c′r+d′ with 0 ≤ c′ ≤ y−1 and 0 ≤ d′ ≤ r−1. Then
e = eg
m
= e0d′,c′,c′ , and this implies that d
′ = 0 and that c′ is divisible by both ut′ and sr′.
Since gcd(sr′, ut′) = 1, we deduce that c′ is divisible by lcm{sr′, ut′} = sr′ut′ = y, and
hence c′ = 0 so m = 0, a contradiction. Thus m = xt and ⟨g⟩ acts regularly on E0.
An analogous argument for the edge e00,0,0 ∈ E1 shows that ⟨g⟩ also acts regularly on
E1, and as |E0| = |E1| = xt = x′t′, the subgroup ⟨gxt⟩ fixes V ∪E pointwise, and hence
is trivial. Thus ⟨g⟩ is edge-bi-regular, and the proof is complete. ✷
Remark 4.17. Figure 3 gives a description of the structure of the graph
Γ = KK(r, r′, s, t, t′, u) = rK
(t)
sr′,ut′ + r
′
K
(t′)
sr,ut
from Definition 4.15. The vertex-subset V2 admits two ⟨g⟩-invariant partitions: first the
partition {B1, . . . , Br} where the Bi are the intersections with V2 of the components of
[E0] = rK
(t)
sr′,ut′ , and second {C1, . . . , Cr′} where the Ci are the intersections with V2
of the components of [E1] = r
′
K
(t′)
sr,ut. The vertex-subsets V1, V3 admit the ⟨g⟩-invariant
partitions {U1, . . . , Ur} and {W1, . . . ,Wr′}, where the Ui and Wi are the intersections
with V1, V3 of the components of [E0], [E1], respectively.
J. Chen et al.: The graphs with a symmetrical Euler cycle 21
. . .
...
. . .
...
. . .
...
W1
W2
...
W
r′
C1
C2
...
C
r′
B1B2. . .Br
U1U2. . .Ur
Figure 3: Γ = rK
(t)
sr′,ut′ + r
′
K
(t′)
sr,ut, where [Bi ∪Ui] = K(t)sr′,ut′ and [Cj ∪Wj ] = K
(t′)
sr,ut.
5 Coset graphs and cyclic edge-regular actions
In [8, Section 1.1], a new coset graph construction was given for arc-transitive, not neces-
sarily simple, graphs, which proved helpful for analysing such graphs. Here we develop a
similar theory of coset graph representations for edge-transitive bipartite graphs with two
vertex-orbits. It extends the theory in [1, Section 2] and [3, Section 3.2] for simple graphs
of this kind. We use this theory in Subsection 5.1 to prove Theorem 1.1 in the case of
edge-regular actions.
Construction 5.1. Let G be a group with proper subgroups L,R, J , such that L ̸= R,
J ≤ L∩R and J is core-free in G, and let λ := |L∩R : J |. Define an incidence structure
BiCos(G,L,R, J) = (V,E, I), called a bi-coset graph, by setting
V = [G : L] ∪ [G : R] = {Lx | x ∈ G} ∪ {Rx | x ∈ G}
E = [G : J ] = {Jx | x ∈ G}
I = {(Lx, Jy) | Lx ∩ Jy ̸= ∅} ∪ {(Rx, Jy) | Rx ∩ Jy ̸= ∅} ⊆ V × E.
Also set s := |L : L ∩R| and t := |R : L ∩R|.
We prove that BiCos(G,L,R, J) is a G-edge-transitive bipartite graph and obtain var-
ious of its other properties.
22 Art Discrete Appl. Math. 5 (2022) #P3.03
Proposition 5.2. Let G,L,R, J, λ, s, t be as in Construction 5.1, and denote the graph
BiCos(G,L,R, J) by Γ.
(a) For x, y, z ∈ G, Jy = [Lx, Jy,Rz] is an edge if and only if Jy = [Ly, Jy,Ry], (that
is, Lx = Ly and Rz = Ry). Thus the edges of Γ are precisely Jy = [Ly, Jy,Ry],
for y ∈ G.
(b) Then Γ is a bipartite graph which is bi-regular with valencies s, t, and constant
edge-multiplicity λ.
(c) The group G, acting by right-multiplication, is an edge-transitive group of automor-
phisms of Γ with vertex-orbits [G : L] and [G : R], and L,R, J are the stabilisers of
the vertices L,R and edge J , respectively.
(d) The base graph of Γ is Γ0 := BiCos(G,L,R,L ∩ R), and Γ = Γ(λ)0 . In particular
Γ is simple if and only if L ∩R = J .
(e) The graph Γ is connected if and only if G = ⟨L,R⟩; and Γ ∼= K(λ)s,t if and only if
G = LR.
Proof. (a) Let x, y, z ∈ G. Then Jy = [Lx, Jy,Rz] is an edge if and only if (Lx, Jy) ∈ I
and (Rz, Jy) ∈ I, that is, Lx ∩ Jy ̸= ∅ and Rz ∩ Jy ̸= ∅. This in turn is equivalent to
Lxy−1 ∩ J ̸= ∅ and Rzy−1 ∩ J ̸= ∅, and since J ≤ L∩R, this is equivalent to xy−1 ∈ L
and zy−1 ∈ R, that is to say Lx = Ly and Rz = Ry so Jy = [Ly, Jy,Ry].
(b) and (c) It is clear from the definition of Γ that [G : L] and [G : R] form the parts of
a bipartition of Γ and each edge is incident with one vertex from each of these sets. Also,
the group G, acting by right-multiplication, preserves the incidence relation I so induces
a group of automorphisms of Γ. The biparts [G : L] and [G : R] are the two G-vertex-
orbits, and by part (a), G is transitive on edges. The subgroups L,R, J are the stabilisers of
vertices L,R and edge J respectively, and since J is core-free in G, G acts faithfully on Γ.
These transitivity properties imply that Γ is biregular with each vertex of [G : L] incident
with |L : L ∩ R| = s vertices and each vertex of [G : R] incident with |R : L ∩ R| = t
vertices, and that Γ has constant edge-multiplicity λ = |L ∩ R : J | (the number of edges
incident with L and R).
(d) From the discussion in the previous paragraph it is clear that the base graph Γ0 of Γ
has edge set identified with [G : L∩R] and is precisely the graph BiCos(G,L,R,L∩R),
and that Γ is a λ-extender of Γ0.
(e) Now Γ is connected if and only if Γ0 is connected, and the latter is true if and only if
G = ⟨L,R⟩, by [3, Lemma 3.7(1)]. Assume now that G = LR. Then as R is the stabiliser
of the vertex R ∈ [G : R] it follows that L is transitive on [G : R] and similarly R is
transitive on [G : L]. Thus L, R is adjacent to each vertex of [G : R], [G : L], respectively
and it follows that Γ0 ∼= Ks,t and Γ ∼= K(λ)s,t . Conversely suppose that Γ ∼= K(λ)s,t . Then
Γ0 ∼= Ks,t, and asG is edge-transitive, the vertex-stabiliserL is transitive on the set [G : R]
of vertices adjacent to L. Then, since R is the stabiliser of R ∈ [G : R] this means that
G = RL = LR. ✷
Next we show that essentially all edge-transitive graphs with two vertex-orbits arise
from Construction 5.1.
J. Chen et al.: The graphs with a symmetrical Euler cycle 23
Γ |G| GV GE GA Conditions
C
(λ)
n nλ
√ √ × n ≥ 3
K
(λ)
s,t stλ ×
√ × gcd(s, t) = 1, st > 1
Table 9: Table for Theorem 5.4.
Proposition 5.3. Let Γ = (V,E, I) be a graph and G ≤ AutΓ such that G is transitive
on E and has two orbits V1, V2 in V , and Γ has no isolated vertices, and constant edge-
multuplicity λ. Then either
(a) Γ ∼= BiCos(G,L,R, J) where for some edge e = [α, e, β] of Γ, the subgroups
L,R, J are the stabilisers of α, β, e respectively, and |L ∩R : J | = λ; or
(b) Γ ∼= rK(λ)2 , where r = |V1| = |V2|.
Proof. If some edge e of Γ is incident with two distinct vertices of Vi, for some i, then
this is true for all edges since Γ is G-edge transitive and G leaves Vi invariant. This is
a contradiction since Γ has no isolated vertices. Hence each edge is incident with one
vertex from V1 and one from V2. Let e = [α, e, β] be an edge with α ∈ V1 and β ∈ V2,
and let L = Gα, R = Gβ and J = Ge. Then we may identify the sets V1, V2, E with
[G : L], [G : R], [G : J ], respectively, with G acting by right multiplication. Note that
Ge fixes the unique vertex of Vi with which it is incident, for each i, and so J = Ge ≤
Gα ∩ Gβ = L ∩ R. With this identification, the edge e = [L, J,R] and each edge can be
expressed as ey = [Ly, Jy,Ry] for some y ∈ G (since G is transitive on E). Thus Lx is
incident with Jz if and only if there exists y such that Lx = Ly and Jz = Jy, and hence
Lx ∩ Jz contains y so is non-empty. Conversely, if Lx ∩ Jz contains an element y, then
Lx = Ly and Jz = Jy. Similar statements hold for incidences between Jx and Rz. Thus,
provided L ̸= R, we see that, under these identifications we have Γ = BiCos(G,L,R, J)
and part (a) holds.
Finally suppose that L = R. Since G is transitive on E, L is transitive on the edges
incident with α, and since L = R fixes β this means that all of these edges are incident
with β. Thus the connected component of Γ induced on {α, β} is K(λ)2 , and we have
Γ ∼= rK(λ)2 , where r = |V1| = |V2|, as in (b). ✷
5.1 Graphs admitting cyclic edge-transitive groups
We use the theory in the previous subsection, to determine all graphs admitting a cyclic
subgroup of automorphisms that is regular on edges. First we consider connected graphs.
We assume that |V | ≥ 3, in the light of Proposition 2.2.
Theorem 5.4. Let Γ = (V,E, I) be a connected graph with |V | ≥ 3, let G ≤ AutΓ be a
cyclic subgroup acting regularly on E, and let Γ have edge-multiplicity λ. Then Γ and |G|
are as in one of the lines of Table 9, and the induced permutation groups GX on vertices
(X = V ), edges (X = E) and arcs (X = A) are transitive if the entry in the column
headed GX is
√
and intransitive if the entry is ×.
Proof. Since |V | ≥ 3 we have Γ ̸∼= K(λ)2 , and since Γ is connected and G-edge-transitive,
G has at most two orbits in V . Suppose first that G has two vertex-orbits, V1 and V2. Then
24 Art Discrete Appl. Math. 5 (2022) #P3.03
by Proposition 5.3, Γ = BiCos(G,L,R, J) with L = Gα, R = Gβ , J = Ge for some
edge e = [α, e, β]. It follows from Proposition 5.2(e) that G = ⟨L,R⟩, and since G is
cyclic this means that G = LR and hence that Γ = K
(λ)
s,t , where s = |L : L ∩ R| and
t = |R : L ∩ R|, and st > 1 since Γ ̸∼= K(λ)2 . Also L ∩ R acts trivially on V and the
edge-stabiliser J acts trivially on E (since G is abelian), and hence J = 1 by Lemma 2.1.
Thus |G| = |E| = stλ, and |L ∩ R| = λ. Now we have |L| = sλ and |R| = tλ
and so |G| = |LR| = lcm{sλ, tλ} = λ lcm{s, t}, and hence st = lcm{s, t}, that is,
gcd(s, t) = 1. Thus all the entries of line 2 of Table 9 hold.
We may now assume that G is transitive on V as well as on E. Suppose that G is
not transitive on arcs. Then the stabiliser of an edge e = [α, e, β] satisfies Ge = 1, and
|G| = |E|. Also, the stabiliserGα is normal inG sinceG is abelian, and soGα acts trivially
on V . This implies that Gα = Gβ and so Gα is transitive on the λ edges incident with α
and β. Since Ge = 1 and Ge ≤ Gα,β = Gα, we have |Gα| = λ, and n := |V | = |G : Gα|
so |G| = nλ, and note that n = |V | ≥ 3. Let Γ0 be the base graph of Γ so Γ = Γ(λ)0 . Then
G/Gα acts transitively on the vertices and edges of Γ0, and Γ0 has |E|/λ = n = |V | edges
and also n vertices. These properties imply that Γ0 has valency 2 and hence is a cycle of
length n. Hence Γ = C
(λ)
n and n ≥ 3, so all the entries of line 1 of Table 9 hold.
Suppose finally that G is transitive (hence regular) on the arcs of Γ. Then the stabiliser
Ge of an edge e = [α, e, β] contains an element g which interchanges α and β. Now
Ge ∩ Gα acts trivially on both V and E (since G is abelian and transitive on V and E),
and hence Ge ∩Gα = 1. Thus Ge = ⟨g⟩ ∼= Z2. Moreover, g ̸∈ Gα, but g normalises Gα
(since G is abelian) and G = ⟨Gα, g⟩ (since Γ is connected). Thus |V | = |G : Gα| = 2,
which is a contradiction. ✷
Now we use this classification to determine all graphs with no isolated vertices which
admit a cyclic subgroup of automorphisms regular on edges. Such graphs have constant
edge-multiplicity.
Corollary 5.5. Let Γ = (V,E, I) be a graph with no isolated vertices, and constant edge-
multiplicity λ, and let G ≤ AutΓ be a cyclic subgroup transitive on E. Then
(a) Γ = rΓ0 where Γ has r connected components, each isomorphic to Γ0, and
(b) for G0 the setwise stabiliser in G of a component Γ0, |G| = r|G0|, and either
Γ0 = K
(λ)
2 , or (Γ0, |G0|) is as in one of the lines of Table 9, and Γ0 is as in one of
the lines of Table 1.
Proof. The result follows from Theorem 5.4 if Γ is connected, so suppose that Γ has r ≥ 2
connected components Σ1, . . . ,Σr, and let Ei be the edge-set of Σi, for 1 ⩽ i ⩽ r. Since
G is edge-transitive it follows that Γ0 := Σ1 ∼= . . . ∼= Σr, so Γ = rΓ0, and {E1, . . . , Er}
forms a G-invariant partition of E. Then since G is cyclic, it induces a regular action on
{E1, . . . , Er}, and the unique subgroup G0 of index r in G is the setwise stabiliser of
Ei for each i. Also G0 is transitive on each Ei and G0 is cyclic, and hence the unique
subgroup J of G0 of index |Ei| is the stabiliser of each edge of each of the Ei. That is to
say, J is the kernel of the action of G on E. It follows from Lemma 2.1 that either J = 1,
or Γ0 = K
(λ)
2 . Thus we may assume that J = 1. In particular G0 acts edge-regularly
and faithfully as a cyclic group of automorphisms of Σi, for each i. Then by Theorem 5.4,
(Γ0, |G0|) is as in one of the lines of Table 9, and so the base graph Γ0 is as in one of the
lines of Table 1. ✷
J. Chen et al.: The graphs with a symmetrical Euler cycle 25
6 Proof of Theorem 1.1
In this section we complete the proof of Theorem 1.1. Let Γ = (V,E) be a connected
graph with |V | ≥ 3, and assume that G ⩽ AutΓ is a cyclic subgroup which is regular or
bi-regular on the edge set E. In particular Γ has no isolated vertices. If G is regular on E
then the possibilities for (Γ, G) are determined in Theorem 5.4 and (see Corollary 5.5(b))
the simple base graph Γ0 of Γ is as in one of the lines of Table 1. So we may assume that
G is bi-regular on E, with two edge orbits E0, E1 of equal size. Let
Πi = [Ei] = (V (Ei), Ei), where i = 0 or 1
be the induced subgraphs. Then Π0,Π1 are edge disjoint graphs, and by our convention,
Γ = Π0 + Π1. By the definition of an induced subgraph, Πi has no isolated vertices, and
by the definition of bi-regular, G induces an edge-regular action on Πi for each i. Hence
by Corollary 5.5, the following holds for each of i = 0 or 1. The graph Πi = riΣi, where
Πi has ri connected components, each isomorphic to Σi, and for Hi the setwise stabiliser
in G of a component Σi, |G| = ri|Hi|, and one of
(1) Σi = C
(λi)
ni , and Hi = Zniλi is transitive on V (Ei), for some ni ⩾ 3, λi ≥ 1; or
(2) Σi = K
(λi)
si,ti , and Hi = Zsitiλi is bi-transitive on V (Ei), with siti > 1 and
gcd(si, ti) = 1, λi ≥ 1; or
(3) Σi = K
(λi)
2 , and by Proposition 2.2, one of lines 1–3 of Table 5 holds for Hi.
It follows from the above discussion (and Corollary 5.5) that G has at most two orbits
in V (Ei) for each i, and since Γ is connected, the sets V (E0) and V (E1) are not disjoint.
Thus V (E0) and V (E1) share at least one G-orbit, and hence G
V has at most three orbits.
We show that Γ and |G| are as in one of the cases of Theorem 1.1 in the following three
subsections, according to the number of G-orbits in V .
6.1 The case where GV is transitive
Suppose in this subsection that GV is transitive, so V = V (Ei) for each i.
Assume first that Π0 = r0K
(λ0)
2 . Then, by Proposition 2.2, line 1 or 2 of Table 5 holds,
so either (i) |G| = 2|E0| = 2r0λ0 with λ0 odd, and the G-action on Π0 is arc-transitive, or
(ii) |G| = |E0| = r0λ0 with λ0 even andG is faithful onE0 and not arc-transitive on Π0. In
either case |G| is even, so G has a unique subgroup X of order 2, and the X-orbits in V are
the components of Π0. Suppose to start with that Π1 = r1K
(λ1)
2 . Then |V | = 2r0 = 2r1
so r0 = r1, and |Ei| = r0λ0 = r1λ1 (by definition, since G is biregular on E), so also
λ0 = λ1 = λ, say, and |G| = 2r1λ or r1λ according as λ is odd or even, respectively. It
follows that the X vertex-orbits are also the connected components of Π1, so a connected
component of Γ has size 2. Since Γ is connected this implies that |V | = 2, which is a
contradiction. Hence Π1 ̸= r1K(λ1)2 . This implies in particular that G acts faithfully on
E1 (by Lemma 2.1). Thus |G| = |E1| since G is abelian, and therefore, since |E0| = |E1|
as G is bi-regular on E, we have |G| = |E0| so (ii) above holds, that is, |G| = r0λ0, λ0 is
even, and G is faithful on E0. Also, since Π1 ̸= r1K(λ1)2 and GV is transitive, it follows
from Corollary 5.5 that Π1 = r1C
(λ1)
n1 for some n1 ≥ 3. Then |V | = 2r0 = r1n1, and
|Ei| = r0λ0 = r1n1λ1, so λ0 = 2λ1.
26 Art Discrete Appl. Math. 5 (2022) #P3.03
Suppose that X fixes a Π1-component Σ setwise. Then V (Σ) is a union of (the vertex-
sets of) some components of Π0, and since Γ is connected it follows that Π1 = Σ, so
V = V (Σ), r1 = 1 and n1 = 2r0 is even. Moreover the two vertices of each component
K
(λ0)
2 of Π0 form an antipodal pair of vertices of Π1 = C
(λ1)
n1 , and X interchanges this
vertex-pair and interchanges in pairs the λ0 = 2λ1 edges of Π0 incident with them. Thus,
setting λ := λ1 and n := n1/2, we have Γ = Γ
(λ)
0 with |G| = 2nλ and Γ0 the Cayley
graph Γ(2n, 1, n) = C2n + nK
(2)
2 of Corollary 4.5(a), and line 2 of Table 2 holds.
On the other hand, suppose that X interchanges the components of Π1 in pairs. Then
r1 is even and the two vertices of a component K
(λ0)
2 of Π0 lie in two distinct components
of Π1. Further, the union of these two distinct components of Π1 is also a union of n1
components of Π0 and hence forms a component of Γ. Since Γ is connected we conclude
that r1 = 2 and r0 = n1 = n, say. Again the groupX interchanges the two Π1-components
C
(λ)
n , where λ := λ1, and for each component K
(2λ)
2 of Π0, X interchanges the 2λ edges
in pairs. Since G is cyclic and transitive on the 2n = |V | vertices of Γ, and since the
index 2 subgroup of G stabilising a component of Π1 is transitive on the n vertices of that
component, it follows that n is odd, and hence |G| = 2nλ and Γ = Γ(λ)0 with Γ0 the Cayley
graph Γ(2n, 2, n) = 2Cn + nK
(2)
2 of Corollary 4.5(b), and line 3 of Table 2 holds.
Thus we may assume from now on that no component of either Π0 or Π1 is K
(λ)
2 for
any λ. Then by Corollary 5.5, since GV is transitive, each Πi = riC
(λi)
ni for some ni ≥ 3.
Thus |V | = r1n1 = r0n0 and |E1| = |E0| = r1n1λ1 = r0n0λ0, and hence in particular
λ1 = λ0. This means that Γ = Γ
(λ0)
0 and Πi = Φ
(λ0)
i , for graphs Φi with edge-multiplicity
1 admitting a cyclic vertex-transitive group H (induced by G) of order n := r1n1 = r0n0.
Thus H is bi-regular on the edge-set of Γ0 with orbits the edge-sets of Φ0 and Φ1. This
means that each Φi may be regarded as a simple Cayley graph for the cyclic group H . If as
simple graphs, the edges sets of Φ0 and Φ1 coincide then, since Γ is connected, r1 = r0 = 1
and n = n0 = n1, and setting λ := λ0, Γ = Circ(n, {1,−1})(2λ) = C(2λ)n , withG = Znλ
acting with two edge orbitsE0 andE1. Thus Γ = Γ
(λ)
0 ,with Γ0 = C
(2)
n and |G| = nλ, and
line 1 of Table 2 holds.
If this is not the case then the edge sets of Φ0 and Φ1 are disjoint (as they are G-
orbits), and hence Γ0 = Circ(n, S), where S = {a,−a, b,−b} ⊂ Zn \ {0} with Φ0 =
Circ(n, {a,−a}) and Φ1 = Circ(n, {b,−b}), for some a, b with |a| ≥ 3, |b| ≥ 3, a ̸= ±b,
(see Section 4.1). Since Γ, and hence also Γ0 is connected it follows (as in the proof of
Lemma 4.4) that gcd(n, a, b) = 1, and hence, setting λ := λ1 = λ0, Γ = Γ
(λ)
0 , where
Γ0 = Circ(n, {a,−a, b,−b}) and |G| = nλ as in line 4 of Table 2 (see Lemma 4.4).
This proves Theorem 1.1 for cyclic vertex-transitive, edge-bi-regular actions.
6.2 The case where GV has two orbits
In this subsection we assume that G has two orbits V1, V2 on vertices. Suppose first that
V = V (Ei) for each i. Then G acts faithfully on each Ei. Assume that Π0 = r0K
(λ0)
2 ,
so by Proposition 2.2, |V | = 2r0, |G| = |E0| = |E1| = r0λ0, and each of the two
G-orbits in V has size r0. We claim that also Π1 = r1K
(λ1)
2 . If this is not the case
then, by Corollary 5.5, Π1 = r1K
(λ1)
s,t with gcd(s, t) = 1 and st > 1. However this
means that the G-vertex-orbits have unequal sizes r1s and r1t, which is a contradiction.
Hence Π1 = r1K
(λ1)
2 , and so |V | = 2r1. Thus r0 = r1 = r, say, and |E1| = r1λ1
J. Chen et al.: The graphs with a symmetrical Euler cycle 27
so λ0 = λ1 = λ, say. If some Π0-component is also a Π1-component, then r = 1
since Γ is connected, and |V | = 2, which is a contradiction. Thus no Π0-component
is equal to a Π1-component. Hence Γ = Γ
(λ)
0 , and the graph Γ0 is connected (since Γ
is connected) with edge-multiplicity 1, and valency 2 (since each vertex is adjacent to
exactly one vertex by an edge from a Πi-component, for each i). Thus Γ0 = Cn, where
n = r0 + r1 = 2r, and the group G induces a cyclic subgroup H ≤ AutΓ0 of order
n/2 = r with two vertex-orbits and two edge-orbits. Thus, as in Lemma 4.2, Γ = C
(λ)
n ,
with n even, Π0 ∼= Π1 ∼= (n/2)K(λ)2 and |G| = nλ/2, and line 1 of Table 3 holds for Γ0.
Therefore, in the case where V = V (Ei) for each i, we may assume that neither of the
Πi has a component K
(λ)
2 for any λ. It follows from Corollary 5.5 that, for each i, Πi =
riK
(λi)
si,ti with gcd(si, ti) = 1 and siti > 1. Without loss of generality we may assume that
theG-vertex-orbits V1, V2 have sizes |V1| = r0s0 = r1s1 and |V2| = r0t0 = r1t1. Also we
have r0s0t0λ0 = |E0| = |E1| = r1s1t1λ1. Let Σ1 be a Π1-component. Suppose first that,
for some vertices ui ∈ Vi∩V (Σ1), for i = 1, 2, there exists an edge e0 = [u1, e0, u2] ∈ E0.
Since Σ1 = K
(λ1)
s1,t1 , we also have an edge e1 = [u1, e1, u2] ∈ E1, and by the transitivity of
G on E0 and E1, it follows that V (Σ1) is contained in the vertex-set of a Π0-component
Σ0, and the same argument gives V (Σ0) ⊆ V (Σ1), yielding equality V (Σ1) = V (Σ0).
This implies that s := s1 = s0, t := t1 = t0, and since Γ is connected, also r1 = r0 = 1
and hence λ1 = λ0 = λ, say. Thus Γ = Γ
(λ)
0 , with Γ0 = K
(2)
s,t and |G| = stλ, and line 2
of Table 3 holds for Γ0.
We may therefore assume that, for u ∈ V1 ∩ V (Σ1), the set Π1(u) of t1 vertices
adjacent to u in Π1 (that is to say, adjacent in Σ1) is disjoint from the set Π0(u) of t0
vertices adjacent to u in Π0. Since G is transitive on E1 and E0, it follows that Gu is
transitive on each of the disjoint sets Π1(u) and Π0(u). Moreover since G is cyclic, Gu is
normal in G and hence all of its orbits in V2 have the same size. Hence t1 = t0 = t, say,
and the number of Gu-orbits in V2 is r0 = |V2|/t0 = |V2|/t1 = r1 = r, say, and r ≥ 2.
An analogous argument with a vertex w ∈ V2 ∩ V (Σ1) yields that s0 = s1 = s, say, and
Gw has all orbits in V1 of length s. Thus λ0 = |E0|/rst = |E1|/rst = λ1 = λ, say, and
it follows that Π0 ∼= Π1 ∼= rK(λ)s,t . Since E0, E1 are disjoint, it follows that Γ = Γ(λ)0 ,
and each Πi = Φ
(λ)
i , with Γ0 = (V, F ) connected and Φi = (V, Fi)
∼= rKs,t such that
E = F (λ), each Ei = F
(λ)
i , and F is the disjoint union F0 ∪ F1. Also G induces a cyclic
bi-transitive, edge-bi-regular group H on Γ0. Note that the H-action on V is equivalent to
the G-action on V , and |H| = rst.
Let L = Hu and K = Hw so the K-orbits in V1 and the L-orbits in V2 are the vertex-
subsets, in V1, V2 respectively, of the components of the Πi, or equivalently the Φi. Since
H = ⟨h⟩ permutes these two families of r subsets cyclically, we may label the subsets as
Uℓ, with ℓ ∈ Z2r, such that V1 = ∪{U2k | 0 ≤ k ≤ r − 1} , V2 = ∪{U2k+1 | 0 ≤
k ≤ r − 1}, with each |U2k| = s, |U2k+1| = t, and U2k ∪ U2k+1 the vertex set of a
Φ1-component, and such that U
h
ℓ = Uℓ+2 for each ℓ ∈ Z2r. Now the vertex-set of the
Φ0-component containing U1 is U2u ∪ U1, for some 2u ∈ Z2r, and the H-action implies
that the Φ0-components have vertex-sets U2(k+u) ∪ U2k+1, for 0 ≤ k ≤ r − 1. It follows
that there are paths in Γ0 from vertices in U0 (with edges alternately in F1 and F0) to
vertices in U2uℓ for all ℓ, and to no other vertices in V1. Since Γ0 is connected, this implies
that gcd(r, u) = 1, and hence there exists v ∈ Zr such that uv ≡ 1 (mod r). Let φ :
V → V be any map which induces bijections U2k → U2kv and U2k+1 → U2kv+1 for each
28 Art Discrete Appl. Math. 5 (2022) #P3.03
k. Then φ permutes the Φ1-components among themselves, and maps the Φ0-component
with vertex-set U2(k+u) ∪ U2k+1 to a graph Ks,t with vertex set U2kv+2 ∪ U2kv+1. Thus
φ induces a graph isomorphism from Γ0 to the graph C2r[sK1, tK1] in Definition 4.6.
Therefore Γ = Γ
(λ)
0 , with Γ0
∼= C2r[sK1, tK1], |G| = rstλ, and line 3 of Table 3 holds
for Γ0.
This completes our analysis of the case where V = V (Ei) for each i, so we may
without loss of generality assume from now on that V (E1) = V1 and (since Γ is connected)
that V (E0) = V = V1 ∪ V2. In particular |V1| > 1 since E1 ̸= ∅. Then, by Corollary 5.5,
we have (i) Π1 = r1 Σ
(λ1)
1 with Σ1 = K2 or Cn and |V1| = 2r1 or nr1 respectively, and
(ii) Π0 = r0Σ
(λ0)
0 with Σ0 = K2 or Ks,t where gcd(s, t) = 1 and st > 1. For case (ii)
we remove the constraint st > 1 and consider the two possibilities for Σ0 together, and
further, we assume that each Π0-component has s ≥ 1 vertices in V1 and t ≥ 1 vertices in
V2, where gcd(s, t) = 1, so |V1| = r0s and |V2| = r0t.
Suppose first that some Π1-edge (that is, an edge in E1) is incident with two distinct
vertices from the same Π0-component, so in particular s > 1. Since G is transitive on E0
and acts as automorphisms of Π1, this holds for all edges in E1, and hence there are no
E1-edges between distinct Π0-components. Since Γ is connected this implies that Π0 is
connected, so r0 = 1, |V2| = t, and |V1| = s = 2r1 or nr1 according as Σ1 = K2 or Cn,
respectively. Consider first Σ1 = K2, so s = 2r1 and hence t is odd and gcd(t, r1) = 1.
Then |E1| = r1λ1 and |E0| = stλ0 = 2r1tλ0, so with λ := λ0 and r := r1 we have
λ1 = 2tλ. Thus Π0 ∼= K(λ)2r,t and Π1 = rK
(2tλ)
2 , and as in Definition 4.10, Γ = Γ
(λ)
0 with
Γ0 = rK
(2t)
2 +K2r,t, |G| = 2rtλ, and line 4 of Table 3 holds for Γ0.
Now consider Σ1 = Cn, so s = nr1 and gcd(t, nr1) = 1. Then |E1| = nr1λ1
and |E0| = nr1tλ0, so with λ := λ0 and r := r1 we have λ1 = tλ, Π0 ∼= K(λ)nr,t and
Π1 = rC
(tλ)
n , and as in Definition 4.10, Γ = Γ
(λ)
0 with Γ0 = rC
(t)
n +Knr,t, |G| = nrtλ,
and line 6 of Table 3 holds for Γ0.
Finally suppose that each edge of E1 is incident with vertices from two different com-
ponents of Π0, so r0 ≥ 2. Thus e ∈ E1 satisfies e = [α1, e, α2] with αi in component Σ0,i
of Π0 for i = 1, 2, and the subgroup H of index r0 in G fixes each of the Π0-components
setwise (since H is normal in G), and the H-orbits in V1 are the s-subsets of V1 lying
in the Π0-components. Thus each of the s vertices of Σ0,1 in V1 is joined by an edge
of E1 to a vertex in Σ0,2. Further since G permutes the Π0-components transitively and
cyclically, it follows that the component, K
(λ1)
2 or C
(λ1)
n , of Π1 containing e meets each
of the Π0-components. Consider first Σ1 = K2. Then there are exactly r0 = 2 compo-
nents of Π0, and r1 = s components of Π1. Thus |V1| = 2s and for the transitive cyclic
group Z2s induced on V1, the stabiliser Z2 of a Π1-component interchanges its two ver-
tices and hence interchanges the two Π0-components. This implies that s is odd. Also
sλ1 = |E1| = |E0| = 2stλ0, so setting λ := λ0 we have λ1 = 2tλ and so Γ = Γ(λ)0 , with
Γ0 = sK
(2t)
2 + 2Ks,t, |G| = 2stλ, and line 5 of Table 3 holds for Γ0.
Now consider Σ1 = Cn. Then Σ1 meets each of the Π0-components in a constant
number u of points, so we have n = ur0 and s = ur1, so gcd(ur1, t) = 1. Now the
stabiliser of a Π1-component is still transitive on the Π0-components, and conversely, and
this holds if and only if gcd(r0, r1) = 1. Also |E1| = nr1λ1 = ur0r1λ1 and |E0| =
r0stλ0 = r0ur1tλ0, so with λ := λ0 we have λ1 = tλ, and so Π1 = r1 C
(tλ)
ur0 ,Π0 =
r0K
(λ)
ur1,t, |G| = ur0r1tλ, and Γ = Γ
(λ)
0 with Γ0 = r1 C
(t)
ur0 + r0Kur1,t, ur0 ≥ 3, r0 ≥ 2,
J. Chen et al.: The graphs with a symmetrical Euler cycle 29
and gcd(r0, r1) = gcd(ur1, t) = 1. So line 7 of Table 3 holds for Γ0.
This completes the analysis of the case where there are two G-vertex-orbits.
6.3 The case where GV has three orbits
In this final subsection we assume that G has three orbits V1, V2, V3 on vertices with, say,
V (E0) = V1 ∪V2 and V (E1) = V2 ∪V3. Then G acts faithfully on each Ei, and by Corol-
lary 5.5, we have Π0 = r0K
(λ0)
s0,t0 and Π1 = r1K
(λ1)
s1,t1 , with gcd(s0, t0) = gcd(s1, t1) = 1,
and to simplify our analysis we assume s0t0 ≥ 1, s1t1 ≥ 1, (identifying K2 with K1,1).
Also we assume, for each i, that each component of Πi has si vertices in V2, so |V2| =
r0s0 = r1s1, and |G| = r0s0t0λ0 = r1s1t1λ1, which gives t0λ0 = t1λ1.
For each i, let Hi be the subgroup of G of index ri. Then the Hi-orbits in V2 are the
si-subsets of vertices in the Πi-components. Let d = gcd(r0, r1) and let H be the index
d subgroup of G. Then for each i, Hi ≤ H and hence the H-orbits in V2 are unions of
Hi-orbits, for each i. Let ∆ be one suchH-orbit and let δ ∈ ∆. Then all paths in Γ starting
from δ and ending in V2 (using edges from E0 or E1 or both) must end at a vertex of ∆.
Since Γ is connected, it follows that ∆ = V2, and hence d = 1, that is, gcd(r0, r1) = 1.
Therefore s := s0/r1 = s1/r0 is an integer (since r0s0 = r1s1); and each Π0-component
meets each Π1-component in exactly s vertices of V2.
Let λ := gcd(λ0, λ1) and set t = λ0/λ and t
′ = λ1/λ, so t0t = t1t
′ (since t0λ0 =
t1λ1). Since gcd(t, t
′) = 1, u := t0/t
′ = t1/t is an integer. Set r := r0, r
′ := r1. Then
Γ = Γ
(λ)
0 and each Πi = Φ
(λ)
i , where Φ0 = rK
(t)
sr′,ut′ and Φ1 = r
′
K
(t′)
sr,ut, and |G| =
rr′suλtt′; and we have gcd(r, r′) = gcd(t, t′) = 1, and gcd(sr, ut) = gcd(sr′, ut′) = 1.
Thus line 8 of Table 3 holds for Γ0.
This graph family covers some special cases: for example if, say, s = r = 1 then
Γ0 = K
(t)
r′,ut′ + r
′
K
(t′)
1,ut; and if in addition u = t = 1 then Γ0 = Kr′,t′ + r
′
K
(t′)
2 .
This completes the proof that all graphs Γ with at least three vertices, and no isolated
vertices, and admitting a cyclic edge-regular or edge-bi-regular group of automorphisms,
are of the form Γ = Γ
(λ)
0 with Γ0 listed in one of the Tables 1, 2, and 3. Conversely it
follows from Lemmas 4.2, 4.4, 4.7, 4.9, 4.11, 4.13, 4.16, and Corollary 4.5, that all the
graphs in these tables admit such groups.
7 Proof of Theorem 1.2
The aim of this section is to complete the proof of Theorem 1.2. So let Γ = (V,E) be a
graph which has a symmetrical Euler cycle
C = (e0, e1, . . . , eℓ−1)
where ℓ = |E|. Thus each ei = [αi−1, ei, αi] where we write αℓ = α0. In particular Γ is
connected, and there exists x ∈ AutΓ such that
x : ei → ei+2, for each i, reading subscripts modulo ℓ.
Then ⟨x⟩ is edge-regular if ℓ is odd, or edge-bi-regular if ℓ is even, and hence, by Theo-
rem 1.1, Γ is one of the graphs in Tables 1, 2, or 3. Our task is to decide which of these
graphs has a symmetrical Euler cycle, and for each of these graphs, to determine the largest
subgroup H(C) of the group D(C) in (1.3) induced by (AutΓ)[C]. Note that x induces the
30 Art Discrete Appl. Math. 5 (2022) #P3.03
element φ2 of (1.4) so H(C) contains φ2. Recall that, for each vertex α, the number of
edges of C incident with α must be even. First we consider some of the examples listed in
Theorem 1.2.
Lemma 7.1. (a) If Γ = C
(λ)
n for some n ≥ 3, λ ≥ 1, then Γ has a symmetrical Euler
cycle C and H(C) = ⟨φ, τ⟩.
(b) If Γ = K
(λ)
s,t for some λ ≥ 1 and st > 1 with gcd(s, t) = 1, then Γ has a symmetrical
Euler cycle C if and only if λ is even, and in this case H(C) = ⟨φ2, φτ⟩.
Proof. (a) Let Γ = C
(λ)
n . By Proposition 3.4, it is sufficient to assume that λ = 1, and in
this case the assertion follows from Lemma 4.2(a).
(b) Now let Γ = K
(λ)
s,t . By our comments above, for an Euler cycle to exist each
vertex must be incident with an even number of edges, so both sλ and tλ are even. Since
gcd(s, t) = 1, this implies that λ is even. Therefore, by Proposition 3.4, it is sufficient
to assume that λ = 2 and to prove that Γ = K
(2)
s,t has a symmetrical Euler cycle C with
H(C) as claimed. We use the notation from Definition 4.8 for the vertices and edges of
Ks,t and the map g, and the convention in (3.1) for edges of Γ. Thus the vertex set is
V = V1 ∪ V2 with V1 = Zs and V2 = Zt. Let e0 = e10,0 and e1 = e21,0, and for each i such
that 1 ≤ i ≤ st− 1, define
e2i := e
1
i,i = e
gi
0 and e2i+1 := e
2
i+1,i = e
gi
1 .
Note that e2i = [i, e2i, i] and e2i+1 = [i+1, e2i, i], where we read the first entries (elements
of V1) modulo s and the last entries (elements of V2) modulo t. Thus e2i, e2i+1 are both
incident with i ∈ V2, and similarly e2i−1, e2i are both incident with i ∈ V1. Further, since
gcd(s, t) = 1, for each edge e of Ks,t, the edge e
1 occurs as e2i for some i, and e
2 occurs
as e2i+1 for some i. Also, since |g| = st and ⟨g⟩ is regular on the edge-set of Kst, it
follows that
C := (e0, e1, . . . , e2st−1)
is an Euler cycle for Γ, that C is preserved by ⟨g⟩, and that g induces the map φ2 in D(C).
Thus C is a symmetrical Euler cycle. Finally, AutΓ contains the following involution y,
where y|V1 : i→ −i for all i ∈ V1 = Zs, and y|V2 : i→ −i for all i ∈ V2 = Zt; and
y : e1i,i → e1−i,−i, and y : e2i+1,i → e2−i,−i−1, for all i.
It is straightforward to check that y preserves C, namely y : e2i → e2j and y : e2i+1 →
e2j−1 where i + j = st; and y induces the map φτ of D(C) as in (1.4). Thus H(C)
contains ⟨φ2, φτ⟩, and since Γ is not vertex-transitive, equality holds. ✷
Lemma 7.1 deals with all the graphs which admit a cyclic edge-regular subgroup, see
Table 1. So we may assume that Γ admits no cyclic edge-regular subgroup. Hence the
length ℓ of C is even, and ⟨x⟩ is bi-regular on E, say with orbits E0, E1. This means that,
in the symmetrical Euler cycle C the edges occur alternately in E0 and E1. Replacing
C by a shift if necessary, we may assume then that E0 = {e2i | 0 ≤ i ≤ ℓ − 1} and
E1 = {e2i+1 | 0 ≤ i ≤ ℓ− 1}.
J. Chen et al.: The graphs with a symmetrical Euler cycle 31
Lemma 7.2. (a) The group ⟨x⟩ has at most two orbits in V .
(b) If ⟨x⟩ has two orbits in V , say V1 and V2, then each edge is incident with exactly one
vertex of V1 and one vertex of V2.
(c) Thus Γ is one of the graphs in lines 2 – 4 of Table 2 or line 3 of Table 3.
Proof. (a) Since E0 = {e2i | 0 ≤ i ≤ ℓ − 1} and E1 = {e2i+1 | 0 ≤ i ≤ ℓ − 1}, it
follows from the definition of C that every vertex is incident with at least one edge in E0
and at least one edge in E1. Then, since ⟨x⟩ is transitive on E0 and E1 we conclude that
⟨x⟩ has at most two orbits in V .
(b) Suppose ⟨x⟩ has two orbits V1 and V2 in V . Then as Γ is connected, at least one
of the edge-orbits, say E0, has edges incident with vertices from each of V1 and V2, so
e0 = [α, e0, β] with, say, α ∈ V1 and β ∈ V2. Thus e2i−1 is incident with α ∈ V1 and
e2i+1 is incident with β ∈ V2, and hence also the edges of E1 are incident with vertices
from both V1 and V2.
(c) The last line of Table 3 is not possible by part (a), graphs in line 1 of Table 2 or lines
1–2 of Table 3 are excluded because they admit cyclic edge-regular subgroups, and graphs
in lines 4–7 of Table 3 are not possible by part (b). ✷
The next two lemmas deal with the remaining graphs from Table 2.
Lemma 7.3. (a) If Γ = Γ
(λ)
0 with Γ0 = Γ(2r, 1, r) = C2r + rK
(2)
2 as in Corol-
lary 4.5(a) (line 2 of Table 2), where r ≥ 2 and λ ≥ 1, then Γ has a symmetrical
Euler cycle C if and only if r is even, and in this case H(C) = ⟨φ2, φτ⟩.
(b) If Γ = Γ
(λ)
0 with Γ0 = Γ(2r, 2, r) = 2Cr + rK
(2)
2 as in Corollary 4.5(b) (line 3 of
Table 2), where r is odd, r ≥ 3, and λ ≥ 1, then Γ has a symmetrical Euler cycle C
and H(C) = ⟨φ2, φτ⟩.
Proof. We use the notation from Definition 4.3. In both cases (a) and (b), Γ0 =
Circ(n, S) = (V,E) with S = {a,−a, r(2)} as in Definition 4.3 (with a = 1 or 2), and we
have V = Z2r and 4r edges ei,a = [i, ei,a, i + a] and ei,r = [i, ei,r, i + r], for i ∈ Z2r.
Also the map g : i → i + 1, ei,a → ei+1,a, ei,r → ei+1,r lies in AutΓ0, by Lemma 4.4,
and is bi-regular on E with ⟨g⟩-orbits Ea = {ei,a | i ∈ Z2r} and Er = {ei,r | i ∈ Z2r}.
We also use the notation from (3.1) for edges of Γ = (V,E(λ)) and from the proof of
Proposition 3.4(a) for the edge-bi-regular automorphism g(λ) of Γ corresponding to the
edge-bi-regular map g on Γ0. The ⟨g(λ)⟩-orbits in E(λ) are E(λ)a and E(λ)r .
(a) Consider first case (a), so a = 1. Then the vertex-action (AutΓ)V is contained in
(Aut[E1])
V ∼= D4r and contains gV of order 2r. Suppose that
C = (e0, e1, . . . , e4rλ−1) is an Euler cycle for Γ with element x inducing φ
2 ∈ H(C),
as above. Then ⟨x⟩ induces Z2r on V , and hence xV = (gi)V for some i such that
gcd(i, 2r) = 1. We show first that r is even. We may assume that the ‘even’ edges e2i
of C lie in E
(λ)
a and the ‘odd’ edges e
(λ)
2i+1 lie in E
(λ)
r . Moreover, replacing C by a cycle
in its sequence class, if necessary, we may assume further that e0 = e
1
0,a and e1 is e
u
1,r or
eur+1,r, for some u, and hence that e2 is e
v
r+1,1 or e
v
r,1, for some v, and e4rλ−1 is e
w
0,r or e
w
r,r,
for some w. Considering the vertices incident with these edges, it follows from ex0 = e2
that {0, 1}gi = {r+1, r+2} or {r, r+1}, and from ex4rλ−1 = e1 that {0, r}g
i
= {1, r+1}.
These conditions together imply that i = r+1, and then from gcd(i, 2r) = 1 we conclude
32 Art Discrete Appl. Math. 5 (2022) #P3.03
that r is even. To complete the proof of part (a) we exhibit a symmetrical Euler cycle C for
Γ such that H(C) = ⟨φ2, φτ⟩ (noting that H(C) ̸= D(C) since Γ is not edge-transitive).
By Proposition 3.4 we may assume that λ = 1. As noted above, in a symmetrical Euler
cycle C preserved by x ∈ AutΓ which induces the map φ2 in D(C), we may assume that
e0 = e0,1, that e1 = e1,r or er+1,r, and that x = g
r+1. So for our construction let us
choose e1 = e1,r, so that, for all i = 1, . . . , 2r − 1,
e2i := e
gi(r+1)
0,1 = ei(r+1),1 and e2i+1 := e
gi(r+1)
1,r = ei(r+1)+1,r.
Then e2i, e2i+1 are both incident with i(r + 1) + 1, and e2i+1, e2i+2 are both incident
with (i + 1)(r + 1). Hence C := (e0, e1, . . . , e4r−1) is an Euler cycle of Γ. The element
gr+1 preserves the cycle C and acts as gr+1 : ej → ej+2, that is gr+1 induces the map
φ2 of D(C). The cycle C is also preserved by the automorphism y ∈ AutΓ defined by
y : − i ↔ i + 1 on V and y : e−i,1 ↔ ei,1, e−i,r ↔ ei+1,r on E. This map induces
φτ ∈ D(C) as in (1.4), and hence H(C) contains ⟨φ2, φτ⟩, and equality holds since Γ is
not edge-transitive. An exactly similar argument works for the case where e1 = er+1,r.
This completes the proof of part (a).
(b) Now consider case (b), so a = 2 and r is odd. By Proposition 3.4 we may assume
that λ = 1. Thus, for the edge-bi-regular automorphism g in the first paragraph of this
proof, gcd(r + 2, |g|) = gcd(r + 2, 2r) = 1, and hence x := gr+2 generates ⟨g⟩ and
⟨x⟩ is edge-bi-regular on Γ. We construct a symmetrical Euler cycle C = (e0, . . . , e4r−1)
preserved by x and on which x induces φ2 in H(C). Let e0 := e0,r and e1 := er,2, and for
1 ≤ i ≤ 2r − 1, let
e2i := e
xi
0 = e
gi(r+2)
0,r = ei(r+2),r and e2i+1 := e
xi
1 = e
gi(r+2)
r,2 = ei(r+2)+r,2.
Then e2i, e2i+1 are both incident with i(r+2)+ r, and e2i+1, e2i+2 are both incident with
(i+1)(r+2). Hence C is an Euler cycle of Γ, and by definition x preserves C and induces
φ2. In addition AutΓ contains an involution y inducing φτ on C, namely y : i→ r − i on
V = Z2r, and y : ei → e4r−i. This is straightforward to check. Hence H(C) contains, and
so is equal to ⟨φ2, φτ⟩. ✷
Lemma 7.4. Suppose that Γ = Γ
(λ)
0 with Γ0 = Circ(n, S) = (V,E) with
S = {a,−a, b,−b} as in Definition 4.3 with V = Zn, |S| = 4 and gcd(n, a, b) = 1,
and that λ ≥ 1.
(a) If either gcd(n, a+b) = 1 or gcd(n, a−b) = 1, and then Γ has a symmetrical Euler
cycle, and H(C) = ⟨φ2, φτ⟩ for all such cycles C.
(b) Moreover, if AutΓ0 is not edge-transitive, then Γ has a symmetrical Euler cycle C if
and only if either gcd(n, a+ b) = 1 or gcd(n, a− b) = 1.
Remark 7.5. We comment on part (b) of Lemma 7.4. The proof depends on the fact that,
for a symmetrical Euler cycle C of Γ = Γ
(λ)
0 (with Γ0 = Circ(n, S) = (V,E) as in line 4
of Table 2), an automorphism x inducing φ2 ∈ H(C) acts on V as a power of the map gV
from Definition 4.3. The assumption that Γ0 is not edge-transitive is sufficient to deduce
this fact. However it is possible for Γ0 to be edge-transitive. For example if n = 5, a =
1, b = 2 then Γ0 ∼= K5. Edge-transitivity occurs more generally, for example, when n
is odd and there exists d ∈ Zn with d2 = −1, and b = ±da so S = {a, da, d2a, d3a}.
J. Chen et al.: The graphs with a symmetrical Euler cycle 33
In this case the ‘multiplicative’ automorphism y : i → di interchanges the two ⟨g⟩ edge-
orbits Ea and Eb defined in Definition 4.3. If Γ0 is a so-called normal Cayley graph, that
is, if ⟨g, y⟩ = Zn ⋊ Z4 is the full automorphism group AutΓ0, then the assertion in case
(b) can be proved by a slightly modified argument. However we do not know if there are
parameters n, a, b for which Γ0 is a non-normal Cayley graph (and hence is edge-transitive)
and the necessary and sufficient condition in part (b) does not hold.
Proof. Here Γ0 = Circ(n, S) = (V,E) with S = {a,−a, b,−b} as in Definition 4.3 with
|S| = 4, V = Zn, and 2n edges ei,a = [i, ei,a, i+ a] and ei,b = [i, ei,b, i+ b], for i ∈ Zn.
The map g : i → i + 1, ei,a → ei+1,a, ei,b → ei+1,b lies in AutΓ0, by Lemma 4.4, and is
bi-regular on E with ⟨g⟩-orbits Ea = {ei,a | i ∈ Zn} and Eb = {ei,b | i ∈ Zn}. We also
use the notation from (3.1) for edges of Γ = (V,E(λ)).
(a) Suppose first that gcd(a+ b, n) = 1 or gcd(a− b, n) = 1. We construct a symmet-
rical Euler cycle for Γ. By Proposition 3.4 it is sufficient to do this for λ = 1, that is, for
Γ = Γ0 = Circ(n, S). So let j = a± b and assume that gcd(j, n) = 1. First let j = a+ b,
let e0 = e0,a, e1 = ea,b and for 1 ≤ i ≤ n− 1, let
e2i := e
gij
0 = eij,a and e2i+1 := e
gij
1 = eij+a,b.
Then C = (e0, . . . , e2n−1) is a symmetrical Euler cycle and g
j acts as φ2 ∈ H(C). In
addition y ∈ Γ defined by y : i → a − i on V , and y : ei → e2n−i, induces φτ on C so
H(C) ≥ ⟨φ2, φτ⟩. Similarly, for j = a − b, let e0 = e0,a, e1 = ea−b,b = ej,b (with e1
incident with {a, a− b}) and for 1 ≤ i ≤ n− 1, let
e2i := e
gij
0 = eij,a and e2i+1 := e
gij
1 = ea−b+ij,b = e(i+1)j,b.
Then again C = (e0, . . . , e2n−1) is a symmetrical Euler cycle and g
j acts as φ2 ∈ H(C).
Again the element y ∈ Γ defined by y : i→ a−i on V and y : ei → e2n−i induces φτ onC
so H(C) ≥ ⟨φ2, φτ⟩. If, for some symmetrical Euler cycle C, H(C) is strictly larger than
⟨φ2, φτ⟩, then H(C) = D(C) and so some element x ∈ AutΓ induces φ on C, and hence
⟨x⟩ is a cyclic edge-regular subgroup of AutΓ. This however contradicts Theorem 1.1 as
Γ = Γ0 does not appear in Table 1. Thus part (a) is proved.
(b) Suppose now that AutΓ0 is not edge-transitive and that Γ has a symmetrical Euler
cycle C = (e0, . . . , e2nλ−1) and x ∈ AutΓ preserves C and acts by x : ei → ei+2 for each
i. Since Γ0 is not edge-transitive, AutΓ0 ≤ Aut[Ea] and hence the groups induced on V
by AutΓ and AutΓ0 are both isomorphic to D2n with index 2 subgroup ⟨gV ⟩ generated by
gV : i→ i+1, with g as in the first paragraph of the proof. Since ⟨x⟩ is cyclic of order nλ
it follows that xV has order n and hence xV = (gV )j for some j coprime to n. Relabelling
the elements of S if necessary, we may assume that e0 = e
u
0,a for some u. Then, the next
edge e1 lies in E
(λ)
b and hence is e
v
a,b or e
v
a−b,b for some v. Assume first that e1 = e
v
a,b.
Then e2 is incident with a+ b while, since e2 = e
x
0 , the pair of vertices incident with e2 are
{0, a}x = {0, a}gj = {j, a + j}. Hence j = a + b or j = b and e2 = ewa+b,a or ewb,a, for
some w, and e3 is incident with 2a+ b or b, respectively. Moreover since e3 = e
x
1 , the pair
of vertices incident with e3 are {a, a + b}x = {a, a + b}g
j
which is {2a + b, 2a + 2b} if
j = a+ b or {a+ b, a+ 2b} if j = b. However, if j = b then e3 should be incident with b,
and b ̸∈ {a+ b, a+2b} since a ̸= 0,−b. Hence j = a+ b so gcd(a+ b, n) = 1. A similar
argument in the case where e1 = e
v
a−b,b leads to the condition gcd(a − b, n) = 1. This
proves the asserted congruence conditions, and the converse follows from part (a). This
completes the proof. ✷
34 Art Discrete Appl. Math. 5 (2022) #P3.03
Finally we consider the remaining graphs from Table 3.
Lemma 7.6. If Γ = Γ
(λ)
0 with Γ0 = C2r[sK1, tK1] as in Definition 4.6 (line 3 of Table 3),
with r ≥ 2, st ≥ 2, gcd(s, t) = 1, and λ ≥ 1, then Γ has a symmetrical Euler cycle C and
H(C) = ⟨φ2, τ⟩.
Proof. We use the notation from Definition 4.6. Thus Γ0 = C2r[sK1, tK1] = (V,E),
and V = VS ∪VT and E = ES ∪ET , where S = Zs, T = Zt. Also we have g, y ∈ AutΓ0
such that ⟨g⟩ is edge-bi-regular with edge-orbits ES , ET and ⟨g, y⟩ ∼= D2rst is bi-transitive
on V with vertex-orbits VS , VT , and regular on E (see Lemma 4.7). By Proposition 3.4 it
is sufficient to construct a cycle C with the required properties in the case λ = 1. So we
assume now that λ = 1 and Γ = Γ0, and we construct C = (e0, . . . , e2rst−1).
From Definition 4.6 the edges in ES , ET are labelled e
2k
i,j and e
2k+1
j,i , respectively,
where 0 ≤ k ≤ r − 1, i ∈ S, j ∈ T . Each integer i such that 0 ≤ i ≤ 2rst − 1 can
be represented uniquely as i = ℓ+ 2rm, where 0 ≤ ℓ ≤ 2r − 1 and 0 ≤ m ≤ st− 1. For
such a representation we define ei = e
ℓ
m,m if ℓ ≤ 2r − 2 and ei = eℓm,m+1 if ℓ = 2r − 1.
If ℓ ≤ 2r − 2, then ei, ei+1 are both incident with the vertex (ℓ,m) where m ∈ T if ℓ is
even or m ∈ S if ℓ is odd. If ℓ = 2r − 1, then ei, ei+1 are both incident with the vertex
(0,m + 1) with m + 1 ∈ S. This proves that C is a cycle of length 2rst, and from its
definition C involves all edges and hence is an Euler cycle. Moreover C is preserved by g
and y, with g inducing φ2 ∈ H(C) and y inducing τ ∈ H(C). If H(C) is strictly larger
than ⟨φ2, τ⟩, then H(C) = D(C), which implies that Γ0 is vertex-transitive, and hence
that s = t, which is a contradiction. This completes the proof. ✷
ORCID iDs
Jiyong Chen https://orcid.org/0000-0002-8484-0300
Cai Heng Li https://orcid.org/0000-0001-8917-9323
Cheryl E. Praeger https://orcid.org/0000-0002-0881-7336
Shu Jiao Song https://orcid.org/0000-0002-2577-8818
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.04
https://doi.org/10.26493/2590-9770.1458.4b9
(Also available at http://adam-journal.eu)
Groups of automorphisms of Riemann and Klein
surfaces, our joint work with Marston Conder
Emilio Bujalance* , Francisco-Javier Cirre†
Departamento de Matemáticas Fundamentales, Facultad de Ciencias,
Universidad Nacional de Educación a Distancia, C/ Juan del Rosal, 16, Madrid, Spain
Received 1 September 2021, accepted 1 December 2021, published online 27 June 2022
Abstract
This work is a survey on the research that we have carried out together with Professor
Marston Conder on groups of automorphisms of Riemann and Klein surfaces over the last
twenty years.
Keywords: Groups of automorphisms, Riemann surfaces, Klein surfaces
Math. Subj. Class.: 30F, 14H
Introduction
The first author met Marston Conder at the St Andrews 1989 Conference on Groups. In the
first conversation we had, Marston pointed out that we lived in antipodal cities, Madrid and
Auckland, a detail that the authors had not realized but which did not stop him from visiting
our university. Our collaboration began ten years later. This collaboration has focused on
the study, from a combinatorial point of view, of the groups of automorphisms of Riemann
and Klein surfaces.
The paper is divided into five sections. In Section 1 we introduce the notation and the
results that are necessary in the rest of the work. In Section 2 we deal with the problem of
deciding whether a group G acting on genus g also acts as a full group on genus g. Another
problem on which many results have been obtained is the determination of the automor-
phism groups of the surfaces of a given genus. The amount of combinatorial data involved
increases rapidly with the genus. In Section 3 we show some results where the computa-
tional methods implemented in programs as MAGMA, in which Marston is an expert, turn
out to be very useful to determine automorphism groups of Klein surfaces. In Section 4 we
*Partially supported by PGC2018-096454-B-I-00
†Corresponding author. Partially supported by PGC2018-096454-B-I-00
E-mail addresses: eb@mat.uned.es (Emilio Bujalance), jcirre@mat.uned.es (Francisco-Javier Cirre)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.04
consider the relation between the automorphism groups of a Klein surface and that of its
Riemann double cover in some special cases. Finally, in Section 5 we study bounds of the
automorphism groups of pseudo-real Riemann surfaces.
1 Preliminaries
The analytic structure of a compact Riemann surface makes it orientable and unbordered.
Non-orientable surfaces and surfaces with boundary also admit a dianalytic structure,
which behaves in many aspects as the analytic structure of a classical Riemann surface,
see [2]. Endowed with this structure, these surfaces are known as Klein surfaces. Groups
of automorphisms of both types of surfaces (either Riemann or Klein) can be studied from
a combinatorial point of view due to the Uniformization Theorem. Indeed, every such com-
pact surface S of algebraic genus g > 1 can be represented as the orbit space H/Γ where H
is the hyperbolic plane and Γ is a discrete subgroup of isometries of the hyperbolic plane
without elliptic elements. Groups as Γ are called surface groups. In this situation, each
(finite) group G of automorphisms of S is described as the factor group Λ/Γ where Λ is a
non-euclidean crystallographic (NEC) group, that is, a discrete subgroup of isometries of
the hyperbolic plane such that the orbit space H/Λ is also compact. Observe that Λ may
contain orientation reversing isometries and elliptic elements. Thus, the action of a group
G on a surface H/Γ is described by an epimorphism θ : Λ → G with ker θ = Γ. These are
called surface-kernel epimorphisms.
The first presentations for NEC groups appeared in [43] and their structure was clarified
by the introduction of signatures in [33]. This is a collection of symbols and non-negative
integers of the form
σ = (γ;±; [m1, . . . ,mr]; {(n11, . . . , n1s1), . . . , (nk1, . . . , nksk)}).
The non-negative integer γ is called the orbit genus of σ. If the sign “+” appears, then we
write sign(σ) = “+”; otherwise sign(σ) = “−”. The integers m1, . . . ,mr are called the
proper periods of σ and the nij are called the link periods of the period cycle (ni1, . . . , nisi).
An empty set of proper periods, (i.e. r = 0), will be denoted by [−], an empty period cycle
(i.e., si = 0) by (−), and the fact that σ has no period cycles (i.e., k = 0) by {−}.
The signature of an NEC group provides a presentation by generators and relations.
We omit it here since it is not necessary for the purposes of this paper. The interested
reader may find it in [43] or [33]. We just point out that since a Fuchsian group contains
no orientation reversing elements, its signature has no period cycles and its sign is always
“+”. Hence we may drop such data and represent its signature simply by (γ;m1, . . . ,mr).
A symmetry on a Riemann surface S is an anticonformal involution τ : S → S. The
orbit space S/⟨τ⟩ can be endowed with a dianalytic structure, making it a Klein surface.
Actually, all Klein surfaces can be obtained in this way, that is, for every Klein surface X
there exist a Riemann surface S and a symmetry τ on it such that X = S/⟨τ⟩. In terms
of algebraic geometry, X is a real algebraic curve and S is its complexification. The pair
(S, τ) is called the double cover of X.
2 On extendability of group actions
An abstract finite group G is said to act on genus g > 1 if it is (isomorphic to) a group of
automorphisms of some compact surface of genus g. We say that G acts as a full group
on genus g if G is (isomorphic to) the full automorphism group of some compact surface
E. Bujalance and F. J. Cirre: Groups of automorphisms of Riemann and Klein surfaces, . . . 3
of genus g. An interesting problem in Riemann and Klein surface theory is to investigate
whether a group G acting on genus g also acts as a full group on genus g. A key point
in the solution are the concepts of maximal NEC group and maximal signature. An NEC
group Λ is said to be maximal if it is not properly contained in another NEC group. In such
a case, for every surface Fuchsian group Γ contained in Λ as a normal subgroup, the group
Λ/Γ acting on the Riemann surface H/Λ is precisely its full automorphism group.
However, if Λ is non-maximal then the action of Λ/Γ on H/Λ could possibly be ex-
tended to the action of a larger group. Indeed, if Λ′ is another NEC group containing Λ
with finite index then Λ′ could possibly contain Γ as a normal subgroup, in which case Λ′/Γ
would be a larger group acting on H/Λ. The determination of maximal and non-maximal
NEC groups is therefore a key point in this topic. This leads to the more cumbersome
concepts of maximal and non-maximal signatures. We avoid their definitions here, which
can be found, for instance, in [21, Chapter 5]. The maximality of a signature σ does not
imply the maximality of every NEC group having signature σ. However, if σ is a maximal
signature then there exists a maximal NEC group Λ with signature σ. The study of maximal
signatures was initiated by Greenberg [28] in the case of Fuchsian signatures. He obtained
a list of those which fail to be maximal. The list was completed by Singerman [42]. The
analogous list for NEC signatures was obtained by Bujalance [4] for the so called normal
pairs, and by Estévez and Izquierdo [24] for the non-normal pairs.
In this section we present the results we have obtained with Marston on the topic of
extendability of group actions. A natural distinction is made depending on the type of
surface: Riemann surfaces in Subsection 2.1 and Klein surfaces in Subsection 2.2. We
would like to point out the interest of other authors on this topic. It is worth mentioning
the paper [23] by Costa and Parlier, where the extendability of group actions on Riemann
surfaces is considered using a somewhat different approach.
2.1 Extendability of group actions on Riemann surfaces
The first case to study is when G is cyclic and the surface is unbordered and orientable,
that is, a Riemann surface. This case was solved in Marston’s first visit to our university
UNED in Madrid, which took place in 1999. The results were published in [14], the two
main ones being the following.
Theorem 2.1. Suppose that the cyclic group Cn acts as a group of automorphisms of a
compact Riemann surface S of genus g>1, corresponding to a surface-kernel epimorphism
from the Fuchsian group Γ with signature σ(Γ)=(γ;m1, . . . ,mr) onto Cn. Let z1, . . . , zr
be the images in G of the canonical elliptic generators x1 . . . , xr, of orders m1, . . . ,mr,
of Γ. Then in each of the following cases, the action of Cn on S can always be extended to
the action of a larger group, so that Cn is not the full group Aut(S) of all automorphisms
of S.
(i) σ(Γ) = (2;−);
(ii) σ(Γ) = (1; t, t), whenever t ≥ 2;
(iii) σ(Γ) = (0; t1, t1, t2, t2), whenever t1 + t2 ≥ 5;
(iv) σ(Γ) = (0; t, t, t), whenever t ≥ 4, provided that Cn has an automorphism of order
3 permuting z1, z2, z3 in a 3-cycle;
(v) σ(Γ) = (0; t1, t1, t2), whenever t1 ≥ 3 and t1+t2 ≥ 7, provided that either z1 = z2
or Cn has an automorphism of order 2 interchanging z1 and z2;
4 Art Discrete Appl. Math. 5 (2022) #P3.04
(vi) σ(Γ) = (0; 3, 4, 12).
Theorem 2.2. Suppose that the cyclic group Cn acts as a group of automorphisms of a
compact Riemann surface S of genus g > 1, corresponding to a surface-kernel epimor-
phism onto Cn from a Fuchsian triangle group Γ with signature (0;m1,m2,m3). Then,
with the exception of cases (iv), (v) and (vi) in the above theorem, the action of Cn on S
can never be extended to the action of a larger group, so that Cn = Aut(S).
The general case of an arbitrary group G was considered shortly after, when the second
author of this survey joined the UNED. The question of extendability was completed and
the results published in [6], where special attention was paid to non-cyclic abelian groups.
The main results are the following, where we distinguish whether the group G acts with
triangular signature or not.
Theorem 2.3. Let G be a finite group acting with a non-maximal and non-triangular Fuch-
sian signature on a compact Riemann surface S of genus g > 1.
(i) If G acts with signature (2;— ), and for a corresponding presentation
G = ⟨a, b, c, d | [a, b][c, d] = · · · = 1 ⟩ the assignment a 7→ a−1, b 7→ ab−1a−1,
c 7→ (b−1cd)c−1(b−1cd)−1 and d 7→ (b−1c)d−1(b−1c)−1 is an automorphism of
G, then G is not the full automorphism group of S.
(ii) If G acts with signature (1; t, t), and for a corresponding presentation
G = ⟨ a, b, x | xt = ([a, b]x)t = · · · = 1 ⟩ the assignment a 7→ a−1, b 7→ b−1 and
x 7→ (ab)−1x−1(ba) is an automorphism of G, then G is not the full automorphism
group of S.
(iii) If G acts with signature (1; t) and for a corresponding presentation
G = ⟨ a, b | [a, b]t = · · · = 1 ⟩ the assignment a 7→ a−1 and b 7→ b−1 is an
automorphism of G, then G is not the full automorphism group of S.
(iv) If G acts with signature (0; t, t, u, u) where t + u ≥ 5, and for a corresponding
presentation G = ⟨ a, b, c, d | at = bt = cu = du = abcd = · · · = 1 ⟩ the
assignment a 7→ b, b 7→ a, c 7→ a−1da and d 7→ bcb−1 is an automorphism of G,
then G is not the full automorphism group of S.
Theorem 2.4. Let G be a finite group acting on a compact Riemann surface S of genus
g > 1 with a triangular signature (0;m1,m2,m3), corresponding to a presentation of
the form G = ⟨ a, b, c | am1 = bm2 = cm3 = abc = · · · = 1 ⟩. Then G is the full
automorphism group of S unless at least one of the following conditions is satisfied (up to
permutation of the periods m1,m2,m3), in which case G ̸= Aut(S) :
(i) G acts with signature (0; t, t, t) where t ≥ 4, and the assignment a 7→ b, b 7→ c and
c 7→ a induces an automorphism of G;
(ii) G acts with signature (0; t, t, u) where t ≥ 3 and t + u ≥ 7, and the assignment
a 7→ b, b 7→ a and c 7→ bcb−1 induces an automorphism of G;
(iii) G acts with signature (0; 2, 7, 7), the conjugates of bc−1bac3 generate a normal
subgroup K of index 56 in G, and G is extendable to a group G′ containing G
as a subgroup of index 9 such that G′ is generated by a and an element α which
normalises K and satisfies α3 = 1, (aα)7 = 1, b = (αaα)−1aα(αaα) and c =
(αaα−1)aα(αaα−1);
E. Bujalance and F. J. Cirre: Groups of automorphisms of Riemann and Klein surfaces, . . . 5
(iv) G acts with signature (0; 3, 3, 7), the conjugates of bac2 generate a normal subgroup
K of index 21 in G, and G is extendable to a group G′ containing G as a subgroup
of index 8 such that G′ is generated by c and an element α which normalises K and
satisfies α3 = 1, (αc)2 = 1, a = αc−2αc2α−1 and b = α−1c2αc−2α;
(v) G acts with signature (0; 3, 8, 8), conjugates of b2ac2 and c−1ba−1b−1ab−1 gen-
erate a normal subgroup K of index 72 in G, and G is extendable to a group G′
containing G as a subgroup of index 10 such that G′ is generated by c and an ele-
ment α which normalises K and satisfies α3 = 1, (αc)2 = 1, a = αc−2αc2α−1 and
b = α−1c2α−1cαc−2α;
(vi) G acts with signature (0; 4, 4, 5), the conjugates of a−1b−1c2 generate a normal
subgroup K of index 20 in G, and G is extendable to a group G′ containing G as
a subgroup of index 6 such that G′ is generated by c and an element α which nor-
malises K and satisfies α4 = 1, (αc)2 = 1, a = α2cαc−1α2 and b = α−1cαc−1α;
(vii) G acts with signature (0; 3, n, 3n) where n ≥ 3, the conjugates of b generate a
normal subgroup K of index 3 in G, and G is extendable to a group G′ containing
G as a subgroup of index 4 such that G′ is generated by c and an element α which
normalises K and satisfies α2 = 1, (cα)3 = 1, a = αc(cα)−1c−1α and b = αc3α;
(viii) G acts with signature (0; 2, n, 2n) where n ≥ 4, the conjugates of b generate a
normal subgroup K of index 2 in G, and G is extendable to a group G′ containing
G as a subgroup of index 3 such that G′ is generated by c and an element α which
normalises K and satisfies α3 = 1, (αc)2 = 1, a = α(αc)α−1 and b = α−1c2α.
As an example of application of the above theorems, we have the following, which
deals with non-cyclic abelian actions.
Theorem 2.5. Let G be a non-cyclic finite abelian group acting on a compact Riemann
surface S of genus g>1 with non-maximal Fuchsian signature σ. Then the following hold:
(a) The signature σ cannot be one of (1; t), (0; 2, 7, 7), (0; 3, 3, 7), (0; 3, 8, 8), (0; 4, 4, 5),
(0; 3, n, 3n) or (0; 2, n, 2n);
(b) If σ = (2;—) or (1; t, t) then G ̸= Aut(S);
(c) If σ = (0; t, t, u, u) then G is a factor group of Ct⊕Cu⊕Cm, where m = gcd(t, u),
and further, if the action of G corresponds to a (partial) monodromy presentation in
terms of commuting generators a, b, c, d subject to relations at = bt = cu = du = 1,
then if the assignment a 7→ b 7→ a, c 7→ d 7→ c is an automorphism of G then
G ̸= Aut(S);
(d) If σ = (0; t, t, t) then G ∼= Ct ⊕ Ck with monodromy presentation of the form
G = ⟨ a, b | at = bt = (ab)t = [a, b] = 1, bk = aks ⟩ for some k dividing t
and some s coprime to t/k, and in this case G ̸= Aut(S) if and only if s2 ≡ 1
(mod t/k) or s2 + s+ 1 ≡ 0 (mod t/k);
(e) If σ = (0; t, t, u) with t ̸= u then u divides t, and G ∼= Ct ⊕ Ck with monodromy
presentation G = ⟨ a, b | at = bt = (ab)u = [a, b] = 1, bk = aks ⟩ for some k
dividing u and some s coprime to t/k, and in this case G ̸= Aut(S) if and only if
s2 ≡ 1 (mod t/k).
6 Art Discrete Appl. Math. 5 (2022) #P3.04
2.2 Extendability of group actions on Klein surfaces
The next step in the problem of extendability of group actions is to consider other types of
surfaces. In this setting, the analysis of NEC group inclusions carried out by Marston in
2013 was the seminal work to study this problem on Klein surfaces.
Recall from Theorems 2.1 and 2.2 that, in the case of Riemann surfaces, the extend-
ability of a cyclic action depends on the signature and on the surface-kernel epimorphism.
In the case of cyclic group actions on non-orientable unbordered surfaces, the situation is
surprisingly quite different, as the next theorem, proved in [8], shows.
Theorem 2.6. A cyclic group acting with non-maximal signature on a non-orientable sur-
face always extends to the action of a larger group.
We also describe some applications concerning non-orientable surfaces whose full au-
tomorphism group is cyclic of largest possible order. For instance, we show that this order
is g + 1, g or g − 1 depending on whether g ≡ 1 (mod 4), g is even or g ≡ 3 (mod 4)
respectively, where g is the algebraic genus of the surface. As a further application, we
determine for each n the smallest algebraic genus of a non-orientable surface on which the
cyclic group of order n acts as the full automorphism group of the surface We call this
the full cross-cap genus of the cyclic group, following the definition by May [38] of the
symmetric cross-cap number as the smallest topological genus of a non-orientable surface
on which the cyclic group acts effectively. We also show that for each integer n > 1, there
exists some g0 such that Cn is the full automorphism group of some non-orientable surface
of algebraic genus g for every g ≥ g0. Finally, we give a table showing the spectrum of
all algebraic genera of the non-orientable Riemann surfaces on which Cn acts as the full
automorphism group, for 2 ≤ n ≤ 10.
An analogous study for bordered surfaces was carried out in 2015, obtaining the fol-
lowing result in [9].
Theorem 2.7. A cyclic group acting with non-maximal signature on a bordered surface
always extends to the action of a larger group.
Some applications are also obtained. We calculate, for a given g > 1 the order of the
largest cyclic group that acts as the full automorphism group of a bordered surface of alge-
braic genus g. The topological type of the surfaces attaining this bound is also determined.
This bound turns out to be the same as for unbordered non-orientable surfaces, and we show
that, in some cases, surfaces of both types attaining the bound come together as the Klein
surfaces associated to two symmetries on the same Riemann surface. We also determine
for each n the smallest algebraic genus of a bordered surface on which the cyclic group of
order n acts as the full automorphism group of the surface. We call this the full real genus
of the cyclic group, following the definition by May [37] of the real genus as the smallest
algebraic genus of a bordered surface on which the cyclic group acts effectively. We deal
separately with orientable and non-orientable surfaces, which leads to the concepts of full
real orientable and full real non-orientable genus of a group.
Let us mention that the concept analogous to the real genus for Riemann surfaces is
called the symmetric genus of a group G (if we allow only orientation preserving automor-
phisms then we are led to the concept of strong symmetric genus). These are classic topics
in Riemann and Klein surface theories which have catalysed a large amount of research.
Let us mention, for Riemann surfaces, the seminal paper [31] by Harvey on cyclic groups,
and [27, 32, 40], among many others. For the real genus, see [26, 39], for instance.
E. Bujalance and F. J. Cirre: Groups of automorphisms of Riemann and Klein surfaces, . . . 7
3 Automorphism groups of Klein surfaces
It was Klein the first who realized that a bordered surface may be seen as the orbit space
of a Riemann surface under the action of a symmetry with fixed points. This initiated the
study of groups of automorphisms of bordered Klein surfaces at the end of the 19th century.
Interest in this topic was resumed seven decades later when the work of Macbeath opened
the door to the combinatorial study of groups of automorphisms. Many results have been
obtained since then.
Lists of all finite groups of automorphisms of bordered surfaces are known for algebraic
genus 2 and 3; see [20, 22]. The amount of combinatorial data involved increases rapidly
with the genus, and this makes it very difficult to deal with higher genus. The main goal
of the paper [13] is to determine, up to topological equivalence, all the finite group actions
on compact (orientable or non-orientable) bordered surfaces of algebraic genus p for 2 ≤
p ≤ 6. Computational methods using the MAGMA program, in which Marston is an expert,
have been used for some of the results.
The lists of groups given in [20, 22] are lists of non-isomorphic abstract groups. In [13]
we are interested in lists of inequivalent topological group actions. This is a finer classi-
fication, as the same abstract group may act on the same surface in different topological
ways.
Instead of looking for all groups acting on low genus, we may look for large group
actions. The largest order of a group acting on bordered surfaces of algebraic genus g is
12(g − 1). Groups attaining this bound are known as M∗-groups. They play the same role
on bordered surfaces as Hurwitz groups play on Riemann surfaces. In [7] we determine,
up to topological equivalence, all the finite group actions of order at least 6(g − 1) on
compact bordered surfaces of algebraic genus g for 2 ≤ g ≤ 101. The topological type of
the surfaces where these actions occur are also given.
Let S be a compact Riemann surface of genus g > 1 and let τ : S → S be a symmetry.
The topological type of the conjugacy classes of all symmetries of S constitute what is
known as the symmetry type of S. The surface S is said to have maximal real symmetry if
it admits a symmetry τ such that the compact Klein surface S/⟨τ⟩ has maximal symmetry,
which means that S/⟨τ⟩ has the largest possible number of automorphisms with respect
to its genus. If τ has fixed points then this maximum number is 12(g − 1). In the first
part of [10] we develop a computational procedure to calculate the symmetry type of every
Riemann surface of genus g with maximal real symmetry, for given small values of g > 1.
We use this to find all of them for 1 < g ≤ 101, and give details for 1 < g ≤ 25 (in an
appendix). In the second part, we determine the symmetry types of four infinite families of
Riemann surfaces with maximal real symmetry.
4 Double covers of Klein surfaces
Let X be a compact Klein surface of algebraic genus g > 1 and let (S, τ) be its Riemann
double cover. As explained in the preliminaries, this means that S is a compact Riemann
surface of genus g which admits an anticonformal involution τ : S → S such that the orbit
space S/⟨τ⟩ is a compact Klein surface isomorphic to X. It is well known that the full group
Aut(X) of automorphisms of X is isomorphic to the group of all conformal automorphisms
of S that commute with τ . It follows that the full group Aut(S) of all automorphisms of
S (conformal or anticonformal) contains the direct product Aut(X) × C2 where C2 is the
cyclic group generated by τ . The following question arises naturally: is Aut(S) equal to
8 Art Discrete Appl. Math. 5 (2022) #P3.04
Aut(X)×C2, or does S admit additional automorphisms? This problem was firstly studied
by May in 1991. We proposed to Marston to work on this topic during our visit to Auckland
in 2016. The results were published in [11].
This question has been considered for bordered Klein surfaces with the largest possible
number of automorphisms, namely 12(g − 1), by May in [36], and for those with the
second largest possible number of automorphisms, namely 8(g − 1), by Bujalance, Costa,
Gromadzki and Singerman in [19]. In both cases the authors showed that the equality
Aut(S) = Aut(X) × C2 holds for almost all surfaces, with one single exception when
|Aut(X)| = 12(g − 1) and five exceptions when |Aut(X)| = 8(g − 1). Subsequently,
Costa and Porto showed in [41] that equality also holds except in a finite number of cases
when |Aut(X)| > 6(g − 1), while if |Aut(X)| = 6(g − 1) then there are infinitely many
Klein surfaces X (with different topological types) such that Aut(S) ̸= Aut(X)× C2.
If |Aut(X)|>6(g−1), then it follows from the Riemann-Hurwitz formula that Aut(X)
is uniformised by an NEC group with quadrangular signature (0;+; [−]; {(2, 2, 2, n)}) with
n = 3, 4 or 5. Using techniques of hyperbolic geometry, Costa and Porto investigated
the case of signature (0;+; [−]; {(2, 2, 2, n)}) where n is an odd prime, and showed the
equality Aut(S) =Aut(X)× C2 holds with one single exception (for each odd prime n).
In [11] we consider the general case where a group G of automorphisms of X is uni-
formised by some NEC group with quadrangular signature (0;+; [−]; {(2, 2, 2, n)}) for an
arbitrary value of n > 2. We investigate the relationship between G and the full automor-
phism groups Aut(X) and Aut(S). Furthermore, we consider actions not only on bordered
Klein surfaces, but also on unbordered non-orientable surfaces.
5 Automorphism groups of pseudo-real surfaces
Marston co-organized the workshop “Symmetries of Surfaces, Maps and Dessins”, held in
the Banff International Research Station from September 24th to September 29th, 2017.
It was there that the idea arose to study with Marston upper bounds for the order of the
automorphism groups of pseudo-real surfaces.
A compact Riemann surface is called pseudo-real if it admits anticonformal automor-
phisms, but none of them has order two. Another term used for such surfaces is asymmetric.
Their importance stems from the fact that in the moduli space of compact Riemann surfaces
of given genus, pseudo-real surfaces represent the points that have real moduli but are not
definable over the reals.
There are no pseudo-real surfaces of genus 0 or 1, since in those cases every reflexible
surface admits an anticonformal automorphism of order two. On the other hand, it was
shown in [15] that there exists at least one pseudo-real surface of genus g for every g > 1.
The pseudo-real surfaces of genus 2 to 4 were classified in [15, 18], and this was extended
for genus 5 to 10 in [3], except that five of the entries in the tables in [3] should be deleted.
Upper bounds on the order of a group of automorphisms of a pseudo-real surface of
given genus g > 1 are studied in [12]. This is motivated by a number of theorems about the
orders of groups of automorphisms of other kinds of surfaces. The most famous of these are
the theorems of Hurwitz (1893) and Wiman (1896) that give an upper bound of 84(g − 1)
on the number of orientation-preserving automorphisms of a compact Riemann surface of
genus g > 1, and an upper bound of 4g + 2 on the order of any such automorphism,
respectively. Other such theorems deal with special cases where the group is abelian, or the
surface is non-orientable, or the automorphism reverses orientation; see [5, 25, 29, 30, 34].
E. Bujalance and F. J. Cirre: Groups of automorphisms of Riemann and Klein surfaces, . . . 9
For every integer g > 1, let M(g) be the order of the largest group of automorphisms
of a pseudo-real surface of genus g. It was shown in [15] that M(g) ≤ 12(g − 1). This
upper bound is attained for infinitely many g, but certainly not for all g > 1, and so it
makes sense to look for more refined bounds on the group order, in general and special
cases. One particular question of interest is to find a sharp lower bound for M(g), akin to
the Accola-Maclachlan bound for general compact Riemann surfaces (see [1, 35]). In [12]
we show that M(g) ≥ 2g for every even g ≥ 2, while M(g) ≥ 4(g − 1) for every odd
g ≥ 3, and we prove that the latter bound is sharp for a very large and possibly infinite
set of odd values of g ≥ 3. Unfortunately we are not yet able to determine the level of
sharpness of the former bound (for even g), so this is left as an open question. We also give
the precise values of M(g) for all g between 2 and 128, together with the signatures for the
actions of the corresponding groups of largest order.
When we restrict to the cases where the group is cyclic or abelian we do find sharp
bounds for every genus g > 1. Let Mab(g) and Mcyc(g) be respectively the orders of
the largest abelian and largest cyclic group of automorphisms of a pseudo-real surface of
genus g, such that the group contains orientation-reversing elements. In [12] we show that
Mab(g) = Mcyc(g) = 2g for every even g ≥ 2, and that Mcyc(g) = 2g − 2 for every odd
g ≥ 3. On the other hand, for odd g > 1 we find that Mab(g) = 2g+6 when g ≡ 1 mod 4,
while Mab(g) = 2g+2 when g ≡ 3 mod 4. In all these cases we give specific details about
the surfaces and groups attaining the bound. It is worth mentioning that we also find similar
bounds when the groups (cyclic or abelian) contain no orientation-reversing elements.
6 A final note
The first author of this survey also has other papers with Marston on related topics of
automorphisms of Riemann or Klein surfaces, see [15, 16, 17] for instance.
Finally, we want to thank Marston for working with us over the years, as we have
learned a lot from him. We are really impressed by his speed in solving the problems we
pose to him. Many times we have asked him about a problem in group theory we had been
thinking about for several days, and the next morning (Spanish time) we had his solution
in the email. Marston has indeed a great capacity for work and a vast knowledge of group
theory and their computational methods. But above all, he has many other human qualities
that make it enjoyable to work with him.
ORCID iDs
Emilio Bujalance https://orcid.org/0000-0002-3407-8936
Francisco-Javier Cirre https://orcid.org/0000-0001-9970-0979
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.05
https://doi.org/10.26493/2590-9770.1450.f39
(Also available at http://adam-journal.eu)
Some remarks on group actions on hyperbolic
3-manifolds
Colin Maclachlan
Department of Mathematical Sciences, University of Aberdeen, Meston Bldg, Aberdeen,
AB24 3UE, U.K.
Alan Reid*
Department of Mathematics, Rice University, Houston, TX 77005, USA
Antonio Salgueiro†
Departamento de Matemática, Universidade de Coimbra, Largo D. Dinis,
3000 Coimbra, Portugal
Dedicated to Marston Conder on the occasion of his 65th birthday
Received 30 August 2021, accepted 28 March 2022, published online 18 July 2022
Abstract
We prove that there are infinitely many non-commensurable closed orientable hyper-
bolic 3-manifolds X , with the property that there are finite groups G1 and G2 acting freely
by orientation-preserving isometries on X with X/G1 and X/G2 isometric, but G1 and G2
are not conjugate in Isom(X). We provide examples where G1 and G2 are non-isomorphic,
and prove analogous results when G1 and G2 act with fixed-points.
Keywords: Hyperbolic manifold, finite group action, arithmetic group.
Math. Subj. Class.: 57M60
*Corresponding author. The author was partially supported by the N. S. F. The author is grateful to Emilio
Bujalence and Javier Cirre for sending him a detailed list of references in connection with actions of p-groups on
Riemann surfaces.
†The author was partially supported by the U.T. Austin - Portugal Mathematics CoLab program (FCT) and the
Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government
through FCT/MCTES.
E-mail addresses: (Colin Maclachlan), alan.reid@rice.edu (Alan Reid), ams@mat.uc.pt (Antonio
Salgueiro)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.05
1 Introduction
The study of finite group actions, both free and with fixed points, on closed Riemannian
manifolds has a long and rich history. In the context of low-dimensional geometry and
topology, two notable examples of this are Kerckhoff’s solution to the Nielsen Realization
Problem for surfaces [19] and in the setting of geometric 3-manifolds, it is known that any
such action is always conjugate to an isometric action (see [7, 12, 14], and [25]) which
formed part of Thurston’s geometrization program for 3-manifolds and 3-orbifolds. In this
paper we will be concerned with (isometric) finite group actions on hyperbolic manifolds
and in particular in dimension 3.
More specifically we will be interested in the following situation: two finite groups
G1 and G2 acting freely or with fixed points by (orientation-preserving) isometries on a
closed orientable hyperbolic 3-manifold X with X/G1 ∼= X/G2 (here and throughout
the symbol ∼= in the context of manifolds or orbifolds will denote isometric). By volume
considerations, it is clear that in this setting, G1 and G2 must have the same order, and it is
also clear that if G1 and G2 are conjugate in Isom(X) then the quotients will be isometric.
The aim of this article is to provide constructions, both general and explicit, of examples of
closed orientable hyperbolic 3-manifolds X , and groups G1 and G2 that are not conjugate
in Isom(X) with X/G1 ∼= X/G2. In fact, our methods provide examples of groups that
are not even isomorphic.
Our first result deals with free actions.
Theorem 1.1. There are infinitely many non-commensurable closed orientable hyperbolic
3-manifolds X , with the property that there are finite groups G1 and G2 satisfying:
(1) G1 and G2 act freely on X by orientation-preserving isometries on X with
X/G1 ∼= X/G2.
(2) |G1| = |G2|, but G1 and G2 are not conjugate in Isom(X).
As mentioned above, since X/G1 ∼= X/G2, it is immediate that |G1| = |G2|. How-
ever, by making the construction explicit we can actually exhibit examples of manifolds
X as in Theorem 1.1 for which G1 is an elementary abelian group of order p
3 (for certain
primes p), and G2 is the non-abelian group of order p
3 containing an element of order p2
(see §4.1).
Examples of closed orientable hyperbolic 3-manifolds X that are fibered over S1 ad-
mitting free actions by finite groups G1 and G2 with X/G1 ∼= X/G2 and for which G1
and G2 are not conjugate in Isom(X) are given [21] (although they are unable to determine
whether these examples fall into infinitely many commensurability classes). By focusing
on very explicit examples, we can also impose topological conditions on the manifold X
and the quotients X/G1 ∼= X/G2, namely we prove the following
Corollary 1.2. (1) There are infinitely many examples of (commensurable) closed ori-
entable hyperbolic 3-manifolds X that fiber over the circle with the property that
there are finite groups G1 and G2 as in the conclusion of Theorem 1.1 such that
X/G1 ∼= X/G2 also fiber over the circle.
(2) There is a hyperbolic 3-manifold X which is a rational homology 3-sphere with the
property that there are finite groups G1 and G2 as in the conclusion of Theorem 1.1
such that X/G1 ∼= X/G2 is also a rational homology 3-sphere.
C. Maclachlan et al.: Some remarks on group actions on hyperbolic 3-manifolds 3
The finite groups G1 and G2 in both cases of Corollary 1.2 are of the type described
before Corollary 1.2 (i.e. elementary abelian p-groups and non-abelian p-groups of order
p3 for certain primes p). As far as the authors are aware, the example of Corollary 1.2(2) is
the first such example of a hyperbolic rational homology 3-sphere as in the conclusion of
Corollary 1.2(2).
By way of contrast, results in [28] and [29] consider the question to what extent G1
and G2 acting with fixed points must be conjugate in Isom(X) (for certain closed Rie-
mannian 3-manifolds X not necessarily hyperbolic), and for example [28, Theorem 8 and
Proposition 13] provides a uniqueness statement in certain settings (e.g. rational homology
3–spheres and most cyclic group actions).
Our methods also provide a construction when the action is not non-free.
Theorem 1.3. There are infinitely many non-commensurable closed orientable hyperbolic
3-manifolds X , with the property that there are finite groups G1 and G2 satisfying:
(1) G1 and G2 act by orientation-preserving isometries on X , have non-empty fixed-
point set, and with X/G1 ∼= X/G2.
(2) |G1| = |G2|, but G1 and G2 are not conjugate in Isom(X).
The examples constructed in the proofs of Theorems 1.1, 1.3 and Corollary 1.2 come
from the class of arithmetic hyperbolic 3-manifolds (see Section 3 and [23] for further
details), and exploit the fact that such manifolds have fundamental groups with large com-
mensurator. The advantages of the arithmetic nature of the construction are first, it provides
infinitely many commensurability classes of examples, and second, the groups G1 and G2
can be made explicit.
As will be clear, the method of proof of Theorem 1.1 (see Section 2) is very general for
arithmetic groups (given a description of maximal groups in the commensurability class),
and although our main focus is hyperbolic 3-manifolds, we sketch some variations of Theo-
rem 1.1 in other dimensions; for example, we provide examples of Riemann surfaces which
admit actions of distinct finite p-groups with conformally equivalent quotients. Although
there is a vast literature on (p-)group actions on Riemann surfaces, we were unable to find
results which have precise overlap with ours, although questions of a similar nature have
been addressed (see [17, 18] and [20] to name a few). However, we do note that the method
of proof of [21] also provides examples of Riemann surfaces with finite groups G1 and G2
acting freely on X with X/G1 conformally equivalent to X/G2.
More care is needed in generalizing the proof of Theorem 1.3, but we expect that this
can also be done.
This paper has its origins in a visit of the third author to the second in the Spring of 2011
whilst the second author held a position at U.T. Austin. The first author sadly passed away
in November 2012, and the paper remained stubbornly unfinished since that time. Because
of Marston’s close personal and mathematical connection to Colin Maclachlan through his
visits to the U.K., and Colin’s to New Zealand, it seemed an appropriate opportunity to fin-
ish the paper, and submit as part of the celebration of Marston’s 65th birthday. The second
author has no doubts that Colin would have been very pleased with this arrangement, and
would have very much enjoyed raising a glass of Scotland’s national drink in Marston’s
honor!
4 Art Discrete Appl. Math. 5 (2022) #P3.05
As was remarked upon above, results similar to Theorems 1.1, 1.3 have since appeared
in the literature (see [21], [22]), but our methods are very different, and so still seem worthy
of publication. The reader will likely note the influence of the first author in this work.
However, since the first author was neither able to verify nor influence the final version of
the paper, any errors in the paper should be attributed to the other two authors.
2 A basic construction
The basic idea of our construction is contained in Proposition 2.1 below. Although our
main focus is in dimension 3, we state it for hyperbolic manifolds of arbitrary dimension.
In Section 3 we provide a detailed discussion how to construct examples of hyperbolic
3-manifolds satisfying the hypothesis of Proposition 2.1 using arithmetic techniques in
dimension 3, and in Section 7 construct examples in dimension 2. In Section 8 we discuss
applying Proposition 2.1 to higher dimensional arithmetic hyperbolic manifolds, but we
have decided to only offer a sketch of a proof. The complete proof requires a detailed
discussion of maximal arithmetic lattices in this setting.
Recall that if Γ ⊂ Isom+(Hn) is a lattice, the commensurator of Γ is the group
Comm(Γ) = {g ∈ Isom+(Hn) | gΓg−1 is commensurable with Γ}. (2.1)
It was proved by Margulis [24], that Γ is an arithmetic lattice if and only if Comm(Γ) is
dense in Isom+(Hn). Furthermore, it is known in this case (see [8], [23, Theorem 11.4],
and Theorem 3.2 below for dimension 3 and [9] more generally) that there are infinitely
many distinct maximal arithmetic lattices commensurable with Γ. On the other hand, if
Γ ⊂ Isom+(Hn) is a non-arithmetic lattice, then [Comm(Γ) : Γ] < ∞. Moreover, in this
case, Comm(Γ) is the unique maximal discrete subgroup of Isom+(Hn) containing Γ.
Notation: For Γ ⊂ Isom(Hn), we denote by Γ+ the subgroup of index at most 2 obtained
as Γ ∩ Isom+(Hn).
Proposition 2.1. Let Γ0 ⊂ Isom(Hn) be a maximal arithmetic lattice, and let Γ1 be a
normal torsion-free subgroup of finite index in Γ0 which is contained in Γ
+
0 . Assume that
there exists g ∈ Comm(Γ+0 ) \ Γ+0 such that gΓ1g−1 ⊂ Γ+0 .
Then there exists ∆ ⊂ Γ+0 for which X = Hn/∆ satisfies the conclusion of
Theorem 1.1.
Proof. Since gΓ1g−1 ⊂ Γ+0 , we deduce, by volume considerations, that gΓ1g−1 has finite
index in Γ+0 equal to [Γ
+
0 : Γ1]. By maximality, the normalizer of Γ1 in Isom(H
n) is
Γ0. Note also that g ̸∈ Γ0, otherwise g ∈ Γ0 ∩ Comm(Γ+0 ) ⊂ Γ0 ∩ Isom+(Hn) = Γ+0
contradicting the hypothesis on g. It follows that gΓ1g
−1 ̸= Γ1.
Since the subgroup Γ1 ∩ gΓ1g−1 has finite index in Γ0, it contains a finite index
subgroup ∆, normal in Γ0, namely the core (i.e. the intersection of all conjugates of
Γ1 ∩ gΓ1g−1 in Γ0). Then X = Hn/∆ is a hyperbolic n-manifold which admits free finite
group actions by the groups G1 = Γ1/∆ and G2 = gΓ1g
−1/∆ with quotients Hn/Γ1 ∼=
M ∼= Hn/gΓ1g−1. Clearly |G1| = |G2| and G1 and G2 act by orientation-preserving
isometries. Furthermore, Isom(X) = Γ0/∆ since Γ0 is maximal. Now G1 ̸= G2 and
G1 is a normal subgroup of Isom(X). It follows that G1 and G2 cannot be conjugate in
Isom(X).
C. Maclachlan et al.: Some remarks on group actions on hyperbolic 3-manifolds 5
Remark 2.2. A version of this Proposition still holds if we assume that Γ1 is not torsion-
free. In this case, to obtain the manifold X as in the conclusion of Theorem 1.3 we must
further require that ∆ as in the proof, is torsion-free. However, standard methods makes
this easy to arrange.
Remark 2.3. In the non-arithmetic setting, the discussion of the commensurator above
shows that there is a unique maximal lattice in the commensurability class; i.e. Γ0 in this
case. Hence, if in Proposition 2.1 we were to assume that Γ0 is non-arithmetic, then there
could be no element g as stated. However, a variation of Proposition 2.1 can be formulated:
Let Γ0 ⊂ Isom(Hn) be a maximal non-arithmetic lattice, and Γ < Γ+0 a proper torsion-
free subgroup of finite index containing a normal subgroup ∆ of finite index that is not
normal in Γ0. Suppose that there exists a subgroup Γ1 with ∆ ⊂ Γ1 ⊂ Γ and g ∈ Γ0 such
that ∆ ⊂ gΓ1g−1 ⊂ Γ with G1 = Γ1/∆ and G2 = gΓ1g−1/∆ not isomorphic. Then the
conclusion of Theorem 1.1 holds.
This construction is very much in the spirit of that given in [21]. Indeed, it is likely that
"many" of the examples in [21] are non-arithmetic, so the above statement would cover the
construction of their examples.
Remark 2.4. We could have defined the commensurator of Γ in Isom(Hn). The group
Comm(Γ) defined at (2.1) is a subgroup of index at most 2 in this larger group. However,
we did not want to constantly keep distinguishing the "orientation-preserving" commensu-
rator and so we retain Comm(Γ) to be as defined at (2.1).
3 Preliminaries on arithmetic hyperbolic 3-manifolds
As mentioned in Section 1, our examples are built using arithmetic hyperbolic mani-
folds. Here we focus on the case of arithmetic Kleinian groups and arithmetic hyperbolic
3-orbifolds, and recall some relevant results and facts that will be needed (see [23] for
further details).
3.1 Arithmetic Kleinian groups
Arithmetic Kleinian groups are obtained as follows. Let k be a number field having exactly
one complex place, Rk its ring of integers and B a quaternion algebra over k which ramifies
at all real places of k. Let ρ : B → M(2,C) be an embedding, O an order of B, and
O1 the elements of norm one in O. Then Pρ(O1) ⊂ PSL(2,C) is a finite co-volume
Kleinian group, which is co-compact if and only if B is a division algebra, which in turn is
equivalent to B not being isomorphic to M(2,Q(
√
−d)), where d is a square-free positive
integer. Following [8], we denote the group Pρ(O1) by Γ1O.
An arithmetic Kleinian group Γ is a subgroup of PSL(2,C) commensurable with a
group Γ1O. In addition, we call Γ ⊂ Isom(H3) arithmetic if it is commensurable with an
arithmetic Kleinian group. We call Q = H3/Γ arithmetic if Γ is arithmetic.
The wide (i.e. up to conjugacy) commensurability class of an arithmetic Kleinian group
is determined by the isomorphism class of B (see e.g. [23, Theorem 8.4.1]). We can refine
this further by noting that if Ramf (B) denotes the finite set of prime ideals P of k where B
is ramified, i.e. BP = B⊗k kP is a division algebra, then the isomorphism class of B (as in
the definition of an arithmetic Kleinian group given above) is determined by Ramf (B). In
particular, using the previous remark, to construct infinitely many commensurability classes
of arithmetic hyperbolic 3-manifolds, it is sufficient to fix the field k and vary Ramf (B).
6 Art Discrete Appl. Math. 5 (2022) #P3.05
Our arguments crucially depend on a fine understanding of maximal arithmetic Kleinian
groups defined using maximal and Eichler orders (intersections of maximal orders) in the
quaternion algebra B (see [8] or [23, Chapter 11] for more details). We will discuss this
further below, however, for convenience, we first provide a "warm-up" version of the gen-
eral construction that may be useful as a template for the reader to bear in mind. This will
construct finite volume non-compact examples that satisfy the conclusion of Theorem 1.1.
3.2 Warm-up construction
The group Γ = PGL(2,Z[i]) is a maximal arithmetic Kleinian group, although it is
not maximal in Isom(H3). The maximal group is obtained as the group generated by
< τ,PGL(2,Z[i]) > where τ is the reflection on H3 obtained by extension of complex
conjugation on C. We will let this group be denoted by Γ0. Let A ⊂ Z[i] be an ideal, and
let
Γ(A) = ker{ϕA : PGL(2,Z[i]) → PGL(2,Z[i]/A)},
which will be a subgroup of the Picard group PSL(2,Z[i]) for most ideals A ⊂ Z[i], e.g.
those of odd norm.
We will focus on ideals A of the form pZ[i] where p ∈ Z is a prime congruent to
3 mod 4, and so the ideal pZ[i] remains prime in Z[i]. It is easy to check that Γ(p) =
Γ(pZ[i]) is torsion-free. Note that since complex conjugation preserves the ideal pZ[i],
Γ(p) is also a normal subgroup of Γ0.
Now it is a simple matter to check that the element σp = P
(
0 −1/√p√
p 0
)
normal-
izes the subgroup Γ0(p) = ϕ
−1
p (Bp) where Bp is the group of upper triangular matrices
in PSL(2,Z[i]/pZ[i]). In particular, σp ∈ Comm(PGL(2,Z[i])). Moreover, and most
importantly in our situation, observe that
σpΓ(p)σ
−1
p = P{
(
1 + ap b
cp2 1 + dp
)
: a, b, c, d ∈ Z[i], and determinant 1}.
With this in hand, we can now apply Proposition 2.1 to the groups Γ0 and Γ1 = Γ(p), with
g = σp.
From above, the group gΓ1g
−1 contains the group ∆ = Γ(p2) which is also normal in
Γ0. Now take X = H
3/∆, and the groups G1 and G2 to be given by Γ1/∆ and gΓ1g
−1/∆
respectively. This finishes the construction of the manifold X as in Theorem 1.1 in this
setting.
To get a version of Theorem 1.3, we continue to use the group Γ0 as above, and tweak
the above construction as follows. In this case we choose the ideal < 1+ i > and construct
the group Γ(1 + i). Again, since complex conjugation preserves the ideal < 1 + i >,
Γ(1 + i) is a normal subgroup of Γ0. However, Γ(1 + i) is not torsion-free since the
element P
(
i 0
0 −i
)
∈ Γ(1 + i). Thus we cannot take this group to be Γ1. However,
setting g = P
(
0 −1/
√
1 + i√
1 + i 0
)
, the group Γ1 can be constructed by passing to a
torsion-free subgroup of Γ(1 + i) ∩ gΓ(1 + i)g−1, and taking its core in Γ0. This is the
group defining the manifold X in Theorem 1.3. We refer the reader to the end of the proof
of Theorem 1.3 in Subsection 6.2 for details of why the groups G1 and G2 are not conjugate
in this case.
C. Maclachlan et al.: Some remarks on group actions on hyperbolic 3-manifolds 7
The discussion that follows in Subsections 3.3 and 3.4 provides the necessary general-
ization of the framework described above that will allow us to pass to the closed case, and
thereby prove Theorem 1.1.
Remark 3.1. In the argument above in the case of Γ(1 + i) and
g = P
(
0 −1/
√
1 + i√
1 + i 0
)
,
then as in the first case of Γ(p), the intersection Γ(1 + i) ∩ gΓ(1 + i)g−1 contains the
principal congruence subgroup Γ((1+i)2) = Γ(2) which is a normal torsion-free subgroup
of PSL(2,Z[i]) (being the fundamental group of a link complement in S3 [5]), so in fact,
in this case one can take ∆ = Γ(2) and so there is no need to pass to any further subgroup
of finite index.
3.3 Orders and maximal groups
The argument of Subsection 3.2 provides the template, and so to arrange that the manifolds
are closed we use the discussions in Subsection 3.1 to replace M(2,Q(i)) with quaternion
division algebras over a number field with one complex place, generalize the groups Γ(p)
and Γ(1 + i) using the principal congruence subgroups described in Subsection 3.4 below,
and generalize the group Γ0(p) and element σp using the theory of Eichler orders and their
normalizers as we now describe.
Thus let O ⊂ B be a maximal order and E ⊂ O an Eichler order, i.e. the intersection
O ∩O′, for some maximal order O′ ̸= O. The Eichler order E is said to be of square-free
level S, where S is a finite set of prime ideals of k, disjoint from Ramf (B), if, locally at
each finite place, EP = OP = O′P if P ̸∈ S and, if P ∈ S, then EP = OP ∩ O′P has
level P so that OP and O′P are adjacent maximal orders in the tree of maximal orders in
BP ∼= M2(kP) (see [23, Chapter 6.5]). When S = ∅, we simply get the maximal order O.
Indeed, it can always be arranged that for square-free level S and P ∈ S we have
EP = TP where
TP =
{(
a b
πc d
)
| a, b, c, d ∈ RP
}
, (3.1)
and RP ⊂ kP is the valuation ring with uniformizer π (see [23, Chapter 6.5]).
Let N(E) and N(O) denote the normalizers of E and O respectively in B∗. Their
images, Pρ(N(E)) and Pρ(N(O)) in PGL(2,C) ∼= PSL(2,C), denoted by ΓE and ΓO
respectively, are arithmetic Kleinian groups and any arithmetic Kleinian group is conjugate
to a subgroup of some such ΓE or ΓO (see [8] and [23, Chapter 11.4]).
Note that since conjugation preserves the norm, the groups ΓO and ΓE normalize the
groups Γ1O and Γ
1
E (the image in PSL(2,C) of E1 the elements of norm one in E). For
convenience we state the following result of Borel [8](see also [23, Chapter 11.4]).
Theorem 3.2. Fix a maximal order O ⊂ B. Then there exist infinitely many distinct sets
of prime ideals Si ⊂ Rk and Eichler orders Ei ⊂ O of level Si such that ΓEi are distinct
maximal arithmetic Kleinian groups.
Remark 3.3. We single out two cases of Theorem 3.2 that we will make use of: if k is
quadratic imaginary or cubic with one complex place and has class number 1, P a k-prime
ideal and S = {P} then the group ΓE can be proved to be maximal (see [8] and [23,
Chapters 6.7 and 11.4]).
8 Art Discrete Appl. Math. 5 (2022) #P3.05
3.4 Principal congruence subgroups
For O a maximal order, we now describe a distinguished class of subgroups of Γ1O, known
as principal congruence subgroups. To that end, let I ⊂ B be a 2-sided integral ideal (see
[23, Chapter 6] for further details). Then I ⊂ O and O/I is a finite ring. Define:
O1(I) = {α ∈ O1 | α− 1 ∈ I}.
The corresponding principal congruence subgroup of Γ1O is Γ(O(I)) = Pρ(O1(I)).
Since I is a 2-sided ideal, it follows that Γ(O(I)) is a normal subgroup of finite index
in Γ1O. Indeed, we can say more.
Lemma 3.4. The principal congruence subgroups Γ(O(I)) are normal subgroups of ΓO.
Thus the normalizer of Γ(O(I)) in PSL(2,C) is ΓO.
Proof. The second statement follows from the first since, as noted above, ΓO is maximal.
Now let α ∈ O1(I) and x ∈ N(O). Then α ∈ 1+ I and x(1+ I)x−1 = 1+ xIx−1. Now
I and xO are elements of the set of two-sided integral ideals, which is an abelian group
(see [23, Chapter 6.7]). This, together with integrality of I gives:
xIx−1 = (xO)I(Ox−1) = (xO)I(xO)−1 = I,
and we deduce that xαx−1 ∈ O1(I).
In addition, for most ideals I , the groups Γ(O(I)) are torsion-free. We record the
following for convenience which needs some additional notation (see [23, Chapters 6.5,
6.6] for details). Apart from a finite set of k-primes T (I), IP = OP . For P ∈ T (I), we
have IP = π
nPOP where π is a uniformizer for kP .
Proposition 3.5. In the notation above, suppose for P ∈ T (I) the following holds:
(1) P /∈ Ramf (B);
(2) P does not ramify in k | Q;
(3) P is not dyadic (i.e. P does not divide 2).
Then Γ(O(I)) is torsion-free.
A group Γ ⊂ Γ1O is called a congruence subgroup if there is some ideal I such that
Γ(O(I)) ⊂ Γ.
As an example of a congruence subgroup that we will exploit (and the analogue of
Γ0(p) in Subsection 3.2), it is shown in [1] that if E ⊂ O is an Eichler order of square-free
level S, then Γ1E ⊂ Γ1O is a congruence subgroup. Using (3.1), we can be more explicit. If
ΓE is of square-free level S, then for each P /∈ S, we have Ep = OP = M2(RP) and for
P ∈ S we have EP = TP where TP is as defined at (3.1).
Then in the notation above, defining I locally by IP = πOP for each P ∈ S and
IP = OP otherwise, it follows that, T (I) = S. Hence, for P ∈ S and α ∈ SL(2, RP) is
such that α− 1 ∈ IP then α ∈ T 1P . It now follows that Γ(O(I)) ⊂ Γ1E .
We will next exploit the feature of having a "large" commensurator which requires an
additional piece of terminology.
For a maximal order O in B, choose an Eichler order E of square-free level S such that
ΓE is maximal (using Theorem 3.2). By maximality, we can find elements g ∈ ΓE such
that g ̸∈ ΓO. Call such an element admissible and note that g ∈ Comm(ΓO).
C. Maclachlan et al.: Some remarks on group actions on hyperbolic 3-manifolds 9
Lemma 3.6. Suppose that Γ1E contains a principal congruence subgroup Γ(O(I)) and that
g is an admissible element of ΓE . Then gΓ(O(I))g−1 is commensurable with, but distinct
from, Γ(O(I)).
Proof. Commensurability is straightforward since g ∈ ΓE . If gΓ(O(I))g−1 = Γ(O(I)),
then g ∈ ΓO by Lemma 3.4. But this contradicts the admissibility of g.
4 Proof of Theorem 1.1
We now focus on constructing explicit examples in dimension 3 using the material from
Section 3.
4.1 General construction
Let k be a number field with exactly one complex place. It will be convenient in the
construction that follows to show that, for certain k, maximal arithmetic subgroups Γ of
Isom(H3) are Kleinian groups, i.e. Γ = Γ+.
Lemma 4.1. If [k : Q] is odd or [k : Q] is even but k has no subfield of index 2, then any
maximal arithmetic Kleinian group defined over k will be maximal in Isom(H3).
Proof. Suppose that Γ is a maximal arithmetic Kleinian group which is properly contained
in Γ0, a maximal discrete subgroup of Isom(H
3). Note that it follows that [Γ0 : Γ] = 2,
otherwise Γ is properly contained in Γ+0 which contradicts maximality.
Let Γ(2) denote the subgroup of Γ generated by squares of elements of Γ. Then Γ(2)
is a finite index characteristic subgroup of Γ, and so is normal in Γ0. Hence the orbifold
H3/Γ(2) admits an orientation-reversing isometry.
By [23, Theorem 8.3.1], the field k can be identified with the field Q(trγ : γ ∈ Γ(2)),
and by [27, Proposition 3.4], the existence of the orientation-reversing isometry on H3/Γ(2)
implies that k is stable under complex conjugation. As k is a field with one complex place,
it follows that [k : k ∩ R] = 2. This contradicts the choice of k.
Proof of Theorem 1.1: In the light of Lemma 4.1, we now assume that [k : Q] is odd, and
let B be a quaternion algebra defined over k, ramified at all real places. The discussion in
Subsection 3.1 immediately implies that all arithmetic hyperbolic 3-manifolds arising from
B/k are closed. We make an additional assumption about Ramf (B) to ensure torsion-
freeness in principal congruence subgroups.
Thus let R denote the finite set of prime ideals P of k such that, either P ∈ R is a
dyadic prime of k or, if pZ = P ∩ Z, then p is ramified in k. We assume that Ramf (B)
contains R.
Let O be a maximal order in B so that ΓO is a maximal arithmetic Kleinian group and,
by Lemma 4.1, also maximal in Isom(H3). Let E ⊂ O be an Eichler order of square-
free level S, chosen so that ΓE is maximal (e.g. as in Theorem 3.2). Then as noted in
Subsection 3.4, Γ1E contains a principal congruence subgroup Γ(O(I)) where I is defined
by S. By definition S is disjoint from Ramf (B) (and so from R), so it follows from
Proposition 3.5 that Γ(O(I)) is torsion-free.
Choose an admissible element g in ΓE (which as noted in Subsection 3.4 exists and
is an element of Comm(ΓO)). By Lemma 3.4, Γ(O(I)) is a normal subgroup of finite
10 Art Discrete Appl. Math. 5 (2022) #P3.05
index in ΓO. Using the fact (recall Subsection 3.3) that Γ
1
E is a normal subgroup of ΓE we
deduce:
gΓ(O(I))g−1 ⊂ gΓ1Eg−1 = Γ1E ⊂ Γ1O ⊂ ΓO.
Furthermore, by Lemma 3.6, gΓ(O(I))g−1 ̸= Γ(O(I)). Now apply Proposition 2.1 with
Γ0 = ΓO and Γ1 = Γ(O(I)).
By Theorem 3.2 we can choose infinitely many sets S, giving infinitely many examples
where Theorem 1.1 holds in a fixed commensurability class. From Subsection 3.1, the
commensurability class of ΓO is determined by the isomorphism class of B which, in turn,
is determined by its ramification set. Hence we have an infinite number of choices of
Ramf (B) subject to the restriction that R ⊂ Ramf (B), and Theorem 1.1 now follows.
4.2 Specific examples
We now refine the construction in Subsection 4.1 to provide more specific examples of
finite groups. In particular we will be able to gain extra control in the construction of a
normal subgroup ∆ as in the proof of Proposition 2.1, and this will allow us to get control
of G1 and G2.
We fix k = Q(x) where x3+x−1 = 0 so that k has one complex place, has discriminant
−31, and has class number 1. Indeed, in what follows, k can be any cubic number field k
with one complex place having class number 1. Let B be a quaternion algebra defined over
k with Ramf (B) = {Q} where the prime ideal Q does not divide any prime p that splits
completely to k (e.g. in this case we can take Q to be the unique prime dividing 5 of norm
53).
Let p ∈ Z be an odd prime that splits completely to k: that there are infinitely many
such primes p is a well-known consequence of Cebotarev Density theorem (see for example
[23, Chapter 0]). Let P ⊂ Rk be a prime dividing such a p, and so in particular NP = p.
Define the two-sided integral ideal I = PO. This can be done locally as follows: For
all k-prime ideals Q let OQ = M2(RQ), and then set IJ = OJ for all primes J ̸= P
and IP = πOP . The Eichler order E is then defined locally by EJ = OJ for J ̸= P and
EP = TP as at (3.1) so that S = {P} and Γ1E contains Γ(O(I)). Since p is odd and p ̸= 31,
P is unramified in k | Q, and Proposition 3.5 applies to show that Γ(O(I)) is torsion-free.
Applying Remark 3.3, the group ΓE is maximal. Indeed, an examination of Borel’s
construction [8] and [23, Chapters 6.7 and 11.4] provides an element α ∈ N(E) with
g = Pρ(α) so that locally at P , g acts as conjugation by the element
(
0 1
πP 0
)
, that is to
say, it acts locally on Γ(O(I)) by (cf. the discussion in Subsection 3.2)
g
(
1 + aπP bπP
cπP 1 + dπP
)
g−1 =
(
1 + dπP c
bπ2P 1 + aπP
)
. (4.1)
With this in place, we now define a two-sided O-ideal I ′ by I ′J = OJ for J ̸= P and
I ′P = π
2
POP so that Γ(O(I))∩gΓ(O(I))g−1 ⊃ Γ(O(I ′)). Now by Lemma 3.4, Γ(O(I ′))
is a normal subgroup of ΓO. Thus the manifold X = H
3/Γ(O(I ′)) admits free actions by
the groups G1 = Γ(O(I))/Γ(O(I ′)) and G2 = gΓ(O(I))g−1/Γ(O(I ′)).
We now identify the groups G1 and G2 explicitly. Note first that since B is unramified
at P we have:
Γ1O/Γ(O(I ′)) ∼= PSL(2, RP/π2PRP)
C. Maclachlan et al.: Some remarks on group actions on hyperbolic 3-manifolds 11
and G1 is the kernel of the reduction homomorphism PSL(2, RP/π
2
PRP) →
PSL(2, RP/πPRP). Since NP = p, G1 is an elementary abelian group of order p3
generated by the images of the matrices
(
1 πP
0 1
)
,
(
1 0
πP 1
)
,
(
1 + πP 0
0 1− πP
)
.
The group G2 has the same order p
3 as G1. To identify G2, we consider the reduction of
gΓ(O(I))g−1 locally modulo π2P . From (4.1), the image modulo the ideal π2PRP consists
of matrices of the form
(
1 + dπP c
0 1 + aπP
)
. From this description it is easy to check
that G2 is non-abelian and contains the element
(
1 1
0 1
)
of order p2. We summarize this
discussion as follows.
Corollary 4.2. For infinitely primes p, we can find a closed hyperbolic 3-manifold Xp such
that Xp admits free, orientation-preserving actions by finite groups of isometries, G1 and
G2 such that
(1) G1 and G2 are finite groups of order p3 with Xp/G1 ∼= Xp/G2.
(2) G1 is isomorphic to an elementary abelian group of order p3, and G2 is the unique
non-abelian group of order p3 which contains an element of order p2.
By varying the prime Q ramifying the quaternion algebra B/k we also obtain infinitely
many commensurability classes of manifolds as in Corollary 4.2. This again follows from
an application of the Cebotarev Density theorem which provides infinitely many rational
primes whose inertial degrees are greater than 1.
5 A rational homology 3-sphere and fibered examples
In the construction of the examples stated in Corollary 1.2, we will make use of the cubic
number field k = Q(x) where x3 − x2 + 1 = 0. This has one complex place, discriminant
−23, and class number 1. Let B be the quaternion algebra defined over k with Ramf (B) =
{Q} where Q is the unique k-prime ideal of norm 5. This determines the commensurability
class of the Weeks manifold (see [23, Chapters 4.8.3 and 9.8.2]), which is the smallest
volume closed orientable hyperbolic 3-manifold [16].
As we describe below, what is important for us is that if O ⊂ B is the unique (up to
B∗-conjugacy) maximal order, the group Γ1O contains subgroups of index 24 that are con-
gruence subgroups of certain levels which are the fundamental groups of a fibered hyper-
bolic 3-manifold or of a rational homology 3-sphere. In Section 9 we include some Magma
[10] computations, which shows, amongst other things, that Γ1O has 11 conjugacy classes
of subgroups of index 24, and unique conjugacy classes with abelianization Z⊕Z/11Z and
Z/7Z⊕ Z/42Z. We will make use of these in what follows. Note that the presentation for
Γ1O that is used in the Magma calculations comes from a description of the orbifold H
3/Γ1O
as (3, 0) Dehn surgery on the knot 52 (see [11, Subsection 5.4]).
5.1 Fibered examples
From [11, Proof of Lemma 9.3], the manifold M arising as the double cover of 0-surgery
on the knot 62 has fundamental group that is a subgroup of Γ
1
O of index 24 and has
12 Art Discrete Appl. Math. 5 (2022) #P3.05
H1(M,Z) ∼= Z ⊕ Z/11Z. As we remarked upon above, Γ1O has a unique such subgroup
(up to conjugacy), which we denote by Γ. As shown in Section 9, Magma computes the
core C of Γ in Γ1O and is the kernel of a homomorphism onto PSL(2,F23), the finite simple
group of order 6072. As is also discussed in [11, Subsection 9.1] there are two k-primes
of norm 23, one of which is ramified in k | Q (recall the discriminant is −23) and one
unramified. Denote these primes by P1 and P2 respectively, and these give rise to the 2-
sided integral ideals I1 = P1O and I2 = P2O. Since B is unramified at both of P1 and
P2, it follows that Γ1O/Γ(O(Ij)) ∼= PSL(2,F23) for j = 1, 2. Putting all of this together
we may deduce that C = Γ(O(Ij)) for one of j = 1, 2. We will not need to explicitly
identify which one, and simply denote the relevant subgroup as Γ(O(I)). In particular,
since H3/Γ(O(I)) → M = H3/Γ it is fibered over the circle. We now apply Proposition
2.1 together with Lemma 4.1, using an admissible element from ΓE where E is the Eichler
order of level S = {Pj} for j = 1 or 2 (using Remark 3.3).
To pass to infinitely many examples, we can use the principal congruence subgroups
Γ(O(In)) for n ≥ 2 an integer. Arguing as in Subsection 4.2, in particular the local
conjugation given by (4.1), shows that Γ(O(In+1)) ⊂ Γ(O(In)) ∩ gΓ(O(In))g−1 and so
the existence of infinitely many fibered examples follows since H3/Γ(O(In)) is fibered for
all n ≥ 1.
Remark 5.1. At present we do not know how to produce infinitely many commensurability
classes. Given [2] we know that every closed hyperbolic 3-manifold has a finite cover that
fibers over the circle. What is needed is that the finite covers can be identified as congruence
subgroups (as in the example above). In principal this should be possible, and note the work
of Agol and Stover [4] in this direction.
5.2 A rational homology 3-sphere
As can be checked there is a unique k-prime ideal P of norm 7, which is unramified in
B. Taking I to be the two-sided integral ideal PO it follows (as in Subsection 4.2) that
Γ1O/Γ(O(I)) ∼= PSL(2,F7). In particular, the Magma computations in Section 9 shows
that there is a unique index 24 subgroup, l[9], which has core a normal subgroup of
index 168 with quotient group being simple. Hence the group l[9] is a congruence
subgroup of Γ1O containing Γ(O(I)) of index 7. From Section 9, we also find that the
abelianization of Γ(O(I)) is Z/7Z ⊕ Z/7Z ⊕ Z/42Z. Hence there is a unique homo-
morphism ϕ : Γ(O(I)) → Z/7Z ⊕ Z/7Z ⊕ Z/7Z. Now arguing as in Subsection 4.2, it
follows that Γ(O(I))/Γ(O(I2)) ∼= Z/7Z⊕Z/7Z⊕Z/7Z and so we take as the manifold
X = H3/Γ(O(I2)). Choosing an admissible element g ∈ ΓE where E is the Eichler or-
der of level S = {P} (using Remark 3.3), the local conjugation argument given by (4.1)
again shows that Γ(O(I2)) ⊂ Γ(O(I))∩gΓ(O(I))g−1. Applying Proposition 2.1 together
with Lemma 4.1 and note from Section 9 that X is a rational homology 3-sphere since the
abelianization of Γ(O(I2)) (viewed as kerϕ) is still finite (albeit gigantic!). This completes
the proof.
6 Proof of Theorem 1.3
Proposition 3.5 shows that the groups Γ(O(I)) are typically torsion-free. The proof of
Theorem 1.3 will exploit a situation where these groups are not torsion-free, and apply
Proposition 2.1 (and see Remark 2.2 following it). To construct examples in this setting,
C. Maclachlan et al.: Some remarks on group actions on hyperbolic 3-manifolds 13
we begin with some preliminary discussion that will help locate elements of order 2 in
principal congruence subgroups.
6.1 Elements of order 2 in Γ1
O
First we recall the following (see [23, Theorem 12.5.4] for details). Let B be a quaternion
algebra over a number field k, let L = k(i) and let O be a maximal order in B. Then
P(B1) < PSL(2,C) contains an element of order 2 if and only if i ̸∈ k and no finite place
at which B is ramified, splits in L | k. Moreover, Γ1O will contain an element of order 2
unless B is unramified at all finite places, and every dyadic place of k splits in L | k.
Indeed, if we consider L →֒ B being given as the subfield k(u) ⊂ B where u is the
image of i, then for a suitable choice of maximal order O, u (or simply abusing notation i)
will project to an element of order 2 in Γ1O.
6.2 Explicit commensurability classes of examples
We now fix k = Q(
√
−2), with ring of integers Rk and p a rational prime with p ≡ 3
mod 8. Hence, pRk = PpP ′p. Let Bp/k be the quaternion algebra ramified at exactly the
places corresponding to Pp and P ′p. This can be described explicitly by the Hilbert Symbol
(see [23, Chapter 2.1])
(−1 , −p
k
)
.
In addition, it can be checked that a maximal order Op can be explicitly described as the
Rk-submodule of Bp given by Rk[1, i, (i+ j)/2, (1 + ij)/2].
Given the discussion in Subsection 6.1 with L = k(i) = Q(eπi/4) we see that L
embeds in Bp. In addition, RL embeds in Op. The reason for this is that it embeds in some
maximal order by [15], and since k has class number 1 there is a unique type of maximal
order in all of the quaternion algebras Bp.
For convenience we suppress the subscript p in what follows. By construction, Γ1O has
an element of order 2 given as the image of i. Let JL (resp. J) denote the principal L-
ideal (resp. k-ideal) generated by i − 1 (resp.
√
−2). Notice that
√
−2 = (1 − i)v where
v = (−
√
2+
√
−2)
2 ∈ R∗L, so
√
−2 ∈ JL, from which it follows that JL ∩ Rk = J . Note
also that 2 is totally ramified in L | k (i.e. 2RL = P42 for some ideal P2 ⊂ RL of norm
2), and (i − 1)RL = JRL. Now take I ⊂ O to be the two-sided ideal JO. The previous
discussion shows that i− 1 ∈ I , and so i determines an element of order 2 in Γ(O(I)).
We also note that as in the discussion in Subsection 3.2, there is an orientation-reversing
involution (an extension of complex conjugation in C to H3) that normalizes Γ1O. This
follows since, by the explicit nature of O, the involution on B given by the extension of
complex conjugation preserves O.
Proof of Theorem 1.3: Using the notation above, we see that B is a division algebra, and so
Γ1O is cocompact. Since there are infinitely many primes p ≡ 3 mod 8, there are infinitely
many isomorphism classes of quaternion algebras and so as before, Subsection 3.1 provides
infinitely many commensurability classes of arithmetic hyperbolic 3-manifolds.
As discussed above, Γ1O is normalized by an orientation-reversing involution, so that
unlike the proof of Theorem 1.1, the group ΓO is maximal in PSL(2,C) but is not maximal
in Isom(H3). Denote the maximal group in Isom(H3) containing ΓO by GO (so that
14 Art Discrete Appl. Math. 5 (2022) #P3.05
G+O = ΓO). With I as above, the group Γ(O(I)) will again play the role of Γ1. The
explicit description given of I and O implies that Γ(O(I)) is also normal in GO.
We now argue as in Subsection 4.2: apply Lemma 3.6 for a suitable choice of Eichler
order E ; namely the Eichler order of square-free level {J}, and just as in Subsection 3.4,
Γ1E contains Γ(O(I)). Remark 3.3 shows that ΓE is a maximal arithmetic Kleinian group.
Choose an admissible element g ∈ ΓE and apply Proposition 2.1. As in Subsection 4.2,
Γ(O(I2)) ⊂ Γ(O(I)) ∩ gΓ(O(I))g−1. This remains normal in GO since I2 = 2O. If
Γ(O(I2)) is torsion-free we use this group as we did previously. Otherwise we can pass to
a torsion-free subgroup and then to its core ∆ in GO, as in the proofs of Proposition 2.1
and Theorem 1.1.
With this we have constructed a hyperbolic 3-manifold X = H3/∆ that admits actions
by groups of orientation-preserving isometries G1 and G2 acting on X with fixed points and
with X/G1 isometric to X/G2. That G1 and G2 are not conjugate subgroups of Isom(X),
or equivalently Γ(O(I)) and gΓ(O(I))g−1 are not conjugate in GO, follows as in the proof
of Theorem 1.1 since Γ(O(I)) is normal in GO. This completes the proof.
Remark 6.1. With reference to the proof of Theorem 1.3, it seems likely that the group
Γ(O(I2)) is torsion-free, but we will not pursue this here.
7 Dimension 2
We now discuss a specific analogue of Theorem 1.1 and the arithmetic constructions given
in Subsection 4.2 in the context of hyperbolic surfaces, or equivalently, on emphasizing
complex structures, Riemann surfaces. We prove the following.
Theorem 7.1. For infinitely many primes p, there exist compact hyperbolic (Riemann)
surfaces Xp with the property that there are finite p-groups G1,p and G2,p which are non-
isomorphic, which act freely on Xp with X/G1,p ∼= X/G2,p (or equivalently X/G1,p and
X/G2,p are conformally equivalent Riemann surfaces).
Borel’s work [8] applies in this setting, and the structure of maximal arithmetic lattices
in Isom(H2) is entirely analogous to what is described in Subsections 3.3, 3.4. So, although
the proof given below can be carried out in more generality, we believe it is instructive
to simply focus on one commensurability class: that given by the indefinite quaternion
algebra B/Q ramified at the primes 2 and 3. If O ⊂ B is a maximal order, then it is
known that Γ+O = ΓO ∩ PSL(2,R) is the (2, 4, 6) Fuchsian triangle group and ΓO is the
group generated by reflections in the faces of this triangle (see [23, Chapter 13.3]). Note the
difference with the Fuchsian case in comparison to the case of arithmetic Kleinian groups is
that N(O) can contain elements of determinant −1, thereby giving elements in PGL(2,R).
However, a version of Remark 3.3 continues to hold in this setting.
Proof. In the notation established above, we let Γ0 = ΓO, and for p ∈ Z a prime different
from 2, 3 we construct the principal congruence subgroup Γ(O(I(p))) where I(p) is the
two-sided integral ideal defined as pO. We now follow the argument in Subsection 4.2:
using a version of Remark 3.3, we may build an Eichler order Ep of level S = {p}, a
maximal group ΓEp , an admissible element gp, and then follow the argument in Subsec-
tion 4.2 to build the required Riemann surface Xp with the free actions of groups G1 and
G2. That the covering groups G1 and G2 are finite p-groups (of order p
3) follows as in
Subsection 4.2.
C. Maclachlan et al.: Some remarks on group actions on hyperbolic 3-manifolds 15
Example 7.2. We take the case of p = 5 in Theorem 7.1. In this case, it can be shown
that Γ1O is a Fuchsian group of signature (0; 2, 2, 3, 3) (see for example [30]) which has
co-area 2π/3. By construction, Γ1O/Γ(O(I(5))) ∼= PSL(2, 5), and using the co-area com-
puted above, determines a 60-fold Riemann surface cover Σ = H2/Γ(O(I(5))) of H2/Γ1O.
Hence Σ has genus 7. The surface X5 we want is then a (Z/5Z)
3 cover of Σ, so of genus
751, which also admits a free action of a non-abelian 5 group of order 53 with quotient
isometric to Σ.
8 Higher dimensions
As mentioned in Section 2 we will only sketch the proof of an analogous result to Theo-
rem 1.1 in higher dimensions. This will make use of so-called arithmetic groups of simplest
type (we refer the reader to [26, Subsection 6.8] for a fuller discussion of these arithmetic
lattices). For convenience, we restrict to one particular family in each dimension n ≥ 4, the
generalizations will be clear. We emphasize that this is a sketch of a proof, and so we will
not designate with "Theorem" as some additional discussion of maximal groups is required
to give a complete proof:
For each n ≥ 4, there are infinitely many non-commensurable closed orientable hyperbolic
n-manifolds X , with the property that there are finite groups G1 and G2 satisfying:
(1) G1 and G2 act freely by orientation-preserving isometries on X with X/G1 ∼=
X/G2.
(2) |G1| = |G2|, but G1 and G2 are not conjugate in Isom(X).
Sketch Proof: We recall some background on certain arithmetic subgroups of Isom(Hn).
Let d be a square-free positive integer, and let fn,d denote the quadratic form:
fn,d = x
2
0 + x
2
1 + . . .+ x
2
n−1 −
√
dx2n.
This has signature (n, 1), and after applying the non-trivial Galois automorphism σ given
by
√
d 7→ −
√
d the resultant quadratic form fσn,d has signature (n+1, 0); i.e. the quadratic
form fn,d is equivalent over R to the quadratic form Jn = x
2
0 +x
2
1 + . . .+x
2
n−1 −x2n, and
the quadratic form fσn,d is equivalent over R to x
2
0 + x
2
1 + . . .+ x
2
n−1 + x
2
n.
Let Fn,d be the symmetric matrix associated to the quadratic form fn,d and let O(fn,d)
(resp. SO(fn,d)) denote the linear algebraic groups defined over k described as:
O(fn,d) = {X ∈ GL(n+ 1,C) : XtFn,dX = Fn,d} and
SO(fn,d) = {X ∈ SL(n+ 1,C) : XtFn,dX = Fn,d}.
For a subring L ⊂ C, we denote the L-points of O(fn,d) (resp. SO(fn,d)) by O(fn,d, L)
(resp. SO(fn,d, L)). Let Rd ⊂ Q(
√
d) denote the ring of integers.
Note that, given this set-up, there exists T ∈ GL(n+1,R) such that T−1O(fn,d,R)T =
O(n, 1), in which case, Isom(Hn) can be identified with the group O+(Jn,R) = O
+(n, 1),
which is the subgroup of O(n, 1) preserving the upper-half sheet of the hyperboloid
Jn = −1. A similar discussion holds for T−1SO(fn,d,R)T = SO(n, 1) and groups
of orientation-preserving isometries. In particular this conjugation provides subgroups
Λ ⊂ O(fn,d, Rd) and Λ+ ⊂ SO(fn,d, Rd) whose images lie in O+(n, 1) and SO+(n, 1)
respectively.
16 Art Discrete Appl. Math. 5 (2022) #P3.05
A subgroup Γ < Isom(Hn) commensurable with the image in Isom(Hn) of the sub-
group of O(fn,d, Rd) (under the conjugation map described above) is an example of an
arithmetic lattice of simplest type. The corresponding arithmetic hyperbolic n-manifold
M = Hn/Γ is also called arithmetic of simplest type. By construction, all the arithmetic
lattices of simplest type we have described above are cocompact.
For an ideal I ⊂ Rd, let Γ(I) denote the principal congruence subgroup of O(fn,d, Rd)
obtained as the kernel of the homomorphism:
πI : O(fn,d, Rd) → O(fn,d, Rd/I),
and note that so long as I is not a dyadic prime ideal, Γ(I) ⊂ SO(fn,d, Rd).
We now fix the ideal I to be considered, namely let p ∈ Z be an odd prime that is
inert to Rd; i.e. pRd remains a prime ideal which we denote by P . In this case, Γ(P) is
torsion-free (see [26, Subsection 4.8]), and the groups Λ and Γ(P)∩Λ+ will play the roles
of Γ0 and Γ1 in Proposition 2.1.
As before, the key point now is a detailed understanding of maximal arithmetic lat-
tices in this setting. As in the case of arithmetic Kleinian groups, these maximal arithmetic
groups arise as normalizers of certain number theoretically defined arithmetic lattices (us-
ing Bruhat-Tits theory), and are again congruence subgroups (see [9] and also [3] and [6]
that deal explicitly with the case of O+(n, 1)). In particular our group Λ is a maximal
discrete subgroup of O+(n, 1) and using the description of maximal discrete groups in the
commensurability class of Λ given in [9] it is possible to find an element g as required by
Proposition 2.1.
9 Magma calculations
In what follows g is the group Γ1O of Section 5. The presentation was computed using
SnapPy [13]. The group Γ from Subsection 5.1 is l[2], and the index 24 subgroup from
Subsection 5.2 is l[9]. The routine CosetAction produces a finite image group i and
a kernel k which is the core of the relevant subgroup. The routine pQuotient is used to
compute the kernel of a homomorphism a to the elementary abelian 7-group (Z/7Z)3.
> g<a,b>:=Group<a,b|b^3,a^2*b^-1*a^-2*b^-1*a^2*b*a^-1*b>;
> print AbelianQuotientInvariants(g);
[ 3 ]
> l:=LowIndexSubgroups(g,<24,24>);
> print #l;
11
> print AbelianQuotientInvariants(l[1]);
[ 30 ]
> print AbelianQuotientInvariants(l[2]);
[ 11, 0 ]
> print AbelianQuotientInvariants(l[3]);
[ 3, 6, 0 ]
> print AbelianQuotientInvariants(l[4]);
[ 2, 2, 6 ]
> print AbelianQuotientInvariants(l[5]);
[ 5, 30 ]
C. Maclachlan et al.: Some remarks on group actions on hyperbolic 3-manifolds 17
> print AbelianQuotientInvariants(l[6]);
[ 3, 6, 6 ]
> print AbelianQuotientInvariants(l[7]);
[ 66 ]
> print AbelianQuotientInvariants(l[8]);
[ 3, 6, 6 ]
> print AbelianQuotientInvariants(l[9]);
[ 7, 42 ]
> print AbelianQuotientInvariants(l[10]);
[ 3, 6, 0 ]
> print AbelianQuotientInvariants(l[11]);
[ 2, 2, 6 ]
> f,i,k:=CosetAction(g,l[2]);
> print Order(i);
6072
> IsSimple(i);
true
> f,i,k:=CosetAction(g,l[9]);
> print Order(i);
168
> IsSimple(i);
true
> print AbelianQuotientInvariants(k);
[ 7, 7, 42 ]
> K:=Rewrite(g,k);
> F,a,b:=pQuotient(K,7,1:Print:=1);
Lower exponent-7 central series for K
Group: K to lower exponent-7 central class 1 has order 7^3
> K1:=Kernel(a);
> print AbelianQuotientInvariants(K1);
[ 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,
13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,
13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,
13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,
13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13,
13, 13, 13, 13, 13, 13, 13, 13, 2743, 2743, 2743, 2743,
2743, 2743, 2743, 2743, 2743, 2743, 2743, 2743, 2743,
2743, 79547, 79547, 79547, 79547, 79547, 79547, 79547,
79547, 79547, 79547, 79547, 79547, 79547, 79547, 79547,
79547, 79547, 79547, 79547, 79547, 79547, 79547, 79547,
79547, 79547, 79547, 79547, 79547, 79547, 79547, 79547,
79547, 79547, 79547, 79547, 79547, 79547, 79547, 79547,
3897803, 3897803, 23386818 ]
> f,i,k:=CosetAction(g,l[1]);
> print Order(i);
1320
18 Art Discrete Appl. Math. 5 (2022) #P3.05
> f,i,k:=CosetAction(g,l[3]);
> print Order(i);
2204496
> f,i,k:=CosetAction(g,l[4]);
> print Order(i);
504
> f,i,k:=CosetAction(g,l[5]);
> print Order(i);
504
> f,i,k:=CosetAction(g,l[6]);
> print Order(i);
2204496
> f,i,k:=CosetAction(g,l[7]);
> print Order(i);
6072
> IsSimple(i);
true
> a:=AbelianQuotientInvariants(k);
> print a;
[ 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 161,
161, 161, 161, 966, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
> print Multiplicity(a,0);
44
> f,i,k:=CosetAction(g,l[8]);
> print Order(i);
2204496
> f,i,k:=CosetAction(g,l[10]);
> print Order(i);
2204496
> f,i,k:=CosetAction(g,l[11]);
> print Order(i);
504
Referring to Subsection 5.1, there is an index 24 subgroup of Γ1O whose core is the principal
congruence subgroup arising from "the other" k-prime of norm 23. This corresponds to the
subgroup l[7] of the Magma output above. Although this gives rise to a manifold that
is also a rational homology 3-sphere, the manifold with fundamental group which is the
principal congruence subgroup (i.e. the core) has first Betti number equal to 44 (as shown
in the Magma output above), and so does not provide rational homology 3-sphere examples
as in Corollary 1.2.
ORCID iDs
Alan Reid https://orcid.org/0000-0003-0367-9453
Antonio Salgueiro https://orcid.org/0000-0002-7290-0708
C. Maclachlan et al.: Some remarks on group actions on hyperbolic 3-manifolds 19
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.06
https://doi.org/10.26493/2590-9770.1391.f46
(Also available at http://adam-journal.eu)
On automorphisms of Haar graphs
of abelian groups
Ted Dobson*
FAMNIT and IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia
Received 21 October 2020, accepted 16 February 2022, published online 18 July 2022
Abstract
Let G be a group and S ⊆ G. In this paper, a Haar graph of G with connection set S
has vertex set Z2 × G and edge set {(0, g)(1, gs) : g ∈ G and s ∈ S}. Haar graphs are
then natural bipartite analogues of Cayley digraphs, and are also called BiCayley graphs.
We first examine the relationship between the automorphism group of the Cayley digraph
of G with connection set S and the Haar graph of G with connection set S. We establish
that the automorphism group of a Haar graph contains a natural subgroup isomorphic to the
automorphism group of the corresponding Cayley digraph. In the case where G is abelian,
we show there are exactly four situations in which the automorphism group of the Haar
graph can be larger than the natural subgroup corresponding to the automorphism group
of the Cayley digraph together with a specific involution, and analyze the full automor-
phism group in each of these cases. As an application, we show that all s-transitive Cayley
graphs of generalized dihedral groups have a quasiprimitive automorphism group, can be
constructed from digraphs of smaller order, or are Haar graphs of abelian groups whose
automorphism groups have a particular permutation group theoretic property.
Keywords: Groups, graphs.
Math. Subj. Class.: 05C15, 05C10
A Haar graph of a group G with connection set S has vertex set Z2 × G and edge
set {(0, g)(1, gs) : g ∈ G and s ∈ S}, where S ⊆ G. Haar graphs are natural bipartite
analogues of Cayley digraphs, and these graphs have appeared in a variety of contexts
and under a variety of names. To the author’s knowledge, Haar graphs were introduced
in [18], where some of their elementary properties were studied, including some results on
isomorphic Haar graphs. Recently there has been a fair amount of work on the isomorphism
problem for Haar graphs [4, 5, 21–23, 25, 44], some of it motivated by applications (see
*The author is indebted to the referees for their careful reading of the paper, and their suggestions for improve-
ments
E-mail address: ted.dobson@upr.si (Ted Dobson)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.06
[7, 27, 41]). Most of this work falls into the two categories of considering the isomorphism
problem for graphs of small valency or determining the structure of groups which are BCI-
groups. Intuitively, these are groups where isomorphism is determined in the “nicest”
possible way.
Our work in this paper was motivated by the isomorphism problem, but in the end we
will not consider this problem here. It is well known that the isomorphism problem for a
Cayley digraph Γ of a group G depends upon the conjugacy classes of natural subgroups of
Aut(Γ) (see [6, Lemma 3.1]). A similar result also holds for Haar graphs [26, Lemma 2.2]
or [4, Theorem C]. Thus, information about a Cayley digraph’s automorphism group is
crucial in determining which other Cayley digraphs of G are isomorphic to it, and similarly,
for Haar graphs. Typically, more information is known about the automorphism group of
Cayley digraphs of a group G (see for example [9,11,13,17]) than of Haar graphs of G, and
for every Haar graph there is a corresponding Cayley digraph with the same connection set.
It is thus of natural interest to determine the relationship, if any, between the automorphism
group of a Haar graph and the automorphism group of its corresponding Cayley digraph
(again, much more is known about the automorphism groups of Cayley digraphs). This is
the main focus of the work in this paper.
We will show in Corollary 2.16 that the automorphism group of a Haar graph of an
abelian group A falls into four natural families. In two of these families, the automorphism
groups are a wreath product and so can be found provided one knows the automorphism
groups of the graphs involved in the wreath products (which are always of smaller order
and so presumably easier to find). In the third family, the automorphism group of the Haar
graph of A is determined, up to conjugacy by |A| natural and explicitly defined permu-
tations, by the automorphism group of the corresponding Cayley digraph. For the fourth
and final family the situation is more interesting in that there does not seem to be a natural
or obvious relationship between the automorphism group of the Haar graph and its cor-
responding Cayley digraph. We do, though, give a group theoretic construction for all of
these graphs, but unfortunately the group theoretic information needed for the construction
does not seem easy to obtain. We should also mention that there is some related work on
finding automorphism groups of Haar graphs - see [46].
As an application, we next consider the implications of the above automorphism group
results to s-arc-transitive Haar graphs. In particular, we characterize s-arc-transitive Cay-
ley graphs of generalized dihedral groups with abelian subgroup of odd order. In all cases
except one (which corresponds to the case in the preceding paragraph where there was
no obvious relationship between the automorphism group of the Haar graph and its cor-
responding Cayley digraph), such s-arc-transitive graphs can be constructed from other
highly symmetric graphs and digraphs of smaller order without the use of graph covers.
There is another problem in the literature related to this work. For a graph Γ, its canon-
ical double cover is the graph K2 × Γ and is denoted B(Γ). So V (B(Γ)) = Z2 × V (Γ)
and E(B(Γ)) = {(0, x)(1, y) : xy ∈ E(Γ)}. If Γ = Cay(G,S) is a Cayley graph, then
B(Γ) = Haar(G,S). Automorphism groups of B(Γ) for Γ a Cayley graph were first stud-
ied in [34] and subsequently by several other authors [28,35,39,43]. The main question is,
for a graph Γ (not necessarily a Cayley graph), is Aut(Γ) = Z2 × Aut(Γ)? If so, such a
graph is stable, and if not, unstable. Corollary 2.16, for example, refines [28, Theorem 3.2]
in the case where Γ is a Cayley graph of an abelian group. Our results also hold for di-
graphs, so one can consider the Haar graph construction for Cayley digraphs of a group G
to be a natural generalization of the canonical double cover of graphs to digraphs.
T. Dobson: On automorphisms of Haar graphs of abelian groups 3
Some words about notation should be mentioned. Haar graphs are special cases of
bi-Cayley graphs of G, which are usually defined as graphs that contain a semiregular
subgroup with two orbits that is isomorphic to G [2, 19, 33, 45]. Bi-Cayley graphs need
not be bipartite, with the prefix bi referring to the two orbits of the semiregular subgroup
isomorphic to G, not to the graph being bipartite, while Haar graphs are always bipartite.
Additionally, some authors refer to Haar graphs as bi-Cayley graphs with the prefix bi
presumably referring to the fact that they are bipartite. To make the terminology issues
more complex, some authors refer to bi-Cayley graphs as defined here as semi-Cayley
graphs. We prefer the term Haar graph to bi-Cayley graph as defined here (this definition
will be formally stated below to hopefully eliminate all ambiguity and henceforth we will
not use the term bi-Cayley graph), simply because this choice of terminology causes less
confusion in that there is only one use of the term “Haar graph” in the literature, at least as
far as the author knows!
1 Basic definitions and results on automorphism groups
All groups and graphs are finite. In this section, we define Cayley digraphs and Haar graphs
and determine a relationship between their automorphism groups. This section is mainly
concerned with Haar graphs of general finite groups G, not necessarily abelian.
Definition 1.1. Let G be a group and S ⊆ G. Define a Cayley digraph of G, denoted
Cay(G,S), to be the digraph with V (Cay(G,S)) = G and A(Cay(G,S)) = {(g, gs) :
g ∈ G, s ∈ S}. We call S the connection set of Cay(G,S).
Note that the map gL : G 7→ G given by gL(x) = gx is always an automorphism of
Cay(G,S) for every group G and connection set S. Thus the group GL = {gL : g ∈ G} ≤
Aut(Cay(G,S)) for every group G and connection set S. Here, if Γ is a graph or digraph,
then Aut(Γ) denotes its automorphism group. Also note that by [6, Proposition 2.1] if
α ∈ Aut(G), then α(Cay(G,S)) = Cay(G,α(S)). In Figure 1, we give an example of a
Cayley digraph of the cyclic group Z7.
•
0
•6
•5
•
4
•
3
• 2
•1''

oo
\\
KK
77

ww
SS //

gg
BB

BB
ww //
gg

SS
Figure 1: The Cayley digraph Cay(Z7, {1, 2, 4}).
4 Art Discrete Appl. Math. 5 (2022) #P3.06
Definition 1.2. Let G be a group and S ⊆ G. Define the Haar graph Haar(G,S) with
connection set S to be the graph with vertex set Z2 × G and edge set {(0, g)(1, gs) : g ∈
G and s ∈ S}.
Some authors use H(G,S) for Haar(G,S). We prefer the somewhat longer but more
descriptive notation. In Figure 2 we give Haar(Z7, {1, 2, 4}). We call Haar(G,S) the
Haar graph corresponding to Cay(G,S). The graph in Figure 2 is the Haar graph corre-
sponding to the Cayley digraph in Figure 1, and is the Heawood graph.
•(0, 0)
•(0, 1)
•(0, 2)
•(0, 3)
•(0, 4)
•(0, 5)
•(0, 6)
• (1, 0)
• (1, 1)
• (1, 2)
• (1, 3)
• (1, 4)
• (1, 5)
• (1, 6)
Figure 2: The Heawood graph as Haar(Z7, {1, 2, 4}).
Clearly every Haar graph is a bipartite graph. Notice also that it is very much allowed
that 1G ∈ S, while for Cayley digraphs this is usually not allowed. This is because the
effect of including 1G in the connection set of a Cayley digraph is to put a loop at each
vertex, and doing this does not usually affect the symmetry properties of Cayley digraphs
(e.g. adding a loop at each vertex does not change the automorphism group of a Cayley
digraph). In some situations, though, allowing 1G ∈ S for a Cayley digraph is not only
advantageous, but crucial (see for example [3]). In this paper we allow loops in Cayley
digraphs.
Definition 1.3. Throughout this paper, if Γ = Haar(G,S), then the natural bipartition
of V (Γ) will be denoted by B, where B = {B0, B1}, B0 = {(0, g) : g ∈ G}, and
B1 = {(1, g) : g ∈ G}.
Notice that the map ĝL : Z2 × G 7→ Z2 × G given by ĝL(i, j) = (i, gj) is an auto-
morphism of Haar(G,S) for every group G and connection set S, and corresponds to the
subgroup GL ≤ Aut(Cay(G,S)).
It is easily determined using results in [1] that Cay(Z7, {1, 2, 4}) has automorphism
group {x 7→ ax + b : a = 1, 2, 4, b ∈ Z7} which is a metacyclic group of order 21. The
Heawood graph is a Haar graph of Z7, as shown in Figure 2. While the Heawood graph
does have a metacyclic subgroup of order 21 corresponding to Aut(Cay(Z7, {1, 2, 4})), the
T. Dobson: On automorphisms of Haar graphs of abelian groups 5
automorphism group of the Heawood graph is actually Z2⋉PGL(3, 2) ∼= PGL(2, 7) which
has order 336 and is an almost simple group. Our first result shows that the automorphism
group of a Haar graph always has a natural subgroup isomorphic to the automorphism
group of its corresponding Cayley digraph. The Heawood graph example, though, shows
the automorphism group of the Haar graph may be much larger.
Lemma 1.4. Let G be a group, and γ ∈ SG. The map γ̂ : Z2 × G 7→ Z2 × G given by
γ̂(i, j) = (i, γ(j)) is an automorphism of Haar(G,S) if and only if γ is an automorphism
of Cay(G,S).
Proof. The permutation γ ∈ SG is in Aut(Cay(G,S)) if and only if whenever g ∈ G
and s ∈ S, γ(g, gs) = (γ(g), γ(gs)) ∈ A(Cay(G,S)). This occurs if and only if there
exists s′ ∈ S such that γ(gs) = γ(g)s′ which is true if and only if (0, γ(g))(1, γ(g)s′) =
(0, γ(g))(1, γ(gs)) ∈ E(Haar(G,S)). This last statement is true if and only if γ̂ ∈
Aut(Haar(G,S)).
We now give a standard notation for the automorphism of Haar(G,S) induced by an
automorphism of Cay(G,S) which will be used henceforth.
Definition 1.5. Let G be a group and S ⊆ G. Let γ ∈ Aut(Cay(G,S)). The auto-
morphism of Haar(G,S) induced by γ as in Lemma 1.4 will be denoted by γ̂. That
is, if γ ∈ Aut(Cay(G,S)) then the automorphism of Haar(G,S) corresponding to γ
is γ̂ : Z2 × G 7→ Z2 × G given by γ̂(i, j) = (i, γ(j)). If H ≤ Aut(Cay(G,S)), then we
define Ĥ = {ĥ : h ∈ H}. In particular, the natural semiregular subgroup of Haar(G,S)
isomorphic to G is denoted ĜL.
In this paper we focus on Aut(Haar(G,S)) in the special case when G is an abelian
group. The reason to restrict our attention to abelian groups is that the automorphism
groups of Haar graphs of abelian groups and nonabelian groups are different. As implicitly
used in [18], for an abelian group A, Aut(Haar(A,S)) contains the element ι : Z2 ×A 7→
Z2 × A given by ι(i, j) = (i+ 1,−j). The group ⟨ι, ÂL⟩ is transitive and so Haar graphs
of abelian groups have transitive automorphism group while Haar graphs of nonabelian
groups need not. See for example [15, Proposition 11]. For Haar graphs of abelian groups
we may use the automorphism ι defined above to say a little more.
Lemma 1.6. Let A be an abelian group, and S ⊆ A. Then
Z2 ⋉Aut(Cay(A,S)) ≤ Aut(Haar(A,S)).
Proof. As A is an abelian group, straightforward computations will show that the map
ι : Z2 × A 7→ Z2 × A given by ι(i, j) = (i + 1,−j) is an automorphism of Haar(A,S).
Set K = Aut(Cay(A,S)). By Lemma 1.4, K̂ ≤ Aut(Haar(A,S)).
Let a ∈ A. Then (a, a+s) ∈ A(Cay(A,S)) if and only if (a, a−s) ∈ A(Cay(A,−S)),
which occurs if and only if (a + s, a) ∈ A(Cay(A,−S)). So Cay(A,−S) can be ob-
tained from Cay(A,S) by reversing the direction of each arc in Cay(A,S). Now, (a, b) ∈
A(Cay(A,S)) if and only if (γ(a), γ(b)) ∈ A(Cay(A,S)) for every γ ∈ Aut(Cay(A,S)).
As (a, b) ∈ A(Cay(A,S)) if and only if (b, a) ∈ A(Cay(A,−S)), we see (γ(a), γ(b)) =
γ(a, b) ∈ A(Cay(A,S)) if and only if (γ(b), γ(a)) = γ(b, a) ∈ A(Cay(A,−S)). We
conclude Aut(Cay(A,S)) = Aut(Cay(A,−S)).
6 Art Discrete Appl. Math. 5 (2022) #P3.06
Define r : A → A by r(j) = −j. Then r ∈ Aut(A) and r(Cay(A,S)) = Cay(A,−S).
Also r−1 = r, and r(Cay(A,−S)) = Cay(A,S). Hence
Aut(Cay(A,S)) = rAut(Cay(A,−S))r.
Thus γ ∈ Aut(Cay(A,S)) if and only if the map j 7→ −γ(−j) is also contained in
Aut(Cay(A,S)). As ιγ̂ι(i, j) = (i,−γ(−j)), we see ι normalizes K̂. Then the group
⟨ι, K̂⟩ = Z2 ⋉Aut(Cay(G,S)) ≤ Aut(Haar(A,S)) as |ι| = 2 and ι normalizes K̂.
In the case where S = −S and Cay(A,S) is a graph, we have a slightly nicer result,
which is contained in the significantly stronger result [31, Lemma 4.2].
Lemma 1.7. Let A be an abelian group and S ⊆ A such that S = −S. Then
Z2 ×Aut(Cay(A,S)) ≤ Aut(Haar(A,S)).
Proof. Simply observe that if S = −S the map i 7→ −i is an automorphism of Cay(A,S)
and so the map (i, j) 7→ (i+ 1, j) is an automorphism of Haar(A,S).
One circumstance in which Aut(Haar(G,S)) is bigger than Z2⋉Aut(Cay(G,S)) is if
Cay(G,S) is connected but Haar(G,S) is not. For example, for n ≥ 2, Cay(Z2n, {±1})
is a 2n-cycle with automorphism group D2n, while Haar(Z2n, {±1}) is a disjoint union
of two 2n-cycles, with automorphism group isomorphic to Z2 ≀ D2n (the group wreath
product is given in Definition 2.3; here it is enough to observe that this group is bigger).
Notice that the vertex sets of these two 2n-cycles are not the sets B0 and B1. A perhaps
more extreme example is Cay(Z2n, S), where S is all odd elements of Z2n. The graph
Cay(Z2n, S) = Kn,n is connected, but Haar(Z2n, S) consists of two disjoint copies of
Kn,n. The necessary and sufficient condition for Haar(G,S) to be connected is SS
−1 =
{st−1 : s, t ∈ S} generates G and is given in [14, Lemma 2.3(iii)]. A more appealing
formulation for Haar graphs of abelian groups is the following:
Lemma 1.8. Let A be an abelian group. Haar(A,S) is disconnected if and only if
S ⊆ a+H for some subgroup H < A and a ∈ A.
Proof. If S ⊆ a + H for some H < A and a ∈ A then for s, t ∈ S we have s − t =
a + h1 − (a + h2) ∈ H < A for some h1, h2 ∈ H , and so Haar(A,S) is disconnected
by [14, Lemma 2.3(iii)]. If Haar(A,S) is disconnected, then ⟨s − t : s, t ∈ S⟩ = H < A
by [14, Lemma 2.3(iii)]. Fix a ∈ S, and let s ∈ S. Then a− s = −h ∈ H and s = a+ h.
Hence S ⊆ a+H .
Before proceeding, we will need some permutation group theoretic terms.
Definition 1.9. Let X be a set and K ≤ SX be transitive. A subset C ⊆ X is a block of
K if whenever k ∈ K, then k(C) ∩ C = ∅ or C. If C = {x} for some x ∈ X or C = X ,
then C is a trivial block. Any other block is nontrivial. The set C = {k(C) : k ∈ K} is a
partition of X , called a block system of K, and is nontrivial if C is nontrivial.
A basic fact about a graph Γ is that Aut(Γ) = Aut(Γ̄), where Γ̄ is the complement of
Γ. But the complement of a Haar graph is not a Haar graph. One could consider “bipartite
complements” (defined below) to avoid this problem, but it still need not be the case that the
automorphism group of the bipartite complement of a Haar graph Γ is the automorphism
group of Γ (although if B is a block system of the automorphism group of the bipartite
T. Dobson: On automorphisms of Haar graphs of abelian groups 7
complement of Γ we do have equality of automorphism groups under bipartite comple-
ments [10, Corollary 4]). Our next result is really an exercise and has certainly appeared as
a comment in the literature [38]. We state and prove this result due to its importance to the
work in this paper.
Lemma 1.10. Let Γ be a vertex-transitive bipartite graph with bipartition B = {B0, B1}.
If Γ is connected then B is a block system of Aut(Γ).
Proof. We prove the contrapositive, and so suppose there exists γ ∈ Aut(Γ) such that
γ(B) = B′ = {B′0, B
′
1} ̸= B. As B is a bipartition of Γ and γ ∈ Aut(Γ), γ(B) = B
′
is also a bipartition of Γ. Let C0 = B0 ∩ B
′
0, C1 = B0 ∩ B
′
1, C2 = B1 ∩ B
′
0, and
C3 = B1 ∩B
′
1. As B ̸= B
′, none of the sets Ci, i ∈ Z4, are empty. If Γ is connected, some
vertex v of C0 ⊂ B0 is adjacent to some vertex w of B1. As C0 is a subset of B0 and B
′
0,
w must be in both B1 and B
′
1, so in C3. But by a symmetrical argument, any vertex of C3
can only be adjacent to vertices in C0, and so there is no path in Γ from v to any vertex of
C2. Hence Γ is disconnected.
Definition 1.11. Let Γ be a bipartite graph with bipartition B = {B0, B1}. The bipartite
complement of Γ is the graph with vertex set V (Γ) and two vertices are adjacent if they are
in different bipartition classes and are not adjacent in Γ.
Corollary 1.12. Let G be a group and S ⊆ G. If Haar(G,S) and Haar(G,G \ S) are
both connected then Aut(Haar(G,S)) = Aut(Haar(G,G \ S)).
Proof. By Lemma 1.10, B is a block system of both Aut(Haar(G,S)) and Haar(G,
G \ S)), and so by [10, Corollary 4] Aut(Haar(G,S)) = Aut(Haar(G,G \ S)).
It is not true that if both Haar(A,S) and Haar(A,A − S) are disconnected then their
automorphism groups are the same. Let A = Z2n, and S = ⟨2⟩ so A − S = 1 + ⟨2⟩.
Then both S + (−S) and (A − S) + [−(A − S)] = ⟨2⟩ and so by Lemma 1.8 neither
are connected. Both of these graphs are isomorphic to two copies of Kn,n, but the vertex
sets of the Kn,n’s are different (in one it is even vertices adjacent to even vertices and odd
vertices adjacent to odd vertices while in the other it is even vertices adjacent to odd vertices
and odd vertices adjacent to even vertices), so their automorphism groups are different, but
permutation isomorphic!
2 Characterization of automorphism groups of Haar graphs of abelian
groups
We now, with some exceptions, focus on abelian groups. We will use the symbol A when
the group under consideration is abelian, and G when a nonabelian group is allowed. We
first consider disconnected Haar graphs. We begin with more permutation group terms.
Definition 2.1. Let X be a set and suppose K ≤ SX is a transitive group which has a block
system C. Then K has an induced action on C, denoted K/C. Namely, for k ∈ K, define
k/C : C 7→ C by k/C(C) = C ′ if and only if k(C) = C ′, and set K/C = {k/C : k ∈ K}.
We also define the fixer of C in K, denoted fixK(C), to be {k ∈ K : k/C = 1}. That is,
fixK(C) is the subgroup of K which fixes each block of C set-wise, and is the kernel of the
induced action of K on C.
8 Art Discrete Appl. Math. 5 (2022) #P3.06
We observe that for an abelian group A, H = Z2 ⋉ Aut(Cay(A,S)) has B as a block
system. Here H/B ∼= Z2 and fixH(B) = K̂, where K = Aut(Cay(A,S)). We shall also
need definitions of wreath products of digraphs and groups.
Definition 2.2. Let Γ1 and Γ2 be digraphs. The wreath product of Γ1 and Γ2, denoted Γ1 ≀
Γ2, is the digraph with vertex set V (Γ1) × V (Γ2) and arcs ((u, v), (u, v
′)) for
u ∈ V (Γ1) and (v, v
′) ∈ A(Γ2) or ((u, v), (u
′, v′)) where (u, u′) ∈ A(Γ1) and v, v
′ ∈
V (Γ2).
Definition 2.3. Let G ≤ SX and H ≤ SY . Define the wreath product of G and H , denoted
G ≀H , to be the set of all permutations of X×Y of the form (x, y) 7→ (g(x), hx(y)), where
g ∈ G and each hx ∈ H .
It is not hard to show that for vertex-transitive digraphs, Aut(Γ1) ≀ Aut(Γ2) ≤
Aut(Γ1 ≀ Γ2). See [12] for more information regarding wreath products.
Let Γ = Haar(A,S) be connected, so by Lemma 1.10 B is a block system of Aut(Γ).
Then F = fixAut(Γ)(B) has induced actions on B0 and B1. Let f ∈ F with f(i, j) =
(i, γi(j)). The induced action of F on B0 is given by f · (0, j) = (0, γ0(j)), and on B1
by f ∗ (1, j) = (1, γ1(j)). We will be considering these induced actions frequently, and
will abuse notation by considering the induced action of F on B0 as an action simply on
A, in which case f · (0, j) = γ0(j) (i.e. we just delete the first coordinate if it is clear
from context), and similarly for the induced action of F on B1: f ∗ (1, j) = γ1(j). We
will also not write the actions formally, and not use the · and ∗ notation, but instead write
FBi , i ∈ Z2 or analogous notation for a subgroup of F . For example, we simply say that
AL is contained in the image of the actions of F on B0 and B1 for the induced action of
ÂL on B0 and B1. Or more simply, ÂL
Bi
= AL as with the above abuse of notation,
âL · (0, j) = aL(0) and âL(1, j) = aL(0) for every a ∈ A.
Definition 2.4. Let G be a group. We will use the notation ḡR for the permutations of
Z2 ×G given by ḡR(0, j) = (0, j) and ḡR(1, j) = (1, jg) in what follows.
It is shown in the proof of [32, Lemma 2.2] that for any group G, S ⊆ G, and g ∈ G,
ḡR(Haar(G,S)) ∼= Haar(G,Sg) ∼= Haar(G,S).
Theorem 2.5. Let A be an abelian group, and S ⊆ A. If Γ = Haar(A,S) is disconnected,
then there is a ∈ A and H < A such that Γ = ā−1R (Haar(A, a + S)) and Aut(Γ)
∼=
ā−1R (SA/H ≀Aut(Haar(H, a+ S)))āR.
Proof. If Γ is disconnected, then by Lemma 1.8 S ⊆ −a + H for some a ∈ A and
H < A. Set H = ⟨S + (−S)⟩ < A (this is written additively as A is abelian). Then
Haar(H, a+ S) is a connected graph which is a component of Haar(A, a+ S) = āR(Γ).
Thus Γ = ā−1R (Haar(A, a+ S)). Also, there are [A : H] components of Haar(A, a+ S),
and Haar(A, a + S) ∼= K̄A/H ≀ Haar(H, a + S), where K̄A/H is the complement of
the complete graph with vertices the cosets of H in A. Then Aut(Haar(A, a + S)) ∼=
SA/H ≀ Aut(Haar(H, a + S)), and Aut(Haar(A,S)) = ā
−1
R Aut(Haar(A, a + S))āR as
ā−1R (Haar(A, a+ S)) = Haar(A,S).
Another way in which the automorphism group of a Haar graph Γ of an abelian group
is bigger than expected is if the graph is connected and the action of F = fixAut(Γ)(B) is
not faithful in its action on a block of B. The next result shows that this situation is easy to
spot by examining the connection set of Γ.
T. Dobson: On automorphisms of Haar graphs of abelian groups 9
Lemma 2.6. Let G be a group, and Γ = Haar(G,S) be connected and vertex-transitive.
Then fixAut(Γ)(B) in its action on B0 or B1 is unfaithful if and only if S is a union of left
cosets of some nontrivial subgroup of G.
Proof. Set F = fixAut(Γ)(B). First observe that as Γ is connected, B is a block system
of Aut(Γ). As Aut(Γ) is transitive, the action of F on B0 is unfaithful if and only if the
action of F on B1 is unfaithful.
Suppose S is a union of left cosets of some subgroup H ̸= 1 of A. Clearly for h ∈ H ,
the map h̄R is an automorphism of Γ as it fixes left cosets of H . Then h̄R is contained in
the kernel of the action of F on B0.
Suppose F in its action on B0 is unfaithful with K ̸= 1 the kernel of the action. Then
K ◁ F and so the orbits of K form a block system C of FB1 . As FB1 contains GL acting
regularly, the blocks of C are left cosets of some subgroup H of G. As K stabilizes (0, 0),
it is clear that if (0, 0) is adjacent to some element (1, gh) for g ∈ G and h ∈ H , then
(0, 0) is adjacent to (1, gh) for every h ∈ H . So S is a union of left cosets of H . As K is
nontrivial, H ̸= 1.
We next explicitly determine the automorphism groups of such Haar graphs of abelian
groups. We will need some additional notation.
Definition 2.7. Let K be a transitive group with block system C. If D is also a block system
of K and each block of D is a union of blocks of C, then we write D/C for the set of blocks
of C whose union is D. Let Γ be a vertex-transitive digraph with K ≤ Aut(Γ). Define the
block quotient digraph of Γ with respect to C, denoted Γ/C, to be the digraph with vertex set
C and arc set {(C,C ′) : C ̸= C ′ ∈ C and (u, v) ∈ A(Γ) for some u ∈ C and v ∈ C ′}.
Theorem 2.8. Let A be an abelian group, S ⊆ A, and Γ = Haar(A,S). If Γ is connected,
and the action of fixAut(Γ)(B) is unfaithful on B1, then there exists a subgroup 1 < H ≤ A
such that Γ ∼= Haar(A/H,S) ≀ K̄β where β = |H| and S is interpreted as a set of cosets of
H in A. Also, Aut(Γ) ∼= Aut(Haar(A/H,S)) ≀ Sβ , and denoting the natural bipartition
of Haar(A/H,S) by D, the action of fixAut(Haar(A/H,S))(D) on D ∈ D is faithful.
Proof. By Lemma 2.6 S is a union of cosets of some subgroup 1 < H ≤ A. We choose
H to be maximal with respect to this property - clearly such a maximal subgroup exists.
Let C be the block system of ⟨ι, ÂL⟩ formed by the subgroup ⟨ĥL : h ∈ H⟩. Notice that
C = {{(i, a+ h) : h ∈ H} : i ∈ Z2, a ∈ A}.
Claim 1: In Γ, for any two distinct blocks C,C ′ ∈ C, either every vertex of C is adjacent
to every vertex of C ′ or no vertex of C is adjacent to any vertex of C ′.
Clearly if C,C ′ ⊆ B0 or B1 then this statement is true as there are no edges with both
endpoints inside any Bi, i = 0, 1. If, say, C ⊂ B0 and C
′ ⊂ B1 and some vertex (0, v) of
C is adjacent to some vertex (1, v′) of C ′, then by definition v′ = v + s for some s ∈ S.
Then C = {(0, w) : w ∈ v +H} and C ′ = {(1, w′) : w′ ∈ v + s +H}. Let (0, w) ∈ C
and (1, w′) ∈ C ′ with w = v + h and w′ = v + s+ h′, h, h′ ∈ H . Then
(0, w)(1, w′) = (0, v+ h)(1, v+ s+ h′) = (0, v+ h)(1+ v+ h+ s+ (h′ − h)) ∈ E(Γ),
as s+ (h′ − h) ∈ s+H ⊆ S, and the claim follows.
It is now straightforward to verify using the definition of the wreath product that Γ ∼=
Haar(A/H,S) ≀ K̄β .
10 Art Discrete Appl. Math. 5 (2022) #P3.06
Claim 2: Haar(A/H,S) ̸∼= Γ2 ≀ K̄r with r ≥ 2, where Γ2 is a connected vertex-transitive
graph.
Suppose not, so Haar(A/H,S) ∼= Γ2 ≀ K̄r for some vertex-transitive graph Γ2 and r ≥
2 is maximum. By [12, Theorem 5.7] and the maximality of r, we have Aut(Haar(A,S)) ∼=
Aut(Γ2)≀Srβ , so Aut(Γ) has a block system E with blocks of size rβ such that Aut(Γ)/E =
Γ2. Now, if Γ2 has an odd cycle, then Γ2 ≀ K̄r has an odd cycle. However, Γ2 ≀ K̄r is a
Haar graph and bipartite. So Γ2 is bipartite, with bipartition {F0, F1}. Observe that both
F0 and F1 are a set of cosets of H in A, and so ∪a+H∈F0(a +H) and ∪a+H∈F1(a +H)
are subsets of A.
We now show that {∪a+H∈F0(a + H),∪a+H∈F1(a + H)} = B. It suffices to show
that if E,E′ ∈ E and E,E′ ∈ F0, then E ∪ E
′ ⊆ Bj , for some fixed j = 0, 1. As Γ is
connected, Γ2 is connected. Thus there is an EE
′ path E0, . . . , Et in Γ2. Then Ei ⊆ Fj if i
is even and Ei ⊆ Fj+1 if i is odd. Similarly, the Ei with even subscripts are all contained in
Bj while the Ei with odd subscripts are contained in Bj+1. As both E and E
′ are contained
in F0, t is even and E ∪ E
′ ⊆ Bj , so {∪a+H∈F0(a+H),∪a+H∈F1(a+H)} = B.
As ι interchanges the blocks of B, ι also interchanges the union of the bipartition sets
∪F0 and ∪F1 of Γ2. Then ι/E is well defined and is not 1 i.e. ι permutes the blocks of E
nontrivially as well. Finally, by [8, Theorem 1.5A] there exists a subgroup K ≤ ⟨ι, ÂL⟩
such that the block of E that contains (0, 0) is the orbit of K that contains (0, 0). As
ι/E ̸= 1, K ≤ ÂL is of order rβ. This implies the blocks of E are orbits of K, and
K = {ℓ̂ : ℓ ∈ L ≤ A}. Also, it should be clear that H ≤ L as every block of C is
contained in a block of E . As r ≥ 2, H < L. But this implies that S is a union of cosets of
L, contradicting the maximality of H . The claim is established.
By [12, Theorem 5.7] and Claim 2, Aut(Γ) ∼= Aut(Haar(A/H,S)) ≀ K̄β . That
the action of fixAut(Haar(A/H,S))(D) on D ∈ D is faithful follows from Claim 2 and
Theorem 2.6.
We now turn to Haar graphs Γ of abelian groups A that are connected and whose con-
nection set is not a union of cosets of some subgroup of A. These two hypotheses imply
that B is a block system of Aut(Γ) and that the actions of fixAut(Γ)(B) on B0 and B1 are
faithful. To investigate further, we will need the terminology and some results concerning
inequivalent permutation representations.
Definition 2.9. A permutation representation of a group K is a homomorphism
ϕ : K → Sn for some n.
Definition 2.10. Let K be a group, and X and Y sets. Let α : K 7→ SX and
ω : K 7→ SY be permutation representations of K. We say α and ω are equivalent permu-
tation representations of K if there exists a bijection λ : X 7→ Y such that λ(α(k)(x)) =
ω(k)(λ(x)) for all x ∈ X and k ∈ K. In this case, we will say that α(K) and ω(K) are
permutation equivalent.
The examples of Cay(Z7, {1, 2, 4}) and its corresponding Haar graph, the Heawood
graph, provide the next way in which the automorphism group of a Haar graph of an abelian
group can be larger than its expected automorphism group. Namely, the full automorphism
group of the Heawood graph is Z2 ⋉PGL(3, 2), and PGL(3, 2) acts permutation inequiv-
alently on the two blocks B0 and B1 of B. It turns out that this is not a coincidence, as we
will show. Before turning to that result, we prove a preliminary result which will be crucial.
T. Dobson: On automorphisms of Haar graphs of abelian groups 11
It does not depend upon A being abelian or even a group, so we will use the symbol X in
place of A or G. We keep the notation Bi = {(i, x) : x ∈ X}, i = 0, 1.
Lemma 2.11. Let X be a set, and let F ≤ SZ2×X have B0 and B1 as orbits. Additionally,
assume that the actions of F on B0 and B1 are faithful and the action of F on B0 is
permutation equivalent to the action of F on B1. Then there exists ϵ̄ ∈ SZ2×X such that
ϵ̄(B0) = B0, ϵ̄
B0 = 1 and every element of ϵ̄−1F ϵ̄ has the form (i, j) 7→ (i, g(j)), where
g ∈ FB0 .
Proof. Let α : F 7→ FB0 and ω : F 7→ FB1 be permutation representations of FB0 and
FB1 , respectively. As F is faithful on B0 and B1, both α and ω are faithful permutation
representations of F . As the action of F on B0 is permutation equivalent to the action of
F on B1, there exists a bijection λ : B0 7→ B1 such that λ(α(g)(0, j)) = ω(g)(λ(0, j)) for
every j ∈ X and g ∈ F . Let β : B0 7→ B1 be defined by β(0, j) = (1, j) for every j ∈ X ,
so that β is a bijection. Set ϵ = λβ−1 so ϵβ = λ and ϵ ∈ SB1 . Substituting ϵβ for λ we
have ϵβ(α(g)(0, j)) = ω(g)(ϵβ(0, j)) or equivalently, β(α(g)(0, j)) = ϵ−1ω(g)ϵ(β(0, j))
for every j ∈ X . As ϵ is a bijection and ω a faithful permutation representation of F , it is
straightforward to verify that the map δ : F 7→ SB1 given by δ(g) = ϵ
−1ω(g)ϵ is a faithful
permutation representation of F .
Now, as α(g)(0, j) ∈ B0 and similarly, δ(g)(1, j) ∈ B1 for every j ∈ X , α(g) and
δ(g) induce permutations ᾱ(g) and δ̄(g) in SX defined by α(g)(0, j) = (0, ᾱ(g)(j)) and
δ(g)(1, j) = (1, δ̄(g)(j)) for all j ∈ X . Then
β(α(g)(0, j)) = β(0, ᾱ(g)(j)) = (1, ᾱ(g)(j))
and
ϵ−1ω(g)ϵ(β(0, j)) = δ(g)(1, j) = (1, δ̄(g)(j))
for all j ∈ X . As β(α(g)(0, j)) = ϵ−1ω(g)ϵ(β(0, j)) for every j ∈ X , we see that
ᾱ(g)(j) = δ̄(g)(j) for all j ∈ X , and so ᾱ(g) = δ̄(g) for all g ∈ F . Define ϵ̄ : Z2 ×X 7→
Z2 ×X by ϵ̄(0, j) = (0, j) and ϵ̄(1, j) = ϵ(1, j). Let g ∈ F . Then
ϵ̄−1gϵ̄(0, j) = α(g)(0, j) = (0, ᾱ(g)(j))
and
ϵ̄−1gϵ̄(1, j) = ϵ−1ω(g)ϵ(1, j) = (1, δ̄(g)(j)) = (1, ᾱ(g)(j)),
and the result follows.
Theorem 2.12. Let G be a group, S ⊆ G, Γ = Haar(G,S), and let F ≤ Aut(Γ) be
the largest subgroup of Aut(Γ) that fixes B0 and B1 set-wise. Suppose F satisfies the
following conditions:
(1) F acts faithfully on both B0 and B1, and
(2) the action of F on B0 is permutation equivalent to the action of F on B1.
Let L be the group of all elements of SZ2×G of the form (i, j) 7→ (i, ℓ(j)) where ℓ ∈ F
B0 ,
and g ∈ G such that StabF (1, g) = StabF (0, 1G). Then L = ḡ
−1
R F ḡR.
12 Art Discrete Appl. Math. 5 (2022) #P3.06
Proof. Clearly every element of F can be written as (i, j) 7→ (i, ℓi(j)), where ℓ0, ℓ1 ∈ SG.
By Lemma 2.11 there exists ϵ̄ ∈ SZ2×G such that ϵ̄
B0 = 1 and every element of ϵ̄−1F ϵ̄
has the form (i, j) 7→ (i, ℓ(j)), where ℓ ∈ FB0 . Let ϵ ∈ SG such that ϵ̄(1, j) = (1, ϵ(j)).
Define gR : G → G by gR(x) = xg, and set GR = {gR : g ∈ G}. We next show that
ϵ ∈ GR.
Of course, ĜL ≤ F , and ĜL has the form (i, j) 7→ (i, ℓ(j)), where ℓ ∈ ĜL
B0
. As
ϵ̄B0 = 1, we have ϵ̄−1ĝLϵ̄ = ĝL for every g ∈ G. Hence ϵ̄ centralizes ĜL, and so ϵ
centralizes GL as ϵ̄
B0 = 1. As the centralizer of GL in SG is GR (this well known fact can
be deduced from [8, Theorem 4.2A(ii)]), we have ϵ ∈ GR. As ϵ ∈ GR, there exists g ∈ G
such that L = ḡ−1R F ḡR. Finally, as F = ḡRLḡ
−1
R and StabL(0, 1G) = StabL(1, 1G),
F = ḡRLḡ
−1
R stabilizes (1, g).
Corollary 2.13. Let A be an abelian group with S ⊆ A such that Γ = Haar(A,S)
is connected, and consequently Aut(Haar(A,S)) has B as a block system. Set F =
fixAut(Γ)(B). Then one of the following is true:
(1) the induced action of F on B1 is not faithful,
(2) the induced action of F on B0 is permutation inequivalent to the induced action of
F on B1, or
(3) Aut(Haar(A,S)) = ā−1R [Z2 ⋉Aut(Cay(A, a+ S))]āR for some a ∈ A.
Proof. We may assume without loss of generality that the action of F on B1 is faithful.
As Aut(Γ) is transitive as A is abelian, we have F is faithful on B0. With that assumption
made, we may then assume without loss of generality that the actions of F on B0 and B1
are permutation equivalent. Set L = {(i, j) 7→ (i, g(j)) : g ∈ FB0} so StabL(0, 0) =
StabL(1, 0). Applying Theorem 2.12, there is a ∈ A such that L = āRF ā
−1
R . By
[32, Lemma 2.2], āR(Γ) = Haar(A, a + S). By Lemma 1.4, Aut(Cay(A, a + S)) ∼= L,
so Aut(Haar(A, a+ S)) = Z2 ⋉Aut(Cay(A,−a+ S). The result follows as Aut(Γ) =
ā−1R Aut(Haar(A, a+ S))āR.
Remark 2.14. Suppose Haar(A,S) is connected and the induced action of F on B1 is
faithful. If a, b ∈ A with a ̸= b, then āR(Haar(A,S)) ̸= b̄R(Haar(A,S)) as otherwise
b̄−1R āR ∈ Aut(Haar(A,S)) and the induced action of fixAut(Cay(A,S))(B) is not faithful
on B0.
Remark 2.15. While Haar(A,S) ∼= Haar(A, a + S) for every a ∈ A, Cay(A,S) is
not necessarily isomorphic to Cay(A, a + S). Indeed, let n be an odd positive integer,
A = Zn, S = {±1}, and a = 1. Then Cay(A,S) is a cycle, while Cay(A, a + S) =
Cay(Zn, {0, 2}) is a directed cycle together with a loop at every vertex.
The following result is a combination of Theorems 2.5 and 2.8, and Corollary 2.13, and
summarizes the possibilities for Aut(Haar(A,S)) with A abelian.
Corollary 2.16. Let A be an abelian group, S ⊆ A, and Γ = Haar(A,S). Then one of
the following is true:
(1) Γ is disconnected, then there is a ∈ A and H < A such that Γ = ā−1R (Haar(A,
a+ S)) and Aut(Γ) ∼= ā−1R (SA/H ≀Aut(Haar(H, a+ S)))āR,
T. Dobson: On automorphisms of Haar graphs of abelian groups 13
(2) Γ is connected, and the action of fixAut(Γ)(B) is unfaithful on B1. There exists a
subgroup 1 < H ≤ A such that Γ ∼= Haar(A/H,S) ≀ K̄β where β = |H| and S is
interpreted as a set of cosets of H in A, and Aut(Γ) ∼= Aut(Haar(A/H,S)) ≀ Sβ .
Additionally, denoting the natural bipartition of Haar(A/H,S) by D, the action of
fixAut(Haar(A/H,S))(D) on D ∈ D is faithful,
(3) Aut(Γ) ∼= ā−1R Z2 ⋉Aut(Cay(A, a+ S))āR for some a ∈ A, or
(4) the action of fixAut(Γ)(B) on B1 is faithful but the actions on B0 and B1 are not
equivalent permutation groups.
We now give a group theoretic description of the graphs in (4) of Corollary 2.16.
Theorem 2.17. Let A be an abelian group and S ⊆ A such that Γ = Haar(A,S) is
connected and S is not a union of cosets of some subgroup of A. Let F = fixAut(Γ)(B),
H = FB0 , and L = StabH(b), where b ∈ B0. If the actions of F on B0 and B1 are in-
equivalent, then there exists σ ∈ Aut(H) which is an involution and maps L to a subgroup
R of H which is not conjugate in H to L.
Proof. The hypothesis implies that the action of F on B0 and B1 is faithful by Lemma 2.6.
As ι ∈ Aut(Γ), conjugation of F by ι induces automorphisms σ and δ which map FB0
to FB1 , and FB1 to FB0 , so we may view FB1 as being contained in H as the image of
δ. As ι has order 2, so does σ. Also, as the action of F on B0 and B1 are not equivalent,
by [8, Lemma 1.6B] σ(L) = R is not the stabilizer of a point in B0, so R is not conjugate
in H to L.
We next give a partial converse of the previous result. We begin with a more general
construction of bipartite graphs than that of the Haar graphs.
Definition 2.18. Let G be a group, let L,R ≤ G, and let D be a union of double cosets
of R and L in G, that is D = ∪iRdiL. Define a bipartite graph Γ = B(G,L,R;D) with
bipartition V (Γ) = G/L∪G/R (here G/L and G/R are the sets of left cosets of L and R
in G) and edge set E(Γ) = {{gL, gdR} : g ∈ G, d ∈ D}. We call Γ the bi-coset graph of
G with respect to L, R and D.
The bi-coset graphs B(G,L,R;D) were introduced in [14], where they were shown
to be well-defined bipartite graphs whose automorphism group contain a natural subgroup
isomorphic to G. Haar graphs are bi-coset graphs with L = R = 1. In [14, Lemma 2.6]
a sufficient condition to ensure that a bi-coset graph is vertex-transitive is given, and then
several more specific circumstances were given where this condition was satisfied. We will
need another such more specific circumstance.
Lemma 2.19. Let G be a group with L ≤ G and σ ∈ Aut(G) an involution such that
σ(L) = R. The bi-coset graph Γ = B(G,L,R;S) is vertex-transitive with σ ∈ Aut(Γ)
provided S = LDR with σ(D) = D−1 = D.
Proof. By [14, Lemma 2.6] we need only show that S−1 = σ(S). Then
σ(S) = σ(LDR) = σ(L)σ(D)σ(R) = RD−1L = R−1D−1L−1 = (LDR)−1 = S−1.
14 Art Discrete Appl. Math. 5 (2022) #P3.06
Definition 2.20. Let K be a group and H ≤ K. The largest normal subgroup of K
contained in H is called the core of H in K. If the core of H in K is trivial, then we say
H is core-free in K.
It is well known that the left action of a group K on a subgroup H ≤ K is faithful if
and only if H is core-free in K.
Theorem 2.21. Let A be an abelian group. Suppose AL ≤ K ≤ SA with L = StabK(a),
a ∈ A, σ ∈ Aut(K) an involution, and R = σ(L) ≤ K which is not conjugate in K to L
and satisfies R ∩ AL = 1. Then the bi-coset graph Γ = B(K,L,R;LR) ∼= Haar(A,S)
for some S ⊆ A, and there is a subgroup H ≤ Aut(Γ) such that H fixes B, H ∼= K, the
action of H on B0 is faithful, and the action of H on B0 = K/L is inequivalent to the
action of H on B1 = K/R.
Proof. By [14, Lemma 2.3] the left multiplication action of K on V (Γ) is contained in
Aut(Γ), fixes B, and is transitive on B0 and B1. Denote by H the corresponding subgroup
of Aut(Γ). Then HB0 is permutation equivalent to K ≤ SA as their stabilizers are the
same, namely L. It is clear that the left multiplication action of K on B0 is faithful as L
is core-free in K as L is the stabilizer of a point in K ≤ SA. Similarly, as σ ∈ Aut(K)
and σ(L) = R, R is core-free in K (as it is the isomorphic image of a core-free subgroup
of K) so the left multiplication action of K on K/R is faithful as well. Note that as
left multiplication of R by an element of K fixes R if and only if it is contained in R,
StabHB1 (R) = R. We see that the action of H on B1 is faithful, and the action of H on
B0 is inequivalent to the action of H on B1. Setting D = {1K} we see by Lemma 2.19
that σ ∈ Aut(Γ) as D−1 = D, and so Γ is vertex-transitive.
To see Γ is isomorphic to a Haar graph of A, let à ≤ Aut(Γ) be the subgroup of
Aut(Γ) induced by left multiplication by elements of AL in K. Of course, Ã is transitive
on B0 as ALL = K. Suppose ã1R = ã2R for some ã1, ã2 ∈ Ã. This occurs if and
only if there are a1, a2 ∈ A such that (a1)LR = (a2)LR, which occurs if and only if
(a−12 a1)LR = R. As AL ∩ R = 1, we see (a1)L = (a2)L. Then à is faithful on B1. As
à ∩ R = 1, ÃB1 is semiregular. By [42, Proposition 4.1] à has one orbit of size |A|, and
so is transitive. The result follows by [14, Lemma 2.5].
Theorem 2.21 is only a partial converse to Theorem 2.17 as it is possible that the graph
constructed in the result (or any graph constructed in a similar way) will have an auto-
morphism group larger than the group K in the statement. For example, if LR = K is
transitive, the graph B(K,L,R;LR) constructed will be a complete bipartite graph with
automorphism group K2 ≀ Sn with n = |A|. The next example shows that this can occur.
Example 2.22. Let K = S6, L = StabS6(x)
∼= S5 and σ be the outer automorphism of
S6 of order 2. Set R = σ(L). Then B(K,L,R;LR) ∼= K6,6.
Proof. We show that L is transitive on R. That is, we will show that σ(S5) is transitive,
where we view S5 as the stabilizer of the point 0 in the set Z6 on which we view S6
permuting. Now, the three cycle (3, 4, 5) is mapped by σ to a product of two disjoint 3-
cycles (see for example [20]). So σ(3, 4, 5) has two orbits of size 3. This implies that
σ(S5) is transitive, or σ(S5) has two orbits of size 3, in which case its maximum order is
|S3| · |S3| = 36. But S5 has order 120, so σ(S5) is transitive and the result follows.
Note that the graph B(K,L,R;LR) as constructed in the previous example does not
satisfy the hypothesis of Theorem 2.21 as R ∩AL ̸= 1.
T. Dobson: On automorphisms of Haar graphs of abelian groups 15
Definition 2.23. Let n ≥ 3 be an integer, q a prime-power, and set
L = PG(n− 1, q) = {rv⃗ : r ∈ Fq and v⃗ is in the vector space F
n
q },
the set of lines (or projective points) in Fnq . Let H be the set of hyperplanes in F
n
q . Define
a graph B(PG(n− 1, q)) to have vertex set L ∪H and edges {L,H} where L ⊆ H .
Note that the ‘B’ used in defining the above graphs has an entirely different meaning
from the ‘B’ used in the previous example. No confusion can arise though as the parameters
of the families are completely different. Also, it is known that B(2, 2) is isomorphic to the
Heawood graph. Our next example, which is also fairly well-known, shows it is a member
of a much larger family of graphs with many of the same properties.
Example 2.24. The graph B(PG(n − 1, q)) is isomorphic to a Haar graph of the cyclic
group of order (qn − 1)/(q − 1). Additionally, if n ≥ 3, F = fixAut(B(PG(n−1,q)))(B) in
its induced actions on B0 and B1 are inequivalent representations of PGL(n, q) with the
actions being on points and hyperplanes.
Proof. It is well-known that PGL(n, q) contains a regular cyclic subgroup of order
(qn − 1)/(q − 1) - see for example [24, Theorem 3] or [29, Theorem 1.1]. It is easy
to see that FB0 ∼= FB1 contains PGL(n, q) as a point contained in a hyperplane is mapped
to a point contained in a hyperplane by an element of PGL(n, q). That F is not larger
than PGL(n, q) basically follows from the Fundamental Theorem of Finite Geometry.
By [18, Lemma 4.2] we see B(PG(n − 1, q)) is isomorphic to a Haar graph, and, as
n ≥ 3, points and hyperplanes have different dimensions. It is then not hard to see that
the stabilizer in PGL(n, q) of a line does not stabilize any hyperplane. Thus the induced
actions of F on B0 and B1 are inequivalent by [8, Lemma 1.6B].
3 Applications to arc-transitive graphs
The study of s-arc-transitive graphs was initiated in a celebrated paper by Tutte [40]. There
has been strong and consistent interest in s-arc-transitive graphs for several decades. Per-
haps the most important tool in this area is Praeger’s Normal Quotient Lemma [36]. This
lemma shows how to reduce an s-arc-transitive graph Γ to an s-arc-transitive quotient of Γ
provided one can find N ◁ Aut(Γ) that has at least three orbits. If Aut(Γ) is quasiprimi-
tive, then one can study such groups and graphs using the O’Nan-Scott Theorem [8, Theo-
rem 4.1A] and Praeger’s quasiprimitive counterpart [37]. So other techniques are necessary
to deal with the case when Γ only has normal subgroups with exactly two orbits. In this
case, if Γ is disconnected, then there is an obvious reduction to s-arc-transitive graphs of
smaller order. If Γ is connected, then there are edges between the two orbits and so no
edges inside the orbits. Hence Γ is bipartite, and if the subgroup N of Aut(Γ) fixing the
parts of the bipartition set-wise contains a semiregular subgroup isomorphic to G, then Γ
is a Haar graph of G by [14, Lemma 2.4]. Thus the study of Haar graphs is essential to the
study of s-arc-transitive graphs.
Definition 3.1. Let s ≥ 1, and Γ a digraph. An s-arc of Γ is a sequence x0, x1, . . . , xs of
vertices of Γ such that (xi, xi+1) ∈ A(Γ), 0 ≤ i ≤ s − 1, and xi ̸= xi+2, 0 ≤ i ≤ s − 2.
The digraph Γ is s-arc-transitive if Aut(Γ) is transitive on the set of s-arcs of Γ.
As an arc-transitive graph without isolated vertices is vertex-transitive, we restrict our
attention to vertex-transitive Haar graphs.
16 Art Discrete Appl. Math. 5 (2022) #P3.06
Definition 3.2. Let Γ be a digraph, and s ≥ 1. A sequence of arcs a1, . . . , as ∈ A(Γ) is an
alternating s-arc if there exists vertices x0, . . . , xs ∈ V (Γ), xi ̸= xi+2, and for 1 ≤ m ≤ s
the arc am = (xm−1, xm) if m is odd while am = (xm, xm−1) if m is even. An alternating
s-arc-transitive digraph is a digraph whose automorphism group is transitive on the set of
alternating s-arcs. The vertices x0, . . . , xs are the vertex-sequence of the alternating s-arc
a1, . . . , as.
• • • • •
• • • • •
// // // //
// oo // oo
Figure 3: A 4-arc (top) versus an alternating 4-arc (bottom).
Clearly if s = 1, then an alternating s-arc is simply an s-arc. If s ≥ 2 then an alternating
s-arc can be obtained from an s-arc by reversing the direction of every other arc - see
Figure 3. Now choose an s-arc, and fix an orientation of this s-arc. An s-arc-transitive
graph is transitive on the set of all s-arcs in Γ with this fixed reorientation, so an s-arc-
transitive graph is trivially alternating s-arc-transitive. The next result gives a relationship
between alternating s-arc-transitive Cayley digraphs and s-arc-transitive Haar graphs.
Theorem 3.3. Let G be a group, S ⊆ G, and s ≥ 1. If Cay(G,S) is an alternating
s-arc-transitive digraph and Haar(G,S) is vertex-transitive, then Haar(G,S) is s-arc-
transitive. Conversely, if Aut(Haar(G,S)) = Z2 ⋉ Aut(Cay(G,S)) and Haar(G,S) is
s-arc-transitive, then Cay(G,S) is alternating s-arc-transitive.
Proof. Suppose Cay(G,S) is alternating s-arc-transitive and Haar(G,S) is vertex-transitive.
Let P1 = (i, x0), (i+1, x1), . . . , (i+s, xs) and P2 = (j, y0), (j+1, y1), . . . , (j+s, ys) be
two s-arcs in Haar(G,S). As Aut(Haar(G,S)) is transitive, replacing P1 or P2 by ι(P1)
or ι(P2), for appropriate ι ∈ Aut(Haar(G,S)), if necessary, we may assume without loss
of generality that i = j = 0. Then there exist t1, . . . , ts ∈ S such that for 1 ≤ m ≤ s
xm =
{
x0t1t
−1
2 . . . t
−1
m−1tm, if m is odd,
x0t1t
−1
2 . . . tm−1t
−1
m , if m is even.
Similarly, there exist u1, . . . , um ∈ S such that for 1 ≤ m ≤ s
ym =
{
y0u1u
−1
2 . . . u
−1
m−1um, if m is odd,
y0u1u
−1
2 . . . um−1u
−1
m , if m is even.
The arcs am = (xm, xmtm) if m is even and am = (xm−1t
−1
m , xm−1) if m is odd,
0 ≤ m ≤ s − 1, are then contained in A(Cay(G,S)), and Q1 = a0, . . . , as−1 is an
alternating s-arc in Cay(G,S). Similarly, the arcs bm = (ym, xyum) if m is even and
bm = (ym−1u
−1
m , ym−1) if m is odd, 0 ≤ m ≤ s−1, are then contained in A(Cay(G,S)),
and Q2 = b0, . . . , bs−1 is an alternating s-arc in Cay(G,S). As Cay(G,S) is alternat-
ing s-arc-transitive, there exists γ ∈ Aut(Cay(G,S)) such that γ(Q1) = Q2. Then
γ̂ ∈ Aut(Haar(G,S)), and γ̂(P1) = P2. The result follows.
T. Dobson: On automorphisms of Haar graphs of abelian groups 17
Conversely, suppose Aut(Haar(G,S)) = Z2⋉Aut(Cay(G,S)) and Haar(G,S) is s-
arc-transitive. Let P1 = a1, . . . , as and P2 = b1, . . . , bs be alternating s-arcs in Cay(G,S)
with vertex-sequences x0, x1, . . . , xs and y0, y1, . . . , ys, respectively. Then there exist
t1, . . . , ts ∈ S such that for 1 ≤ m ≤ s we have
xm =
{
x0t1t
−1
2 . . . t
−1
m−1tm, if m is odd,
x0t1t
−1
2 . . . tm−1t
−1
m , if m is even.
Similarly, there exist u1, . . . , um ∈ S such that for 1 ≤ m ≤ s
ym =
{
y0u1u
−1
2 . . . u
−1
m−1um, if m is odd,
y0u1u
−1
2 . . . um−1u
−1
m , if m is even.
Corresponding to these two alternating s-arcs, there are two s-arcs
(0, x0), (1, x1), . . . , (s, xs) and (0, y0), (1, y1), . . . , (s, ys) in Haar(G,S). As Haar(G,S)
is s-arc-transitive, there exists δ ∈ Aut(Haar(G,S)) such that
δ((0, x0), (1, x1), . . . , (s, xs)) = (0, y0), (1, y1), . . . , (s, ys).
Then δ(0, x0) = (0, y0) so δ ∈ fixAut(Haar(G,S))(B). As Aut(Haar(G,S)) =
Z2 ⋉ Aut(Cay(G,S)), δ = γ̂ for some γ ∈ Aut(Cay(G,S)). Then γ(xi) = yi and
so γ(P1) = P2. So Cay(G,S) is alternating s-arc-transitive.
We next consider alternating s-arc-transitivity when Cay(G,S) contains arcs and edges.
We will need a preliminary lemma.
Lemma 3.4. Let Γ be a digraph such that every vertex has in- and out-degree at least two,
and s ≥ 2. If Γ is alternating s-arc-transitive, then Γ is alternating (s− 1)-arc-transitive.
Proof. Let a1, . . . , as−1 and b1, . . . , bs−1 be alternating (s − 1)-arcs in Γ with vertex-
sequences x0, . . . , xs−1 and y0, . . . , ys−1. As xs−1 and ys−1 have in- and out-degree at
least two, a1, . . . , as−1 and b1 . . . , bs−1 can be extended to alternating s-arcs a1, . . . , as
and b1, . . . , bs. As Γ is alternating s-arc-transitive, there is γ ∈ Aut(Γ) such that
γ(a1, . . . , as) = b1, . . . , bs.
So γ(a1, . . . , as−1) = b1, . . . , bs−1 and Γ is alternating (s− 1)-arc-transitive.
Corollary 3.5. Let G be a group, S ⊆ G such that Aut(Haar(G,S)) =
Z2 ⋉Aut(Cay(G,S)). Haar(G,S) is s-arc-transitive if and only if
(1) S = S−1 and Cay(G,S) is s-arc-transitive, or
(2) S ∩ S−1 = ∅ and Cay(G,S) is alternating s-arc-transitive.
Proof. Suppose Haar(G,S) is s-arc-transitive. By Theorem 3.3, Cay(G,S) is alternat-
ing s-arc-transitive. Also, |S| ≥ 2 as Aut(Haar(G,S)) = Z2 ⋉ Aut(Cay(G,S)), so by
Lemma 3.4 we have that Cay(G,S) is arc-transitive. This implies Cay(G,S) cannot con-
tain both edges and arcs, so S = S−1 or S ∩ S−1 = ∅. If S ∩ S−1 = ∅, then (2) follows.
Otherwise, S = S−1 so Cay(G,S) is a graph, and is alternating s-arc-transitive if and only
if it is s-arc-transitive and (1) follows.
For the converse, we have already observed that an s-arc-transitive graph is alternating
s-arc-transitive, and so the result holds if S = S−1. If S∩S−1 = ∅, then the result follows
by Theorem 3.3.
18 Art Discrete Appl. Math. 5 (2022) #P3.06
Definition 3.6. Let Γ be a digraph. we say that Γ is a strict digraph if for every arc
(a, b) ∈ A(Γ), the arc (b, a) ̸∈ A(Γ).
Note that if G is a group and S ⊆ G such that S ∩ S−1 = ∅, then Cay(G,S) is a strict
digraph.
Definition 3.7. Let A be an abelian group. The group Z2 ⋉ A ∼= ⟨ι, ÂL⟩ is a generalized
dihedral group.
Definition 3.8. A transitive permutation group G ≤ Sn is quasiprimitive if every nontrivial
normal subgroup of G is transitive.
We now characterize s-arc-transitive Cayley graphs of generalized dihedral groups with
abelian normal subgroup of odd order.
Theorem 3.9. Let s ≥ 2 and Γ be an s-arc-transitive Cayley graph of a generalized
dihedral group G with a normal abelian subgroup A of odd order n and index 2 in G.
Then one of the following is true:
(1) Γ is disconnected,
(2) Aut(Γ) is quasiprimitive or primitive,
(3) Γ ∼= Kn,n,
(4) Γ is isomorphic to a Haar graph corresponding to an s-arc-transitive Cayley graph
of A,
(5) Γ is isomorphic to a Haar graph corresponding to an alternating s-arc-transitive
Cayley strict digraph of A, or
(6) Γ is isomorphic to a Haar graph of A and its corresponding Cayley digraph need
not be s-arc-transitive. In this case, B is a block system of Aut(Γ) and the action of
fixAut(Γ)(B) on B1 is faithful and inequivalent to the action of fixAut(Γ)(B) on B0.
Proof. We assume (1) - (5) do not hold, and show (6) holds. As Aut(Γ) is neither quasiprim-
itive or primitive, there exists 1 < N ◁Aut(Γ) that is not transitive. Let C be the nontrivial
block system of Aut(Γ) formed by the orbits of N . As Γ is connected, there is some edge
in Γ between C1 and C2 ∈ C with C1 ̸= C2. As Γ is edge-transitive, there can be no edges
inside any block of C.
Case 1: N has two orbits. As there are no edges inside any block of C, Γ is a connected
bipartite graph with bipartition C. So Aut(Γ)/C ∼= Z2 has order 2. As GL has order
2n = |G|, we see fixGL(C) has order n. As the only proper normal subgroups of G
of odd order are contained in A as A is of odd order, fixGL(C) contains a semiregular
subgroup with two orbits isomorphic to A. Then Γ is isomorphic to a Haar graph of A
by [14, Lemma 2.5]. So we may assume without loss of generality that Γ = Haar(A,S)
(and C = B).
If the action of fixAut(Γ)(B) is not faithful, then by Theorem 2.16 we have Γ ∼=
Haar(A/H,S) ≀ K̄|H| for some maximal subgroup 1 < H ≤ A. If |A/H| = 1, then
Γ ∼= Kn,n ∼= Haar(G,A), contradicting our assumption that (3) does not hold. If |A/H| ≥
2, then it is straightforward to show that no such wreath product is 2-arc-transitive as
|V (Haar(A/H,S))| ≥ 4. To see this, first observe that Z2 × A/H is a block system of
T. Dobson: On automorphisms of Haar graphs of abelian groups 19
Aut(Γ) as Γ ∼= Haar(A/H,S) ≀K̄|H| and H was chosen to be maximal. Then some 2-arcs
in Γ are of the form ((0, ah), (1, ah), (0, ah′)) for some a ∈ G and h, h′ ∈ H with h′ ̸= h,
and some 2-arcs in Γ are of the form (0, ah), (1, ah), (0, bh) where h ∈ H , and a, b ∈ G
with aH ̸= bH . As Z2×A/H is a block system of Aut(Γ), no automorphism of Γ can map
a 2-arc of the first kind, whose vertices are in two blocks of Z2×A/H , to a 2-arc of the sec-
ond kind, whose vertices are in three blocks of Z2×A/H . Thus the action of fixAut(Γ)(B)
is faithful. If the action of fixAut(Γ)(B) on B1 is equivalent to the action of fixAut(Γ)(B)
on B0, then by Theorem 2.16 we may assume Aut(Γ) = Z2 ⋉Aut(Cay(A,S)), in which
case (4) or (5) would occur by Corollary 3.5. Then (6) follows from Theorem 2.16.
Case 2: If N has at least three orbits, then, as Γ is connected, Γ is a cover of some s-arc-
transitive graph by the Praeger Normal Cover Lemma [36], so N is semiregular. Let M be
the largest subgroup of N that is normal in Aut(Γ) and is contained in ÂL. Let D be the
block system of Aut(Γ) formed by the orbits of M .
Suppose M = 1. As C is the set of left cosets of some subgroup of G, and as the
square of every element of G is contained in A, C consists of blocks of size 2. So N is a
semiregular group of order 2. As A has odd order, ÂL/D ∼= A is a semiregular subgroup
of order n = |A| permutating n blocks, and so is regular. Then ⟨ÂL,M⟩ is transitive,
ÂL ∩ M = 1, and as M has order 2, ⟨ÂL,M⟩ is abelian. We conclude that ⟨ÂL,M⟩ ∼=
ÂL × M is abelian. Thus Γ is an s-arc-transitive Cayley graph of an abelian group, and
as n is odd, we have Γ is isomorphic to a Cayley graph of Z2n. By [30, Theorem 1.1],
Γ = K2n, Γ = Kn,n or Γ is Kn,n with a 1-factor deleted. If Γ = Kn, then Aut(Γ) is
primitive, a contradiction. By hypothesis, Γ ̸= Kn,n. Finally, Kn,n with a 1-factor deleted
is 2-arc-transitive but not 3-arc-transitive and is isomorphic to Haar(A,A−{0}). So Kn,n
with a 1-factor deleted is the Haar graph corresponding to the 2-arc-transitive Cayley graph
Kn = Cay(Zn,Zn − {0}), also a contradiction.
Suppose M ̸= 1. Then M has at least as many orbits as N . If G/M is abelian, then
the commutator subgroup G′ of G is contained in M , and as G′ ∼= A, [G : G′] = 2, M has
at most two orbits, a contradiction. Thus G/M is nonabelian. Then A/M is semiregular in
G/M with two orbits, so Γ/D is isomorphic to a Haar graph of A/M by [14, Lemma 2.5].
Then Γ/D is connected, and so B/D is a block system of Aut(Γ/D), and so of Aut(Γ)/D.
Then B is a block system of Aut(Γ), and this case reduces to the one above with N =
fixAut(Γ)(B) having two orbits.
Note that Kn,n does not satisfy 3.9(4) as Kn,n = Haar(A,A) but Cay(A,A) has
loops, and even if the loops were deleted, Kn is 2-arc-transitive but not 3-arc-transitive
while Kn,n is 3-arc-transitive. So including Kn,n in the conclusion of the above theorem
is not superfluous.
Also, the Heawood graph is 4-arc-transitive by the Foster Census [16], while its corre-
sponding Cayley digraph Cay(Z7, {1, 2, 4}) is arc-transitive but not 2-arc-transitive. This
follows as by [1, Theorem 2] Aut(Cay(Z7, {1, 2, 4})) has automorphism group G = {x 7→
ax + b : a ∈ {1, 2, 4}, b ∈ Z7}, and it is arc-transitive. However, as G has order 21,
Cay(Z7, {1, 2, 4}) cannot be 2-arc-transitive as there are 42 2-arcs in Cay(Z7, {1, 2, 4}).
Finally, while we have shown that there are graphs satisfying Theorem 3.9(6) holds,
and some of the other possibilities obviously hold, it is not clear that for each of the six
possibilities, there are corresponding graphs.
20 Art Discrete Appl. Math. 5 (2022) #P3.06
ORCID iDs
Ted Dobson https://orcid.org/0000-0003-2013-4594
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.07
https://doi.org/10.26493/2590-9770.1388.c56
(Also available at http://adam-journal.eu)
On the direct products of skew-morphisms
Jun-Yang Zhang*
School of Mathematical Sciences, Chongqing Normal University,
Chongqing 401331, P.R. China
Received 25 September 2020, accepted 26 January 2021, published online 18 July 2022
Abstract
A skew-morphism ϕ of a finite group G is a permutation on G fixing the identity el-
ement of G and for which there is an integer-valued function π on G such that ϕ(gh) =
ϕ(g)ϕπ(g)(h) for all g, h ∈ G. For two permutations α : A → A and β : B → B on
the sets A and B, their direct product α × β is the permutation on the Cartesian product
A × B given by (α × β)(a, b) =
(
α(a), β(b)
)
for all (a, b) ∈ A × B. In this paper,
necessary and sufficient conditions for a direct product of two skew-morphisms to still be
a skew-morphism are given.
Keywords: Finite group, skew-morphism, direct product.
Math. Subj. Class.: 20B25, 20B05, 05C25.
1 Introduction
All groups considered in this paper are finite. For a permutation α on a set A, we use |α|
to denote the order of α. The direct product α × β of two permutations α : A → A and
β : B → B is defined as the permutation on the Cartesian product A× B of sets A and B
given by (α× β)(a, b) =
(
α(a), β(b)
)
for all (a, b) ∈ A×B.
A skew-morphism ϕ of a group G is a permutation on G such that ϕ(1) = 1 and
ϕ(gh) = ϕ(g)ϕπ(g)(h) for all g, h ∈ G, where π is a function from G to the set Z|ϕ|
of nonnegative integers smaller than |ϕ|, called the power function of ϕ. Furthermore,
we say ϕ is of skew-type k provided π takes on exactly k values in Z|ϕ|, and call the set
Kerϕ = {g ∈ G|π(g) = 1} the kernel of ϕ. Clearly, a skew-morphism ϕ of G is of skew-
type 1 if and only if it is an automorphism of G. Therefore skew-morphisms of groups can
be viewed as a generalization of automorphisms of groups.
*Supported by the Basic Research and Frontier Exploration Project of Chongqing (No. cstc2018jcyjAX0010);
the Science and Technology Research Rrogram of Chongqing Municipal Education Commission (No. KJQN
201800512); and the National Natural Science Foundation of China (No. 11671276).
E-mail address: jyzhang@cqnu.edu.cn (Jun-Yang Zhang)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.07
It is well known that any automorphism of a group G gives rise to a semidirect product
of G and a cyclic group. In a similar way, a skew-morphism ϕ of a group G induces a so
called skew-product of G and a cyclic group. To see this, we use LG := {Lg|g ∈ G} to
denote the left regular representation of G. Then for any g, h ∈ G, we have
(ϕLg)(h) = ϕ(gh) = ϕ(g)ϕ
π(g)(h) = (Lϕ(g)ϕ
π(g))(h)
and so ϕLg = Lϕ(g)ϕ
π(g). Therefore, ⟨ϕ⟩LG ⊆ LG⟨ϕ⟩. Since G is finite and L(G) ∩
⟨ϕ⟩ = {1}, we have ⟨ϕ⟩LG = LG⟨ϕ⟩, which implies that T := LG⟨ϕ⟩ is a subgroup of
the group Sym(G) of all permutations on G, while T is said to be a skew-product of LG
and ⟨ϕ⟩. Conversely, suppose that T is any group admitting a complementary factorization
T = GY with Y = ⟨y⟩ cyclic. Then for any g ∈ G, there exist unique g′ ∈ G and
yi ∈ ⟨Y ⟩ such that yg = g′yi. Define ϕ : G → G by ϕ(g) = g′, and π : G → Z|ϕ| by
π(g) = i. Then ϕ is a skew-morphism of G with associated power function π.
Based on the above reasons, skew-morphism can be a powerful tool for studying prob-
lems which can be reduced to complementary factorizations of a finite group into two
groups, where one of them is cyclic. Actually, the definition of skew-morphism just comes
from such problems. The concept of a skew morphism of a group was introduced by
Jajcay and J. Siráň [10] about two decades ago in their study of the theory of regular
Cayley maps, which are embeddings of graphs on surfaces that admit a group of auto-
morphisms acting regularly on the vertices of the embedded graph. Subsequently, several
articles using skew-morphism as a tool for study of regular Cayley maps were published
[4, 5, 7, 11, 12, 15, 16, 17, 18, 20, 21, 22]. Recently, several papers devoted to the theory
of skew-morphisms itself emerged, which focus on cyclic groups [2, 8, 13, 14], dihedral
groups [9, 19, 23] and simple groups [1, 3]. In [6], a general theory of skew-morphisms
and its connection with complementary products was presented.
Although skew-morphisms of groups are a generalization of group automorphisms,
they fail to have some good properties which possessed by the latter. For example, the com-
position of two skew-morphisms of a group is often not a skew-morphism. In
[6, Lemma 7.2], it was shown that a direct product of a skew-morphism of a group A and
the identity permutation on a group B is a skew-morphism of the Cartesian product A×B.
However, unlike automorphisms, direct products of skew-morphisms are not always skew-
morphisms. The aim of this paper is to deduce necessary and sufficient conditions for a
direct product of two skew-morphisms to still be a skew-morphism.
The paper is organized as follows. In Section 2, some preliminary results on skew-
morphisms are listed. In Section 3, the main results of this paper are presented and proved.
2 Preliminaries
In this section, we introduce some known result for later use. First let us fix a number
of notations. We use gcd{n1, n2, . . . , nt} and lcm{n1, n2, . . . , nt} to denote the greatest
common divisor and least common multiple of the t integers n1, n2, . . . , nt respectively.
For a skew-morphism ϕ of G, we use Ox to denote the orbit of any x ∈ G under the action
of ϕ, and |Ox| to denote the length of Ox. If ϕ preserves the subset S of G, then we use
ϕ|S to denote the restriction of ϕ to S.
The following two propositions give some basic properties of skew-morphisms and
their power functions.
J.-Y. Zhang: On the direct products of skew-morphisms 3
Proposition 2.1 ([10]). Let ϕ be a skew-morphism of a group G and let π be the power
function of ϕ. Then the following hold:
(i) the set Kerϕ = {a ∈ G|π(a) = 1} is a subgroup of G;
(ii) π(g) = π(h) if and only if (Kerϕ)g = (Kerϕ)h;
(iii) ϕk(gh) = ϕk(g)ϕ
k−1∑
i=0
π(ϕi(g))
(h) for all positive k and all g, h ∈ G;
(iv) π(gh) ≡
π(g)−1
∑
i=0
π(ϕi(h)) (mod |ϕ|) for all g, h ∈ G.
Proposition 2.2 ([23]). Suppose that G = ⟨xi
∣
∣ 1 ≤ i ≤ t⟩ and ϕ is a skew-morphism of
G with associated power function π. Then
(i) |ϕ| = lcm{|Ox1 |, |Ox2 |, . . . , |Oxt |};
(ii) for any c ∈ G, if π(c) ≡ 1 (mod |Oxi |) for all i = 1, 2, . . . , t, then c ∈ Kerϕ.
Proposition 2.2 has a direct corollary as follows.
Corollary 2.3. Let ϕ be a skew-morphism of G preserving two subgroups A and B of G
and let s and t be the orders of ϕ|A and ϕ|B respectively. Then |ϕ| = lcm{s, t}.
The following proposition is straightforward.
Proposition 2.4. Suppose that H is a subgroup of G. Let ϕ be a skew-morphism of G
preserving H . Then the restriction of ϕ to H is a skew-morphism of H .
3 Main results
In this section, we deduce necessary and sufficient conditions for a direct product of two
skew-morphisms to still be a skew-morphism. For convenience, the direct (or semidirect)
product of groups refers to their internal direct (or semidirect) product. Therefore, if G =
A× B (or G = A⋊ B) is the direct product (or semidirect product) of two groups A and
B, then both of these two groups can be seen as subgroups of G and any element of G can
be uniquely written as ab for some a ∈ A and b ∈ B. If α and β are permutations on the
groups A and B respectively, then the direct product α × β of α and β is the permutation
on G = A×B given by (α× β)(ab) = α(a)β(b) for all ab ∈ G.
The following lemma plays a key role in the proof of our main theorem and it has some
interesting corollaries.
Lemma 3.1. For a semidirect product G = A ⋊ B of groups A and B, let ϕ be a skew-
morphism of G preserving both A and B and let π be the associated power function of ϕ.
Then π(a) ≡ 1 (mod |Ob|) for any a ∈ A and any b ∈ B. In particular, A ≤ Kerϕ if ϕ|A
is an automorphism of A.
Proof. Take arbitrary a ∈ A and b ∈ B. Then
ϕ(a)ϕπ(a)(b) =ϕ(ab) = ϕ(bb−1ab) = ϕ(b)ϕπ(b)(b−1ab)
=ϕ(b)ϕπ(b)(b−1ab)ϕ(b)−1ϕ(b).
(3.1)
4 Art Discrete Appl. Math. 5 (2022) #P3.07
Since ϕ preserves A and A ◁ G, we have ϕ(a), ϕ(b)ϕπ(b)(b−1ab)ϕ(b)−1 ∈ A. Since
ϕ preserves B, we get ϕ(b), ϕπ(a)(b) ∈ B. Note that A ∩ B = {1}. By (3.1), we
have ϕ(a) = ϕ(b)ϕπ(b)(b−1ab)ϕ(b)−1 and ϕπ(a)(b) = ϕ(b). It follows that π(a) ≡ 1
(mod |Ob|). In particular, if ϕ|A is an automorphism of A, then π(a) ≡ 1 (mod |Ox|) for
all x ∈ A. Then by Proposition 2.2(ii), we obtain a ∈ Kerϕ and thus A ≤ Kerϕ.
Corollary 3.2. Let G = A ⋊ B be a semidirect product of groups A and B and let ϕi,
i = 1, 2 be two skew-morphisms of G preserving both A and B. If ϕ1|A = ϕ2|A and
ϕ1|B = ϕ2|B , then ϕ1 = ϕ2.
Proof. Lemma 3.1 implies ϕi(ab) = ϕi(a)ϕi(b) for any a ∈ A and b ∈ B. Since ϕ1|A =
ϕ2|A and ϕ1|B = ϕ2|B , we have ϕ1(a) = ϕ2(a) and ϕ1(b) = ϕ2(b). It follows that
ϕ1(ab) = ϕ2(ab) and hence ϕ1 = ϕ2.
Corollary 3.3. Let G = A × B be a direct product of groups A and B, and let ϕ be a
skew-morphism of G preserving both A and B. Then Kerϕ = Kerϕ|A × Kerϕ|B .
Proof. Since ϕ preserves both A and B, it follows from Proposition 2.4 that ϕ|A is a skew-
morphism of A and ϕ|B is a skew-morphism of B. Let π be the associated power function
of ϕ. Take any x ∈ Kerϕ|A. Then π(x) ≡ 1 (mod |Oa|) for all a ∈ A. By Lemma 3.1,
π(x) ≡ 1 (mod |Ob|) for all b ∈ B. Then by Proposition 2.2(ii), we get x ∈ Kerϕ
and therefore Kerϕ|A ⊂ Kerϕ. Similarly, Kerϕ|B ⊂ Kerϕ. On the other hand, for any
xy ∈ Kerϕ with x ∈ A and y ∈ B and any a ∈ A, we have
ϕ(xa) =ϕ(xyay−1)
=ϕ(xy)ϕ(ay−1)
=ϕ(x)ϕ(y)ϕ(a)ϕ(y−1)
=ϕ(x)ϕ(a)ϕ(y)ϕ(y−1).
Since xa ∈ A and ϕ preserves A, we have ϕ(y)ϕ(y−1) = 1 and ϕ(xa) = ϕ(x)ϕ(a).
Therefore x ∈ Kerϕ|A. Similarly, y ∈ Kerϕ|B . Summarizing all the above we have
Kerϕ = Kerϕ|A × Kerϕ|B .
Corollary 3.4. Let G = A × B be a direct product of groups A and B, and let ϕ be a
skew-morphism of G with associated power function π. If ϕ preserves both A and B, then
π(ab) ≡ π(a)π(b) (mod |ϕ|) for all a ∈ A and b ∈ B.
Proof. Take arbitrary a ∈ A and b ∈ B and let i be any nonnegative integer. Since ϕ
preserves both A and B, it follows from Lemma 3.1 that π(ϕi(a)) ≡ 1 (mod t) and
π(ϕi(b)) ≡ 1 (mod s) where s and t are the orders of the restriction of ϕ to A and B
respectively. Therefore, by Proposition 2.1(iv), we have
π(ab) ≡
π(a)−1
∑
i=0
π(ϕi(b)) ≡ π(a) ≡ π(a)π(b) (mod s)
and
π(ab) ≡
π(a)−1
∑
i=0
π(ϕi(b)) = π(b) ≡ π(a)π(b) (mod t).
By Corollary 2.3, |ϕ| = lcm{s, t}. It follows that π(ab) ≡ π(a)π(b) (mod |ϕ|).
J.-Y. Zhang: On the direct products of skew-morphisms 5
The following is our main theorem.
Theorem 3.5. Let G = A×B be a direct product of groups A and B, and let α, β be skew-
morphisms of A and B with associated power functions πα and πβ respectively. Then the
direct product α×β of α and β is a skew-morphism of G if and only if πα(a) ≡ πβ(b) ≡ 1
(mod d) for all a ∈ A and b ∈ B where d = gcd{|α|, |β|}.
Proof. For convenience, write ϕ = α × β. Then ϕ(ab) = α(a)β(b) for any ab ∈ G.
Furthermore, ϕ preserves both A and B, and ϕ|A = α and ϕ|B = β. Set |α| = s, |β| = t
and |ϕ| = m. Then d = gcd{s, t}. By Corollary 2.3, we get m = lcm{s, t}.
First we prove the necessity. Suppose that ϕ is a skew-morphism of G with the asso-
ciated power function π. By Lemma 3.1, we have π(a) ≡ 1 (mod |Ob|) and π(b) ≡ 1
(mod |Oa|) for any a ∈ A and b ∈ B. It follows that π(a) ≡ 1 (mod t) and π(b) ≡ 1
(mod s). Since d = gcd{s, t}, we have π(a) ≡ π(b) ≡ 1 (mod d). Clearly, π(a) ≡
πα(a) (mod s) and π(b) ≡ πβ(b) (mod t). Then noting that d | s and d | t, we have
πα(a) ≡ πβ(b) ≡ 1 (mod d).
Next, we prove the sufficiency. Suppose that πα(a) ≡ πβ(b) ≡ 1 (mod d) for all
a ∈ A and b ∈ B. Let u, v be a pair of integers such that us+ vt = d. Define a function π
from G to Zm by the rule:
π(ab) ≡
(vt
d
(
πα(a)− 1
)
+ 1
)(us
d
(
πβ(b)− 1
)
+ 1
)
(mod m) (3.2)
for any ab ∈ G. Since πα(a) ≡ πβ(b) ≡ 1 (mod d), we have
vt
d
(
πα(a)− 1
)
+ 1 ≡ 1 (mod t), (3.3)
vt
d
(
πα(a)− 1
)
+ 1 = πα(a)−
us
d
(
πα(a)− 1
)
≡ πα(a) (mod s), (3.4)
us
d
(
πα(b)− 1
)
+ 1 ≡ 1 (mod s), (3.5)
and
us
d
(
πα(b)− 1
)
+ 1 ≡ πβ(b) (mod t). (3.6)
By equations (3.2), (3.4) and (3.5), we get π(ab) ≡ πα(a) (mod s). By equations (3.2),
(3.3) and (3.6), we get π(ab) ≡ πβ(b) (mod t). It follows that
ϕ(a1b1a2b2) =ϕ(a1a2b1b2)
=α(a1a2)β(b1b2)
=α(a1)α
πα(a1)(a2)β(b1)β
πβ(b1)(b2)
=α(a1)β(b1)α
πα(a1)(a2)β
πβ(b1)(b2)
=α(a1)β(b1)α
π(a1b1)(a2)β
π(a1b1)(b2)
=ϕ(a1b1)ϕ
π(a1b1)(a2b2).
Therefore ϕ is a skew-morphism of G with associated power function π.
By Proposition 2.3, Corollary 3.3 and Corollary 3.4, we have the following theorem,
the proof for which is trivial and so is omitted.
6 Art Discrete Appl. Math. 5 (2022) #P3.07
Theorem 3.6. Let G = A × B be a direct product of groups A and B, and let α, β be
skew-morphisms of A and B with associated power functions πα and πβ respectively. If
the direct product ϕ = α×β of α and β is a skew-morphism of G, then the following hold:
(i) |ϕ| = lcm{|α|, |β|};
(ii) the associated power function π of ϕ are given by π(ab) ≡ πα(a)πβ(b) (mod |ϕ|)
for all a ∈ A and b ∈ B;
(iii) Kerϕ = Kerα× Kerβ.
ORCID iDs
Jun-Yang Zhang https://orcid.org/0000-0002-0871-2059
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.08
https://doi.org/10.26493/2590-9770.1392.f9a
(Also available at http://adam-journal.eu)
Classifying edge-biregular maps
of negative prime Euler characteristic*
Olivia Reade†
The Open University, Milton Keynes, UK
Jozef Širáň‡
The Open University, Milton Keynes, UK and
Slovak University of Technology, Bratislava, Slovakia
Dedicated to Marston Conder on the occasion of his 65th birthday.
Received 31 October 2020, accepted 29 March 2020, published online 18 July 2022
Abstract
An edge-biregular map arises as a smooth normal quotient of a unique index-two sub-
group of a full triangle group acting with two edge-orbits. We give a classification of all
finite edge-biregular maps on surfaces of negative prime Euler characteristic.
Keywords: Biregular map, automorphism group, non-orientable surface, Euler characteristic.
Math. Subj. Class.: 20B25, 20F05, 57M60, 05C25
1 Introduction
A map is a 2-cell embedding of a connected graph on a surface (which may be orientable
or not but without boundary components). As such it is possible to form the barycentric
subdivision of a map, and the resulting regions are the flags of the map. An automorphism
*Both authors attended the Symmetries of Discrete Objects 2020 conference held in Rotorua, New Zealand
and would like to thank the organisers Marston Conder and Gabriel Verret for arranging the event and for their
generous hospitality. It was at this Symmetries of Discrete Objects workshop that some of the above research
was conducted and partial results were presented. Both authors would like to thank the anonymous referee whose
comments made us aware of both errors and omissions in our original manuscript, leading us to make significant
improvements to this paper.
†Corresponding author. The author is grateful for the travel grants from both the Open University and also
from the London Mathematical Society which enabled her to attend the conference.
‡The author gratefully acknowledges support from the APVV Research Grants 17-0428 and 19-0308, as well
as from the VEGA Research Grants 1/0567/22 and 1/0206/20.
E-mail addresses: olivia.jeans@open.ac.uk (Olivia Reade), jozef.siran@open.ac.uk (Jozef Širáň)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.08
of a map is a structure-preserving bijection of the set of flags onto itself, and the set of all
automorphisms of a map form its automorphism group.
Informally, how much symmetry a map displays may be measured by the size of its
automorphism group. If a map automorphism fixes a flag, then, by connectivity, it must fix
all flags and hence be the identity; this means that the action of the automorphism group of
a map is semi-regular on flags. A largest ‘level of symmetry’ exhibited by a map therefore
arises when the automorphism group is regular, that is semi-regular and transitive, on the
set of flags. Such maps are fully regular; they are, in some sense, the most symmetric maps
[13]. Elements of the automorphism group of a fully regular map can be identified with
flags in a one-to-one manner by fixing a flag and assigning to every other flag the unique
automorphism taking the fixed flag to the other one.
An obvious consequence of full regularity of a map is that all its vertices have the
same valency and all the faces have boundary walks of the same length; we speak about
a map of type (k, ℓ) in the case of valency k and face length ℓ. As it turns out, every
fully regular map of type (k, ℓ) is a smooth normal quotient of a (k, ℓ)-tessellation of a
simply connected surface by ℓ-gons, k of which meet at every vertex. Equivalently, in a
group-theoretic language, the automorphism group of a fully regular map of type (k, ℓ) is
a torsion-free normal quotient of the automorphism group of a (k, ℓ)-tessellation; the latter
is known as the full (2, k, ℓ)-triangle group.
There are other classes of ‘highly symmetric’ maps which are not necessarily fully
regular but which have automorphism groups that can still be described as ‘large’. Perhaps
the most natural candidate for a map to qualify for such a distinction would be when a
group of automorphisms of the map has two (necessarily equal-sized) orbits and is regular
on each. Following the example of existing graph-theoretic language [7, 11, 15], we use the
term biregular to describe such maps. As in the case of fully regular maps, automorphism
groups of biregular maps of type (k, ℓ) also arise as torsion-free normal quotients, but this
time of index-two subgroups of the full (2, k, ℓ)-triangle groups. As we shall see, a full
(2, k, ℓ)-triangle group may contain up to 7 subgroups of index two (depending on the
parity of k and ℓ). This potentially gives rise to 7 different kinds of biregular maps, and we
now give two examples.
Perhaps the most well known kind of biregularity arises from the subgroup of a full tri-
angle group formed by orientation-preserving automorphisms, leading to a widely studied
class of maps known as orientably-regular [13] or rotary [14]. We note that orientability of
a map is equivalent to admitting local orientations at vertices that are consistent when pass-
ing between incident vertices along an edge on the carrier surface of the map. An opposite
phenomenon, namely, when a map of type (k, ℓ) admits an assignment of local orientations
at vertices which disagree when traversing along an arbitrary edge on the surface and, at
the same time, exhibits the highest ‘level of symmetry’ with respect to this property, was
studied in [6]. These maps, called bi-rotary in [6], arise as normal torsion-free quotients of
a different index-two subgroup of the full (2, k, ℓ)-triangle group.
In this paper we will focus on still another kind of biregular maps, those admitting
an action of the map automorphism group with two orbits on edges, accordingly called
edge-biregular. It turns out that edge-biregular maps of type (k, ℓ) arise as torsion-free
normal quotients from a unique index-two subgroup of the full (2, k, ℓ)-triangle group. In
such a map, edges emanating from a vertex in their cyclic order on the carrier surface must
necessarily alternate between the two orbits, so that k must be even. Similarly, ℓ must also
be even.
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 3
Edge-biregular maps have been investigated in great detail in [12], including their clas-
sification on surfaces of non-negative Euler characteristic and also a classification of such
maps with a dihedral automorphism group. Our aim here is to complement the results of
[12] by deriving a classification of edge-biregular maps on surfaces with negative prime Eu-
ler characteristic, extending thereby also the existing classification results for fully regular
maps [2] and bi-rotary maps [6] on these surfaces.
To this end, in section 2 we introduce biregular maps in the wider context of symmetric
maps of a given type and establish notation and some basic facts about edge-biregular
maps. Section 3 contains a summary of our results on classification of edge-biregular
maps on surfaces of negative prime Euler characteristic and a setup of proofs; these are
subsequently given in sections 4 and 5. Finally section 6 contains further observations and
remarks.
2 Algebra of edge-biregular maps
It is well known (see for example [10] or [3], or the survey [13]) that fully regular maps of
a given type (k, ℓ) can be identified with torsion-free normal quotients of the full (2, k, ℓ)-
triangle group with presentation
⟨ R0, R1, R2 |R
2
0, R
2
1, R
2
2, (R0R2)
2, (R1R2)
k, (R0R1)
ℓ ⟩ (2.1)
This group is isomorphic to the full automorphism group of a universal (k, ℓ)-tessellation
of a simply connected surface, where the tiles are regular l-gons, and the valency of the
underlying graph is k. One usually considers a universal (k, ℓ)-tessellation also as a geo-
metrically regular tiling of a sphere, a Euclidean plane, or a hyperbolic plane, depending
on whether the value of 1/k + 1/ℓ is greater than, equal to, or smaller than 1/2.
As alluded to in the introduction, biregular maps arise as smooth, that is torsion-free,
normal quotients of subgroups of the full (2, k, ℓ)-triangle groups of index two. Any such
subgroup is completely determined by a non-empty subset of {R0, R1, R2}, consisting of
the involutory generators not contained in the subgroup. There are 7 such subsets, and
the number of resulting subgroups of index two depends on the parities of k and ℓ. The
index two subgroup avoiding each of R0, R1, R2 occurs in every one of the triangle groups,
regardless of parity of k and ℓ. If both k and ℓ are odd then this is the only subgroup of
index two in the full (2, k, ℓ)-triangle group. If exactly a given one of k, ℓ is even, there are
two further index two subgroups, and so including the aforementioned subgroup, 3 such
subgroups in total. If both k and ℓ are even, the full (2, k, ℓ)-triangle group contains 7
subgroups of index two.
The subgroup of (2.1) of index two containing none of the three involutory generators,
informally denoted ⟨R0, R1, R2⟩ here, is known as the (ordinary) (2, k, ℓ)-triangle group
comprising all orientation-preserving automorphisms of a universal (k, ℓ)-tessellation. It is
generated, for instance, by the products U = R2R1 and V = R1R0 representing rotations
of the tessellation about a vertex and about the centre of a face incident to the vertex, and
has (irrespective of parity of k, ℓ) a presentation of the form
⟨R0, R1, R2 ⟩ = ⟨U, V |U
k, V ℓ, (UV )2 ⟩ (2.2)
Smooth normal quotients of this subgroup give rise to orientably-regular maps in the ter-
minology of [13], also known as rotary maps in [14]. The second type of biregular maps
mentioned in the introduction, the bi-rotary maps of [6], arise as smooth quotients of the
4 Art Discrete Appl. Math. 5 (2022) #P3.08
index-two subgroup ⟨R0, R1, R2 ⟩ of (2.1) containing R0 but avoiding R1 and R2, for ℓ
even. Using the notation A = R0, Z = R1R2, and the square brackets for a commutator
of two elements as usual, this subgroup admits a presentation
⟨R0, R1, R2 ⟩ = ⟨A, Z |A
2, Zk, [A,Z]ℓ/2 ⟩ (2.3)
In this paper we will focus on biregular maps corresponding to torsion-free normal quo-
tients of the subgroup ⟨R0, R1, R2 ⟩ of the full (2, k, ℓ)-triangle group (2.1) for both k and
ℓ even, distinguished by inclusion of R0 and R2 and avoidance of R1. It may be checked
that the subgroup ⟨R0, R1, R2 ⟩ is the only one among the seven index-two subgroups of
(2.1) inducing two edge orbits on a universal (k, ℓ)-tessellation for k, ℓ even. With the help
of new variables X = R0, Y = R2, S = R1R0R1 and T = R1R2R1 it can be shown that
a complete presentation of this subgroup is
⟨R0, R1, R2 ⟩ = ⟨X, Y, S, T |X
2, Y 2, S2, T 2, (XY )2, (ST )2, (TY )k/2, (SX)ℓ/2 ⟩
(2.4)
If N is a torsion-free normal subgroup of ⟨R0, R1, R2 ⟩ of finite index, the corresponding
quotient group H = ⟨R0, R1, R2 ⟩/N has a presentation of the form
H = ⟨x, y, s, t |x2, y2, s2, t2, (xy)2, (st)2, (ty)k/2, (sx)ℓ/2, . . . ⟩ (2.5)
with x = XN , y = Y N , s = SN and t = TN . By the correspondence between
maps and groups, any finite group H together with a presentation as in (2.5) is a group
of automorphisms of a finite biregular map, the biregularity of which is derived from the
index-two subgroup ⟨R0, R1, R2 ⟩ of (2.1). Maps exhibiting this kind of biregularity were
referred to in [12] as edge-biregular.
The obvious motivation for this terminology comes from the already mentioned two-
orbit action of the subgroup ⟨R0, R1, R2 ⟩ on edges of a universal (k, ℓ)-tessellation. By
taking a torsion-free normal quotient as in (2.5) this two-orbit action projects onto edges of
the quotient map as shown in Figure 1.
In order to introduce a ‘reference system’ within the map on the diagram one may fix
and shade an arbitrarily chosen flag f , let x and y represent the reflections of H in the
sides of f as depicted, and call the quadruple of flags f , xf , yf and xyf surrounding an
edge ‘distinguished’. The pattern of the shaded flags in this diagram demonstrate how the
group of automorphisms H of an edge-biregular map ‘spreads around’ the distinguished
quadruple and hence acts transitively on faces and vertices. At the same time, Figure 1
displays two orbits of H on edges, those which are bold in one orbit, with dashed edges
indicating the other orbit; notice also the two H-orbits on flags formed by quadruples of
shaded and unshaded flags surrounding respectively bold and dashed edges.
Observe that it is a choice, consistent with the choice of notation, which orbit of edges
is coloured bold, and which is dashed. As shown in Figure 1, the generators x and y cor-
respond to automorphisms acting locally as reflections respectively along and across the
distinguished bold edge. Meanwhile s and t correspond to reflections respectively along
and across the distinguished dashed edge, that is the dashed edge which is surrounded
by the quadruple of unshaded flags sharing a boundary with f . We call this, and the
corresponding presentation (2.5), the canonical form of the edge-biregular map M and
denote it M = (H;x, y, s, t). A pair of edge-biregular maps M = (H;x, y, s, t) and
M ′ = (H ′;x′, y′, s′, t′) given in the canonical form are isomorphic if there is a group
isomorphism H → H ′ taking z onto z′ for every z ∈ {x, y, s, t}.
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 5
f
xf
yf
sf
tf
Figure 1: The images under x, y, s, t of the distinguished flag f of an edge-biregular map.
Whether an edge-biregular map M = (H;x, y, s, t) is also fully regular will depend
on the existence (or otherwise) of an involutory automorphism of M lying outside H and
fusing the two H-orbits of edges together. By the above, this condition is equivalent to the
existence of an automorphism of the group H which interchanges x with s and y with t.
The map M = (H;x, y, s, t) is thus fully regular if and only if M and M ′ = (H; s, t, x, y)
are isomorphic as maps, otherwise we say these maps are twins.
To avoid unnecessary work, all our forthcoming results will be up to duality. The dual
map M∗ of an edge-biregular map M = (H;x, y, s, t) is also an edge-biregular map and
is formed by interchanging x with y and s with t in the presentation (2.5) for H , thereby
swapping the vertices with the faces and vice versa, to give M∗ = (H; y, x, t, s). If M
is isomorphic to M∗ the map is self-dual, which (by the map isomorphism condition) is
equivalent to the existence of an automorphism of the group H swapping x with y and s
with t.
Except for the case of characteristic −2, all our edge-biregular maps M = (H;x, y, s, t)
will be carried by non-orientable surfaces. Since each canonical generator of H corre-
sponds to a reflection on the carrier surface, it follows that M is supported by an orientable
surface if and only if every relator in the presentation of H has an even length in terms of
x, y, s, t. This is easily seen to be equivalent to the statement that the carrier surface of M
is non-orientable if and only if H is generated by any three products, of two involutions
each, provided that every involution out of x, y, s, t appears in at least one product.
Having assumed that our edge-biregular maps M = (H;x, y, s, t) arise from the index-
two subgroup ⟨R0, R1, R2 ⟩ of full triangle groups by torsion-free normal subgroups im-
plies that all the four canonical involutory generators of H are distinct. For completeness
6 Art Discrete Appl. Math. 5 (2022) #P3.08
we address also the situations when some of the generators in the set {x, y, s, t} are either
trivial or equal to each other. These are commonly known as ‘degeneracies’ and have been
treated in detail in [12]; they give rise to maps on surfaces with boundary or to maps with
semi-edges or to maps of spherical type with 2 ∈ {k, l}. Recall that a semi-edge has one
of its endpoints attached to a vertex but the other one is ‘dangling’ and not incident to any
vertex; we will regard edges and semi-edges as different objects.
While we are not considering maps on surfaces with boundary here, edge-biregular
maps with semi-edges are relevant for us. Leaving the trivial case of a semi-star (a spherical
one-vertex map with some number of attached semi-edges) aside, suppose that an edge-
biregular map of type (k, ℓ) with both entries even contains a semi-edge as well as an edge.
By [12], edges and semi-edges must then alternate around each vertex on the carrier surface
of the map, and consequently the deletion of all semi-edges results in a fully regular map
of type (k/2, ℓ/2). Conversely, every edge-biregular map containing both an edge and a
semi-edge arises from a fully regular map that is not a semi-star by inserting a semi-edge
into each corner.
To arrive at an algebraic explanation of this edge/semi-edge phenomenon in edge-
biregular maps, consider such a map M = (H;x, y, s, t) in a canonical form and assume
that all the dashed edges in Figure 1 would collapse to semi-edges. This is equivalent to
identifying the flags marked sf and tf into a single flag, which means regarding s and t as
identical automorphisms. It now follows from (2.5) that, up to twinness, an edge-biregular
map M = (H;x, y, s, t) of type (k, ℓ) for even k and ℓ, containing both edges and semi-
edges, may be identified with its group of automorphisms H presented in the form
H = ⟨x, y, s, t |x2, y2, s2, t2, (xy)2, st, (ty)k/2, (sx)ℓ/2, . . . ⟩ (2.6)
Note the difference from (2.5) in the power at the product st, implying s = t in (2.6).
Conversely, let a fully regular map M of type (k/2, ℓ/2) for k, ℓ even and distinct from
a semi-star be given by its (full) automorphism group, that is, by a torsion-free normal
quotient of (2.1) presented in the form
⟨ r0, r1, r2 | r
2
0, r
2
1, r
2
2, (r0r2)
2, (r1r2)
k/2, (r0r1)
ℓ/2 . . . ⟩ (2.7)
Then, a presentation of the group H as in (2.6) for the corresponding edge-biregular map
of type (k, ℓ) arising from M by inserting a semi-edge into every corner of M is obtained
from the one in (2.7) simply by letting x = r0, y = r2, and s = t = r1.
3 Results, and proofs setup
In this section we state our classification results for edge-biregular maps on surfaces of
negative prime Euler characteristic. We note that a classification of such maps on surfaces
with a non-negative Euler characteristic (a sphere, a projective plane, a torus and a Klein
bottle) was derived in [12].
As already alluded to, edge-biregular maps on surfaces without boundary split into
two families, according to whether they contain semi-edges or not. Also, for the purpose
of this section we may disregard the semi-star maps. Let us first consider edge-biregular
maps containing both edges as well as semi-edges. By the facts summed up at the end
of the previous section, edge-biregular maps of type (k, ℓ) for even k and ℓ containing
both edges as well as semi-edges and carried by a particular surface are in a one-to-one
correspondence with fully regular maps of type (k/2, ℓ/2) on the same surface. It follows
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 7
that a classification of all fully regular maps on a particular surface automatically implies
a classification of edge-biregular maps with semi-edges on the same surface, including
presentations of the corresponding groups of automorphisms. And since a classification of
fully regular maps on surfaces of negative prime Euler characteristic is available from [2],
a corresponding classification of edge-biregular maps with semi-edges on these surfaces
follows by the outlined procedure. In the interest of saving space we will not go into any
further detail and we also omit a formal statement of the related presentations, referring the
reader to [2] or to [13].
We now pass onto classification of edge-biregular maps without semi-edges on surfaces
with negative prime Euler characteristic. From the work in [8] and [12], we know that if
such a map M = (H;x, y, s, t) is given in a canonical form with the group of automor-
phisms H presented as in (2.5), then all the four generators are non-trivial and distinct. We
state our main results separately for odd primes p (in which case, of course, all the maps
are carried by non-orientable surfaces) and then for p = 2.
Theorem 3.1. A surface of negative odd prime Euler characteristic χ = −p supports, up
to duality and twin maps, only the following pairwise non-isomorphic edge-biregular maps
M = (H;x, y, s, t) with no semi-edges, given in their canonical forms:
(1) A single-vertex map of type (4(p+1), 4) with H = Hp(1) = ⟨y, t⟩ isomorphic to the
dihedral group of order 4(p+ 1) with canonical presentation
Hp(1) = ⟨x, y, s, t |x
2, y2, s2, t2, (xy)2, (st)2, xys, s(yt)p+1⟩
(2) A two-vertex map of type (2(p + 2), 4) with H = Hp(2) = ⟨x, t⟩ isomorphic to the
dihedral group of order 4(p+ 2) and canonical presentation
Hp(2) = ⟨x, y, s, t |x
2, y2, s2, t2, (xy)2, (st)2, xys, s(xt)p+2⟩
(3) The maps Mp,j of type (k, ℓ) where k = 4κ and ℓ = 2λ for odd and relatively
prime κ and λ, such that p = 2κλ − 2κ − λ, where H = Hp,j is of order kl/2 and
presented as follows, for any positive integer j < λ such that j2 ≡ 1 mod λ, with
a = (j − 1)(λ+ 1)/2, and u = sx, v = ty:
Hp,j = ⟨x, y, s, t |x
2, y2, s2, t2, (xy)2, (st)2,
uλ, v2κ, [s, v2], [x, v2], tutuj , vκuas ⟩
(4) When p ≡ 5 mod 9, that is when p = 9m − 4 we have a map of type (8, 6m) and
the presentation of the corresponding group H = Hp which has order 24m is of the
form
Hp = ⟨x, y, s, t |x
2, y2, s2, t2, (xy)2, (st)2, (sx)3m, (ty)4, (sxy)2t, txty⟩
(5) When p = 3, we have a map of type (4, 6) and the corresponding group H = H(3) ∼=
D6 ×D6 has canonical presentation
H(3) = ⟨x, y, s, t |x
2, y2, s2, t2, (xy)2, (st)2, (ty)2,
(sx)3, (xyt)3, (sty)3, (xyst)2⟩
8 Art Discrete Appl. Math. 5 (2022) #P3.08
Moreover, the only fully regular map in the above list is H(3).
Theorem 3.2. A surface of Euler characteristic χ = −2 supports, up to duality and twin
maps, only the following non-isomorphic edge-biregular maps M = (H;x, y, s, t) with no
semi-edges:
(1) One self-dual map of type (8, 8) with a single vertex and a single face, with the group
H = H2,1 presented as
H2,1 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)4, (sx)4, ysxs, txsx, xtyt, syty⟩
∼= D8
(2) One map of type (4, 12) with three vertices and a single face, with the group H =
H2,2 having canonical presentation
H2,2 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)6, xyt, t(sx)3⟩ ∼= D12
(3) One self-dual map of type (6, 6) with two vertices and two faces, with the group
H = H2,3 being as follows
H2,3 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)3, (sx)3, styx⟩ ∼= D12
(4) Six maps of type (4, 8) with four vertices and two faces, with groups H2,i ∼= D8×C2
for i = 4, 5, 6, 7, 8, 9 presented in the form
H2,4 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)4, (tx)2, ysxs⟩
H2,5 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)2, (tx)2, y(sx)2⟩
H2,6 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)2, (stx)2, y(sx)2⟩
H2.7 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)2, (tx)2, ty(sx)2⟩
H2.8 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)2, (stx)2, tyxsx⟩
H2.9 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)2, (stx)2, tys⟩
(5) Three maps of type (4, 6) with six vertices and four faces, with groups of automor-
phisms H2,i for i = 10, 11, 12 given by
H2,10 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)3, t(sy)2, y(xt)2⟩ ∼= S4
H2,11 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)3, (ysx)2, (tx)2⟩
∼= D6 × V4
H2,12 = ⟨ x, y, s, t | x
2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)3, (ys)2, (tx)2⟩
∼= D6 × V4
The maps supported by the groups H2,1, H2,3, H2,4, H2,7, H2,11 and H2,12 are orientable
while the other six are not, and the only fully regular ones out of the twelve are those
supported by the groups H2,1, H2,3, H2,7, H2,10 and H2,12.
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 9
We make no claim of a lack of redundancy in these presentations - in fact quite the
opposite: some of the above presentations have unnecessary relators which have been re-
tained. This is deliberate in order to better demonstrate the interplay between the canonical
generators, and also to make evident the presence, or absence, of full regularity or self-
duality in the underlying map.
In order to set the stage for proofs in the next three sections, throughout we let M =
(H;x, y, s, t) be an edge-biregular map of type (k, ℓ) for even k and ℓ, with no semi-edges,
and of characteristic −p (meaning that the carrier surface of M has Euler characteristic
χ = −p). All our working will be up to duality, so that we may without loss of generality
assume that k ≤ ℓ. We also know by [8] and [12] that in this situation the four generating
involutions of the associated group of automorphisms H are mutually distinct. Note that
this instantly implies that k, ℓ ≥ 4; moreover, since maps of type (4, 4) necessarily live on
surfaces with Euler characteristic 0, we will assume that k ≥ 4 and ℓ ≥ 6.
Referring to the diagram in Figure 1 which may be assumed to display a fragment of
M , the stabiliser in H of the distinguished face is ⟨s, x⟩, isomorphic to the dihedral group
Dℓ of order ℓ, while the H-stabiliser of the distinguished vertex is ⟨t, y⟩, isomorphic to
the dihedral group Dk of order k. The map thus has
|H|
ℓ faces and
|H|
k vertices. The H-
stabiliser of the distinguished bold and dashed edge, respectively, is ⟨x, y⟩ and ⟨s, t⟩, in
both cases isomorphic to the Klein four-group V4 ∼= C2 × C2; hence the map has
2|H|
4
edges. The Euler-Poincaré formula applied to the number of vertices, edges and faces of
M gives |H|( 1k −
1
2 +
1
ℓ ) = χ = −p, or, equivalently,
|H| = νp where ν = ν(k, ℓ) =
2kℓ
kℓ− 2(k + ℓ)
(3.1)
It may be checked that our lower bounds k ≥ 4 and ℓ ≥ 6 imply ν ≤ 12. Combining
this with (3.1) one obtains the upper bound |H| ≤ 12p for the group of automorphisms
H of the map M under consideration. Also, by (2.5) and distinctness of the involutory
generators, the group H contains two distinct subgroups isomorphic to V4; in particular,
the order of H must be divisible by 4.
From this point on we split the proof into four parts. In Section 4 we will consider the
case when p divides the order of H , first addressing all odd primes, and then the special
case when p = 2. Section 5 will deal with the opposite case, and the analysis will be
divided according to whether the 2-part Fitting subgroup of H is cyclic or dihedral.
4 The case when p divides the order of H
Let us recall that we are considering an edge-biregular map M = (H;x, y, s, t) of type k, ℓ
for even k, ℓ such that k ≤ ℓ, k ≥ 4, ℓ ≥ 6, carried by a surface of characteristic −p for
some prime p. Suppose p is a divisor of |H|. This, together with (3.1) and the facts summed
up at the end of Section 3, implies that |H| = νp where ν ∈ {4, 6, 8, 12}. Moreover, one
may check that in our range for the even entries in the type of our biregular map M one
has: ν(k, ℓ) = 12 only for (k, ℓ) = (4, 6); ν(k, ℓ) = 8 only for (k, ℓ) = (4, 8); ν(k, ℓ) = 6
only when p = 2 and for types (6, 6) and (4, 12); and finally ν(k, ℓ) = 4 only for (k, ℓ)
equal to (6, 12) or (8, 8), corresponding to the cases when p = 3 and p = 2 respectively.
The case when p = 2 is treated as a special case in the proof of Theorem 3.2 at the end
of this section. Henceforth, unless stated otherwise, we assume that p is an odd prime. We
sum up the above observations for odd p in the form of a lemma.
10 Art Discrete Appl. Math. 5 (2022) #P3.08
Lemma 4.1. If p is an odd prime which divides |H|, then either |H| = 12p for the type
(4, 6), or |H| = 8p for the type (4, 8), or else p = 3 and |H| = 4p for type (6, 12). □
Note that our previous considerations apply also to the situation when χ = −1, that is,
Lemma 4.1 can be applied also to the exceptional case when p = 1. In fact, a more detailed
look into this case will be useful later. Remembering that we need four distinct involutory
generators for H if |H| ∈ {4, 8, 12} this leaves us with only two options, namely, |H| = 8
when (k, ℓ) = (4, 8), and |H| = 12 when (k, ℓ) = (4, 6). If |H| = 8 then ℓ = 8 and hence
H ∼= D8. If, on the other hand, |H| = 12 then ℓ = 6, which means that H contains a
subgroup isomorphic to D6. There is only one such group H of order 12, namely D12. By
the classification of [12] each of these dihedral groups support an edge-biregular map on a
surface with χ = −1. Since we do not need more detailed information about such maps,
we just state the following conclusion here:
Lemma 4.2. If M = (H;x, y, s, t) is an edge-biregular map on a surface of Euler char-
acteristic χ = −1, then H is isomorphic to a dihedral group of order 8 or 12. □
We continue by showing that assuming p | |H| leads to contradictions for p ≥ 13. To
do so we first exclude the existence of maps M with types as above for which the group H
would be a semidirect product of a particular form.
Lemma 4.3. Let p be an odd prime which does not divide κ or λ where λ ≥ 3. Suppose
M = (H;x, y, s, t) is an edge-biregular map of type (2κ, 2λ) such that H ∼= Cp ⋊ Dν .
Then ν = 2λ ≡ 4 mod 8, and κ ∈ {2, λ} while the supporting surfaces for these maps
have χ = p(1− λ/2) and χ = p(2− λ) respectively.
Proof. Suppose that an edge-biregular map of type (2κ, 2λ) exists with H ∼= Cp ⋊ Dν .
With the usual notation, for any edge-biregular map we have ⟨x, y⟩ ∼= ⟨s, t⟩ ∼= V4, and
hence 4 must divide |Dν |. In particular this means that Dν has non-trivial centre, and this
leaves us with only two congruence classes to consider for ν, namely 0 mod 8, where the
central element of Dν is a square, and 4 mod 8, where the central element of the dihedral
group is not a square.
We use additive notation in the first coordinates to indicate the Abelian cyclic group Cp
and multiplicative notation for the dihedral part of the semi-direct product, where Φ: Dν →
Aut(Cp), which maps α → ϕα, is the associated homomorphism. Thus each element of
H , and hence each of the canonical generators {x, y, s, t}, will be written (a, α) for some
a ∈ Cp and some α ∈ Dν .
For any involution α ∈ Dν , the automorphism ϕα must have order dividing two. Since
Aut(Cp) ∼= Cp−1 is cyclic there are only two such elements, the identity and the unique
involution, so for g, a generating element of Cp, we have ϕα must either fix g or invert
g. Hence for any β ∈ Dν we will have ϕβ being one of these two automorphisms. Also
(a, α)2 = (a + ϕα(a), α
2) so the element (a, α) is an involution in H if and only if both
a+ ϕα(a) = 0 and α2 = 1. Note that if a ̸= 0 then, in order for (a, α) to be an involution,
we must have ϕα(a) = −a, that is ϕα must be inverting.
Triples of commuting involutions, that is the non-trivial elements of a copy of V4 in the
group H , have the form: {(a, α), (b, β), (a+ϕα(b), αβ)} such that a+ϕα(b) = b+ϕβ(a),
and where α, β and αβ are all involutions in Dν , one of which must necessarily be central.
So, if both a and b are non-zero in Cp, then a − b = b − a, so 2a = 2b, and hence a = b,
so the third element of the triple must be (0, αβ). It is important to note at this point that
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 11
in this case αβ would fix any g ∈ Cp. If instead we suppose that a ̸= 0 and b = 0 then
a+ ϕα(b) = b+ ϕβ(a) becomes a = ϕβ(a), in other words ϕβ must fix every element of
Cp.
For much of this proof we only need consider the action generated by the dihedral part
of the elements in question, rather than the element itself, and so we abuse the notation to
denote those in the kernel of Φ, which fix the cyclic elements, by “+" and members of the
other coset, whose elements invert the cyclic group, by “−". Hence any copy of V4 in H
contains three distinct non-identity elements written: (a,−), (a,−), (0,+) for some fixed
a ∈ Cp (which could itself be zero); or, exceptionally, (0,+), (0,+), (0,+).
We now consider what combination of forms can occur within the set of four distinct
canonical involutory generators for H , namely {x, y, s, t}, taking each case in turn.
It will be useful to note that assuming H is generated by any number of involutory
elements, each of which has the form (0,+) or (a,−) for a given fixed a ∈ Cp, leads to
a contradiction. Indeed since KerΦ must have index two in Dν , given any two elements α
and β which are not in the kernel, we have αβ ∈ KerΦ. Hence, in this case, the product
of elements (a, α)(a, β) must have the form (0, αβ) that is (0,+). Also then, for γ ∈
KerΦ, the product (0, γ)(a, α) must have the form (0 + a, γα) that is (a,−), and similarly
(a, α)(0, γ) must have the form (a,−). Regardless of the orders of any such products, this
restrictive set of forms makes it clear that a is just a place-holder marking when the dihedral
part of the element is not in the kernel of Φ, and as such this fixed a cannot contribute
towards generating Cp in the first coordinate. So, this is in contradiction to H ∼= Cp ⋊Dν .
We now highlight another property which will be of use:
Since ⟨x, s⟩ is a group isomorphic to D2λ, henceforth we identify the dihedral parts of
⟨x, s⟩ with the corresponding elements for a particular copy of D2λ ≤ Dν . Now 2λ > 4
so the central involution of the dihedral group Dν , which we now call z, will never occur
as the second coordinate of x or s (for otherwise we would have ⟨x, s⟩ ∼= V4).
If λ is odd then ⟨x, s⟩ has trivial centre, and every rotation in D2λ is a square element in
Dν . This means that either D2λ ≤ KerΦ or (all of the rotations but) none of the involutions
in D2λ are in KerΦ. In particular, the canonical involutions generating ⟨s, x⟩ must have
forms with the same symbol, be that + or −, in the second coordinate.
If λ is even then ⟨x, s⟩ has non-trivial centre and z ∈ D2λ, and, according to the
congruence class of ν modulo 8, this z may or may not be a square in Dν .
Up to twinness of maps, we now divide the argument into two cases.
The first case to consider is when x has the form (a,−), for some a which may be zero:
Suppose s has the form (b,−). But sx, which has order λ, is then denoted (b − a,+).
Now p does not divide the face length 2λ, so this forces b = a, and hence s is also an (a,−).
Now since t commutes with s, and y commutes with x, we know that each of t and y will
also have forms within the restrictive set {(0,+), (a,−)}, contradicting H ∼= Cp ⋊Dν .
Suppose instead that s has the form (0,+). Thus λ must be even and so z ∈ D2λ. This
splits into two cases, according to the congruence class of ν.
When ν ≡ 0 mod 8, z is a square element in Dν , and hence z ∈ KerΦ. Remember
that a triple of commuting involutions must include an element (. . . , z) which in this case
has the form (0,+). Hence the set {s, t, st} must be {(0,+), (0,+), (0,+)}. In this case,
along with y, which is either (0,+) or (a,−), our canonical generators are once again all
contained within the restrictive set of forms {(0,+), (a,−)}, contradicting H ∼= Cp ⋊Dν .
When ν ≡ 4 mod 8, the centre z ∈ D2λ is not a square element in Dν , and hence there
is the possibility of z /∈ KerΦ, which we will now explore. (If the central element was in the
12 Art Discrete Appl. Math. 5 (2022) #P3.08
kernel we would be in the same situation as immediately above, and so the map would not
exist.) The group ⟨s, x⟩ ∼= ⟨r, f |rλ, f2, (rf)2⟩ = D2λ ≤ Dν , can only have an inverting
central element if λ = 2(2m + 1) and r /∈ KerΦ. Up to the choice of notation for f , a
reflection, and r, which generates a cyclic group of order λ, this happens when exactly one
of these two generating involutions f, rf is in the kernel, say rf ∈ KerΦ, thereby ensuring
r /∈ KerΦ and hence z = r2m+1 /∈ KerΦ. With this assumption x = (a, f) and s = (0, rf),
which forces y ∈ {(a, z), (0, fz)} and t ∈ {(b, z), (b, rfz)}. Notice in particular that t has
the form (b,−). If y = (a, z) then the non-divisibility of κ by p forces a = b and thus the
canonical generators would be restricted to a set which cannot generate H . So, for such an
edge-biregular map to survive, y = (0, fz) and the choices for t with b ̸= a would give
edge-biregular maps of type (4, 2λ) and (2λ, 2λ) respectively. In particular it is clear that
no larger dihedral group than D2λ can be found in the second coordinates and so 2λ = ν.
Meanwhile the element xt has the form (a− b,+) and hence the cyclic part Cp of H will
also be generated by the set of canonical generators. These maps therefore exist, as do their
twins (and indeed duals) while the characteristic for the corresponding supporting surfaces
can be found using the Euler-Poincaré formula.
Until now we have been considering the case when x has the form (a,−), for some a
which may be zero, and latterly when we also had s being (0,+). The final case to address
is when x has the form (0,+):
To avoid being a twin of the maps considered immediately above, we may now assume
that s also has the form (0,+). But then, since p does not divide κ, the order of yt, the
two canonical generators y and t introduce a maximum of one new form, say (a,−), to the
set of canonical generators. Hence we are once again left with an overly restrictive set of
generating forms, contradicting H ∼= Cp ⋊Dν .
We are now in position to exclude primes p ≥ 13 from consideration in the case of p
dividing |H|.
Proposition 4.4. If M = (H;x, y, s, t) is an edge-biregular map, and p is a divisor of |H|,
then p ≤ 11.
Proof. We suppose that H = ⟨x, y, s, t⟩ is the group for an edge-biregular map on surface
of Euler characteristic −p and that p | |H| where p ≥ 13. Combining this with Lemma 4.1
we note that the edge-biregular map must have type (4, 8) or (4, 6) forcing ν ∈ {8, 12} and
this implies |H| = νp ≤ 12p < p2. Hence there is only one Sylow p-subgroup of H and
this Sylow subgroup is cyclic, and unique (hence normal) in H .
Since we are restricted to maps of type (4, 8) or (4, 6), and p ≥ 13, the factor group
H/Cp forms a smooth quotient and so would correspond to an edge-biregular map of the
same type as H on a surface with Euler characteristic χ = −1. Applying now Lemma 4.2
one sees that H/Cp is a dihedral group of order 8 or 12. Schur-Zassenhaus now implies
that H is a (non-trivial) semi-direct product H ∼= Cp ⋊Dν . But this is in contradiction to
Lemma 4.3, so we conclude that p ≤ 11.
As the next step we extend Proposition 4.4 to all odd primes greater than 3, while the
case where p = 3 is dealt with separately in Proposition 4.6. The results of the following
two propositions, and those of Theorem 3.2, could be found using a computer, however
here we adopt a more classical approach.
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 13
Proposition 4.5. If M = (H;x, y, s, t) is an edge-biregular map on a surface of Euler
characteristic −p for some odd prime p ≥ 5, then p is not a divisor of |H|.
Proof. The conclusion of Proposition 4.4 leaves us with a few small cases we need to
address, namely, those listed in Lemma 4.1 (together with the corresponding types) for
p ∈ {3, 5, 7, 11}; here we will handle the three larger primes one by one.
• The prime p = 5, type (4, 8) for |H| = 8× 5 and type (4, 6) for |H| = 12× 5.
For type (4, 8) with |H| = 40. By Sylow theorems the Sylow 5-subgroup is normal so
H ∼= C5 ⋊D8, contrary to Lemma 4.3.
The second type to consider here, (4, 6), comes with a group |H| of order 60. We
know H contains a (face stabiliser) subgroup isomorphic to D6, and so H is not Abelian.
Suppose that H has a unique Sylow 5-subgroup, which (by the Schur-Zassenhaus theorem)
implies that H ∼= C5 ⋊ K for some group K of order 12. Out of the five such groups
L, however, only D12 is generated by involutions, and the possibility H ∼= C5 ⋊ D12
contradicts Lemma 4.3. It follows that H contains six Sylow 5-subgroups and so H ∼= A5.
Now A5 contains 15 involutions and 5 Sylow 2-subgroups, meaning that the intersection of
two copies of V4 is either trivial or the two groups are the same. Hence ⟨x, y⟩ = ⟨y, t⟩ =
⟨s, t⟩ ∼= V4 implies |H| = 4, which is a contradiction.
• The prime p = 7, type (4, 8) for |H| = 8× 7 and type (4, 6) for |H| = 12× 7.
For the type (4, 8) we have |H| = 56. If the Sylow 7-subgroup of H is not normal then
H contains 8× 6 = 48 elements of order 7, leaving only eight other elements in the group
H . These eight elements must form the single (and hence normal) Sylow 2-subgroup,
which is the face stabiliser, Dℓ ∼= D8. But then all the involutions of H must lie in Dℓ and
so H would not be generated by involutions. So the Sylow 7-subgroup of H is normal and
hence H ∼= C7 ⋊D8 which contradicts Lemma 4.3 again.
The type (4, 6) and |H| = 84 is excluded by Sylow theorems and the Schur-Zassenhaus
as for the same type but in the case p = 5: A group of order 84 has a unique (normal)
Sylow 7-subgroup, and then the unique possibility is H ∼= C7 ⋊D12, which is impossible
by Lemma 4.3.
• The prime p = 11, type (4, 8) for |H| = 8× 11 and type (4, 6) for |H| = 12× 11.
The type (4, 8) with |H| = 88 is excluded by observing that H contains a unique Sylow
11-subgroup, leaving only the possibility H ∼= C11 ⋊D8 that contradicts Lemma 4.3.
The final case to consider is the one of the type (4, 6) that comes with a group H of
order 132. If the Sylow 11-subgroup in H is normal then, by a similar analysis as done
for p = 5, we would have H ∼= C11 ⋊ D12, which is impossible by Lemma 4.3. So
there are 12 Sylow 11-subgroups and there are 12 × 10 = 120 elements of order 11. If
H contained more than one Sylow 3-subgroup then there would be 8 elements of order 3,
leaving room for only one Sylow 2-subgroup. Now, H is generated by involutions and so
the Sylow 2-subgroup Dk ∼= V4 cannot be unique, otherwise we would have H = Dk.
So, the Sylow 3-subgroup in H , isomorphic to C3, must be unique and hence normal.
Thus, in our case with ℓ = 6, the subgroup ⟨sx⟩ ∼= C3 is normal in H , which implies that
ysxy, tsxt ∈ {sx, xs}. If we suppose tsxt = sx then txt = x so t commutes with s and x
and y. But ⟨x, y⟩ is a Sylow 2-subgroup of H and so, by the distinctness of the generators,
we would have t = xy. Hence H = ⟨sx⟩ ⋊ ⟨x, y, t⟩ = ⟨sx⟩ ⋊ ⟨x, y⟩ which has order 12,
not 132. By symmetry, supposing that y fixes sx leads to a contradiction too. Hence we
have tsxt = xs and ysxy = xs. These give, in turn, txt = sxs and ysy = xsx which,
14 Art Discrete Appl. Math. 5 (2022) #P3.08
when combined, give txtysy = 1. However, recalling that y commutes with both x and t,
this yields xs = 1, a contradiction. This completes the proof.
We now consider the exceptional case when p = 3, which does indeed yield edge-
biregular maps, namely those corresponding to the final part of Theorem 3.1.
Proposition 4.6. If M = (H;x, y, s, t) is an edge-biregular map on a surface of Euler
characteristic −3, then up to duality and twinness, H is of type (4, 6) and has a presenta-
tion of the form
H = ⟨x, y, s, t|x2, y2, s2, t2, (xy)2, (st)2, (yt)2, (sx)3, (xyt)3, (sty)3, (xyst)2⟩ ∼= D6×D6
Proof. When the prime p = 3, we have the following possibilities: type (6, 12) for |H| =
4× 3; type (4, 8) for |H| = 8× 3; and type (4, 6) for |H| = 12× 3.
An edge-biregular map of type (6, 12) on this surface would have |H| = 4×3 = 12 and
so the group H must be the face stabiliser H = Dl ∼= D12 which is dihedral. Referring to
the dihedral classification of edge-biregular maps in [12] we can see that such a map does
not exist.
For type (4, 8) we have |H| = 24 and as H is generated by involutions, the Sylow 2-
subgroup Dℓ ∼= D8 cannot be normal in H . This leaves only two options: H ∼= C3 ⋊D8,
which is clearly impossible, or H ∼= S4. Suppose H ∼= S4 is represented as a per-
mutation group on the set {1, 2, 3, 4}. The non-trivial elements in the three copies of
V4 in S4 then form the sets T1 = {(12)(34), (12), (34)}, T2 = {(13)(24), (13), (24)},
T3 = {(14)(23), (14), (23)} and T4 = {(12)(34), (13)(24), (14)(23)}. Since ⟨s, x⟩ =
Dℓ ∼= D8, we may assume, without loss of generality, that {s, x} = {(12)(34), (24)}. But
then the fact that ⟨y, t⟩ = Dk ∼= V4 leads us to conclude that for the four distinct canonical
generators we must have x, y, t, s ∈ T2 ∪ T4. But these two sets only generate D8, not the
whole of S4, a contradiction.
We proceed with type (4, 6) for a group H of order 36 and let K be its Sylow 3-
subgroup of order 9.
Consider the non-trivial and transitive permutation representation π of H on the set
H/K of the right cosets of K in H , given as follows: To each h ∈ H we assign a per-
mutation πh of the set H/K mapping the coset Kx onto Kxh, for every x ∈ H . Now, as
|H/K| = 4, the representation π is a group homomorphism from H into S4, and its kernel
is a normal subgroup of H distinct from H; note also that π cannot be surjective (by divis-
ibility of the source and target by 4 and 8). The only proper transitive subgroups of S4 are
A4 and the unique subgroup isomorphic to V4, so that |Ker(π)| ∈ {3, 9}. But notice that if
|Ker(π)| = 3, so that H contains a normal subgroup Ker(π) ∼= C3, and Im(π) ∼= A4 then
we have an immediate contradiction since A4 is not generated by involutions.
It remains to consider the case when |Ker(π)| = 9, which means that the kernel coin-
cides with the Sylow 3-subgroup K of order 9 in H , and H ∼= K ⋊ V4.
Suppose, for a contradiction, that K = C9. Now, C3 is characteristic in C9 and hence
normal in H . There is only one copy of C3 in H , and that is ⟨sx⟩. Denoting elements in
H/⟨sx⟩ with bar notation, we then have s̄ = x̄ and hence H/⟨sx⟩ is generated by three
commuting involutions x̄, ȳ and t̄. This has order at most 8, not the required 12, and so is
in contradiction to the order of H .
We may consider C23 as a two dimensional vector space over GF (3) with elements writ-
ten as vectors. The automorphism group for C23 is GL(2, 3) which is known to have just one
conjugacy class of subgroups isomorphic to V4. Thus we let x = (x, x′),
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 15
y = (y, y′), t = (t, t′), s = (s, s′) where the underlined vectors in the first coor-
dinates are elements of C23 , and x
′, y′, t′, s′ ∈ V4 ≤ GL(2, 3). Now, since sx has order 3,
it is clear that s′x′ must be the identity, and so x′ = s′. Also, since the products xy, yt and
ts must all have order 2, we certainly have x′ ̸= y′ ̸= t′ ̸= x′, and so t′ = x′y′.
Suppose the homomorphism Φ: V4 → GL(2, 3) associated with the semi-direct prod-
uct is not injective. In any case, the kernel of Φ cannot be the whole of V4, since this would
yield a direct product H ∼= C23 × V4 which certainly cannot be generated by involutions.
The only remaining option is for two of the elements in V4 to be mapped to a given in-
volution α ∈ GL(2, 3), while their product is mapped to the identity. Now x′ /∈ KerΦ,
otherwise x and s being involutions would then force x = s = 0 which is absurd since
xs must have order 3. So, up to twinness, we may assume, x′y′ ∈ KerΦ. But then this
would force t = 0, meaning t is central in the group H . Now, the elements x and y are
involutions so α acts to invert any non-zero parts of x and y. Also, x and y commute so
x + α(y) = y + α(x), that is x − y = α(x − y), and this forces x = y. Now we have in
fact t = xy and so H = ⟨x, t, s⟩ = ⟨x, s⟩⟨t⟩ ∼= D6 × C2, a contradiction.
We have now shown that the homomorphism mapping from V4 to the associated actions
in GL(2, 3) must be injective, since in any other case we could not generate the whole
group H ∼= C23 ⋊ V4 just with involutions. So we may now identify {x
′, y′, t′, s′} with
their images in GL(2, 3), choosing the canonical copy of V4 in GL(2, 3), which consists
of the diagonal matrices.
In the interests of generating the whole group H ∼= C23 ⋊ V4, it can be checked that
x′ = s′ = −I . To satisfy the other known order properties for the products of pairs of
generating involutions, up to twinness, we find we are restricted to the set of canonical
involutions as follows:
x = (
(
x1
x2
)
,
[
−1 0
0 −1
]
), y = (
(
0
x2
)
,
[
1 0
0 −1
]
),
t = (
(
s1
0
)
,
[
−1 0
0 1
]
), s = (
(
s1
s2
)
,
[
−1 0
0 −1
]
)
where x1 ̸= s1 and x2 ̸= s2. Regardless of whichever allowable choices of values for xi
and si are made, it is clear that ⟨xyt, sty⟩ ∼= C23 . Also we have ⟨y, t⟩ ∼= V4 as expected, and
hence H = ⟨x, y, s, t⟩ = ⟨xyt, sty⟩⋊ ⟨y, t⟩ ∼= C23 ⋊V4. Also H = ⟨xyt, xy⟩×⟨sty, st⟩ ∼=
D6 ×D6 and a presentation for the resulting map will be as follows:
H = ⟨x, y, s, t|x2, y2, s2, t2, (xy)2, (st)2, (yt)2, (xyt)3, (sty)3, (xyst)2⟩ ∼= D6 ×D6.
By this analysis, the map is unique up to duality and twinness. The above group presen-
tation clearly indicates that this edge-biregular map is isomorphic to its twin, and hence is
also a fully regular map. Meanwhile the face length ℓ being 6 is clearer to see when the
additional (consistent) relator (sx)3 is incorporated into the presentation .
The only prime left to be considered in this part is p = 2, and by exploring this case we
will also establish validity of Theorem 3.2.
Proof of Theorem 3.2.
Proof. We now assume p = 2. We recall from the initial workings of this section that
the possible types of map which can occur when ν ∈ {4, 8, 12} giving |H| = 8, 16 and
16 Art Discrete Appl. Math. 5 (2022) #P3.08
24, respectively, for types (8, 8), (4, 8) and (4, 6). We will also need to address the only
remaining case for even k, ℓ such that 4 | |H| = 2ν, namely when ν = 6 and |H| = 12
which occurs when the type (k, ℓ) is (4, 12) or (6, 6). We proceed according to the order
of the group H .
The first possibility is dealt with quickly, because in an edge-biregular map of type
(8, 8) with |H| = 8 we must have H ∼= D8 (the stabiliser of a (single) vertex, say). The
group D8 contains exactly four non-central involutions, and two subgroups isomorphic to
V4, while the central element cannot be equal to any one of the distinct canonical involutory
generators. It may be checked that up to isomorphism (since the resulting map is both self-
dual and fully regular) there is just one way to present H canonically, necessarily equivalent
to the form given by H2,1 in Theorem 3.2.
When we consider ν = 6 for the type (4, 12) we clearly have the group H = ⟨x, s⟩ ∼=
D12, and up to twinness and duality there is only one canonical presentation. This is shown
as H2,2 in Theorem 3.2.
In the case of type (6, 6), we have ν = 6 and the group is again H ∼= D12. This
dihedral group contains two copies of D6, one of which must be ⟨s, x⟩ and the other ⟨t, y⟩.
There is only one cyclic group of order three contained in D12 and so there are only two
possibilities for the elements of order 3, namely sx = yt (which results in contradictions)
or sx = ty which yields the presentation H2,3 in Theorem 3.2.
The next possibility we look at is type (4, 8), with |H| = 16 = 2×8. We know ⟨s, x⟩ =
Dℓ ∼= D8 is normal in H since it has index two. First note that H/Dℓ = ⟨yDℓ, tDℓ⟩ has
order 2 so either y ∈ Dℓ, or t ∈ Dℓ, or yDℓ = tDℓ ̸= Dℓ and so the product ty is in Dℓ.
We deal with the latter case after addressing the first two together, since they are equivalent
up to twinness. Also ⟨sx⟩ ∼= C4, being characteristic in the dihedral group, is normal in H .
Conjugation by the elements x and s clearly invert sx while each of y and t must either fix
or invert sx, resulting in differing canonical presentations for H .
We first suppose, by our choice of the labeling of orbits, that is up to twinness, that
y ∈ Dℓ. If y inverts sx then we have y ∈ {sxs, xsx} and so, by the distinctness of
generators, y = sxs, and the order of ys is four. If t also inverts sx then tsxt = xs. But
then y = stsxt = txt which implies y = x, a contradiction. However, if t fixes sx, which
happens if and only if t fixes x, we obtain
⟨ x, y, s, t | x2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)4, (tx)2, ysxs⟩
which is the presentation of H2,4 in Theorem 3.2. Notice that this map must be supported
by an orientable surface since there are no odd length relators in this presentation.
If, on the other hand, y fixes sx then y = (sx)2, the central element in Dℓ, and hence
y(sx)2 is a relator, forcing the supporting surface to be non-orientable. Another conse-
quence of y = (sx)2 is that (ys)2 may also appear as a relator. If t also fixes sx, we may
arrive at the presentation
⟨ x, y, s, t | x2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)2, (tx)2, y(sx)2⟩
which is H2,5 in Theorem 3.2. Otherwise, t inverts sx, giving
⟨ x, y, s, t | x2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)2, (stx)2, y(sx)2⟩
which is the presentation of the group H2,6 in Theorem 3.2.
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 17
Now let us suppose that ty ∈ Dℓ. As before, we consider when ty fixes sx, that is in
this case ty = (sx)2, the central element in Dℓ. If one (and hence both) of y and t invert
sx then tsxt = xs so txt = sxs. This leads to ty = sxsx = txtx which implies y = xtx
so y = t, a contradiction to the distinctness of the canonical involutions. In the other case
both y and t fix sx. This is equivalent to y commuting with s, and t commuting with x, and
this gives us the following presentation which yields a group of the right order.
⟨ x, y, s, t | x2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)2, (tx)2, ty(sx)2⟩
This is listed as H2.7 in Theorem 3.2. You may notice this presentation has a redundant
relator, which serves to make clear the underlying full regularity of the corresponding map.
On the other hand consider when ty inverts sx, and hence exactly one of y or t must fix
sx. Up to twinness we may assume y fixes sx in which case y is central in H and (ys)2 is
a relator. In this case then t inverts sx and hence (tsx)2, or equivalently (stx)2, is also a
relator.
For ty ∈ Dℓ to invert sx we must have ty ∈ {sxs, x, xsx, s}. The first two options
yield a contradiciton, as we shall now see. If ty = sxs then 1 = tysxs = ysys so
ytysx = sy, that is tsx = sy and hence tx = y. But then ty = x = sxs which means
xs = sx, contradicting the order of sx. If ty = x then 1 = tyx = (tsx)2 so y = sxts, that
is x = yt = sxs, the same contradiction.
It can be checked that the remaining options, namely ty is either xsx or s, avoid the
contradictions which cause the order of the group H to become too small, and they yield
these two remaining canonical presentations.
⟨ x, y, s, t | x2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)2, (stx)2, tyxsx⟩
⟨ x, y, s, t | x2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)4, (ys)2, (stx)2, tys⟩
These are listed respectively as H2.8 and H2.9 in Theorem 3.2. Careful comparisons of the
relators in the above presentations shows that the maps are pairwise non-isomorphic.
The last possibility to address is the type (4, 6) for a group H of order 3 × 8 = 24.
Suppose the Sylow 3-subgroup is not normal in H . Then H ∈ {SL(2, 3), S4, A4 × C2}.
Now, we know H has more than one involution, so it cannot be isomorphic to SL(2, 3).
Also, A4 only contains 3 involutions, all in the subgroup which is isomorphic to V4, but H
is generated by involutions, H so cannot be isomorphic to A4 × C2. Thus, if the Sylow
3-subgroup is not normal in H , then H ∼= S4.
We will now refer to the standard permutation representation of S4 on the set {1, 2, 3, 4}
and remember the sets T1 – T4 introduced in the first part of the proof of Proposition 4.6
for p = 3. Since ⟨s, x⟩ = Dℓ ∼= D6, we may assume, up to choice of notation and hence
without loss of generality, that x and s are (12) and (23), respectively. The condition that
⟨y, t⟩ ∼= V4 means that y = (12)(34), and t = (14)(23), arriving at the non-orientable map
given by the presentation listed as H2,10 in Theorem 3.2:
⟨ x, y, s, t | x2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)3, t(sy)2, y(xt)2⟩ ∼= S4
Now suppose the Sylow 3-subgroup ⟨sx⟩ ∼= C3 is normal in H . Then conjugation by
each of y and t must either fix or invert sx.
Suppose y inverts sx. Then ysxy = xs so (ysx)2 is a relator. If t also inverts sx then
tsxt = xs. Then txt = sxs and hence txtx = sxsx but y commutes with both t and x
18 Art Discrete Appl. Math. 5 (2022) #P3.08
so txtx = ytxtxy = ysxsxy = xsxs is self-inverse. But sxsx has order three so this is a
contradiction. If, on the other hand, t fixes sx then we obtain the presentation of the group
H2,11 from Theorem 3.2:
⟨ x, y, s, t | x2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)3, (ysx)2, (tx)2⟩.
Suppose finally that y fixes sx, in which case ysxy = sx so (ys)2 is a relator. In the
case where t also fixes sx, this leads to the presentation of the group H2,12 from Theo-
rem 3.2:
⟨ x, y, s, t | x2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)3, (ys)2, (tx)2⟩
Otherwise, t inverts sx, giving the presentation
⟨ x, y, s, t | x2, y2, s2, t2, (xy)2, (st)2, (ty)2, (sx)3, (ys)2, (txs)2⟩
which represents the twin map of the edge-biregular map determined by the group H2,11.
It remains to address isomorphism type, full regularity and orientability of the twelve
edge-biregular maps identified above. Obviously, H2,1 ∼= D8, H2,2 ∼= D12 ∼= H2,3 and
H2,7 ∼= S4. Observe that the element t /∈ ⟨s, x⟩ is central in H2,4 and H2,5 while st /∈
⟨s, x⟩ is central in H2,6, and y /∈ ⟨s, x⟩ is central in H2,7, H2,8 and H2,9 and so the
six groups are all isomorphic to D8 × C2. Further, it follows from the derivation of the
remaining two groups that H2,11 = ⟨s, x⟩ ⋊ ⟨t, y⟩ = ⟨s, x⟩ × ⟨t, xy⟩ ∼= D6 × V4 and
H2,12 = ⟨s, x⟩ × ⟨t, y⟩ ∼= D6 × V4. It is easy to check that the twelve presentations are
correct and complete (describing exactly the groups listed, albeit including some redundant
relators). The maps defined by H2,1, H2,3, H2,4, H2,7, H2,11 and H2,12 are orientable
because all their defining relations have even length in the generators x, y, s, t, which does
not apply to the remaining six maps. Finally, the maps associated with H2,1, H2,3, H2,7,
H2,10, and H2,12 are the only fully regular ones out of the above twelve, as their groups
admit an automorphism swapping x with s and y with t. The proof of Theorem 3.2 is now
complete.
5 The case when p does not divide the order of H
Let M = (H;x, y, s, t) be a (finite) edge-biregular map of type (k, ℓ) on a surface of
Euler characteristic −p for some prime p; in view of Theorem 3.2 we will assume that p
is odd and hence the surface is non-orientable. Recall that the group H is assumed to be
presented as in (2.5), and its order together with the type of the map and the characteristic
of the surface are tied by the equation (3.1). We begin with an auxiliary result and omit a
proof since it is almost verbatim the same as the proof of Lemma 3.2 of [4].
Lemma 5.1. If q is a prime divisor of |H| relatively prime to the Euler characteristic, then
Sylow q-subgroups of H are cyclic if q is any odd prime, and dihedral if q = 2. □
From this point on until the end of this section we will assume that p is not a divisor of
the order of H . Since we are working up to duality, instead of k ≤ ℓ we henceforth assume
that the 2-part of k is not smaller than the 2-part of ℓ. Remember also that H contains a
subgroup isomorphic to V4 and so 4 divides |H|. Comparing this condition with equation
(3.1) allows us to set k = 4κ and ℓ = 2λ for integers κ and λ. The stage will be set by
proving solvability of H .
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 19
Proposition 5.2. If p is such that p ∤ |H|, the group H is solvable.
Proof. We start by proving that |Dk ∩ Dℓ| ≤ 4. The group K = Dk ∩ Dℓ is obviously
cyclic or dihedral. Suppose for a contradiction that |K| > 4. Then K contains an element z
of order at least 3, so that z = (sx)m = (yt)n for some m and n. Clearly, z commutes with
sx and also with yt. We also have (xy)z(xy) = yx(sx)mxy = y(xs)my = y(ty)ny =
(yt)n = z, so z commutes with xy as well. Now the map is carried by a non-orientable
surface and so H = ⟨xy, sx, yt⟩ by the theory explained in section 2. Hence z commutes
with all the canonical generators, and thus it is central in H . Specifically, z is central in
Dk ≤ H . It is well-known that the centre of a (non-Abelian) dihedral group is either trivial
or has order 2. But, by our assumption, the order of ⟨z⟩ is greater than 2, a contradiction.
Next we prove that the group H is a product of Dk and Dℓ. By (3.1) the assumption
p ∤ |H| implies that p must be absorbed by the denominator of ν(k, ℓ), that is, kℓ− 2(k +
ℓ) = rp for some integer r. Thus, |H| = 2kℓ/r, but by the first part of the proof we also
have |H| ≥ |Dk||Dℓ|/|Dk ∩Dℓ| ≥ kℓ/4, implying that r ≤ 8. Using k = 4κ and ℓ = 2λ
as agreed before, one has rp = kl− 2(k+ l) = 8κλ− 8κ− 4λ = 4(2κλ− 2κ−λ) = 4cp.
Notice that this, along with the assumption p ∤ |H|, also implies gcd(2κ, λ) = c ≤ 2.
Hence |H| = kℓ/(2c) where c = gcd(2κ, λ) ∈ {1, 2}.
If c = 1, then |H| = kℓ/2 and also λ must be odd, so that |Dk ∩ Dℓ| ≤ 2. We also
have |H| ≥ |Dk||Dℓ|/|Dk ∩ Dℓ| ≥ kℓ/2 and so equality holds throughout and hence
H = DkDℓ if c = 1. In the case when c = 2 we have (2κλ− 2κ− λ) = 2p where p is an
odd prime, so λ must be even; also, gcd(κ, λ/2) = 1. Now, c = 2 implies |H| = kℓ/4 and
|Dk ∩Dℓ| ≤ 4, and as we also have |H| ≥ |Dk||Dℓ|/|Dk ∩Dℓ| ≥ kℓ/4 we conclude that
equality holds throughout and hence H = DkDℓ.
We may now complete the proof by invoking the result of Huppert [9] that the product
of two dihedral groups is solvable.
The fact that H is solvable yields that it has a non-trivial Fitting subgroup F ; recall that
F is the largest nilpotent normal subgroup of H . In particular, F is a direct product of its
Sylow subgroups. By what we know about the Sylow subgroups of H from Lemma 5.1
we have F = F1 × F2, where F1 is cyclic, of odd order (possibly trivial), and F2 (if non-
trivial) is a cyclic or a dihedral 2-group; we will henceforth split our analysis according to
this dichotomy.
As a general remark, observe that we may assume F ̸= H . Indeed, if F = H , then
F1 would have to be trivial (otherwise F could not be generated by involutions) and F2
would have to be non-cyclic (to contain enough distinct involutions), so that H = F = F2
would have to be dihedral. But edge-biregular maps with dihedral groups H have already
been classified in [12]. Without giving details we just state that, as a consequence of the
table displaying the classification results in [12], the only edge-biregular maps of Euler
characteristic −p for an odd prime p determined by a dihedral group of automorphisms are
the first two maps in Theorem 3.1 defined by the groups Hp(1) and Hp(2).
5.1 The case when the Fitting subgroup is cyclic
From now on we will assume that F is cyclic. In such a case F contains either no involution
(if |F | is odd) or a unique involution (if F2 is non-trivial cyclic 2-group). This implies that
F can contain at most one of the four involutions s, t, x, y.
20 Art Discrete Appl. Math. 5 (2022) #P3.08
We will show that F cannot have index 2 in H . Indeed, suppose [H : F ] = 2. If
F contains no generating involution from the set {s, t, x, y}, then the index-2 condition
implies that sx, xy, yt ∈ F . But since H = ⟨sx, xy, yt⟩ we would have F = H , a contra-
diction. Thus, let one of {s, t, x, y} be the unique generating involution of H contained in
F . We may without loss of generality assume that this element is y, as all the other cases
are handled by symmetries in the forthcoming argument. Now ⟨y⟩ is characteristic in F
and therefore normal in H , and so y, being an involution, is also central in H . Further,
from s, t /∈ F we have st ∈ F , and by uniqueness of the involution in F it follows that
y = st.
As now s, x /∈ F , we have sx ∈ F and so F contains the cyclic group ⟨sx⟩. If its order
is odd, then F also contains the cyclic group K = ⟨sx⟩⟨y⟩. But then, recalling centrality of
y in H , conjugation by, say, s inverts every element of K and so L = K ⋊ ⟨s⟩ is a dihedral
group. From s, xs, y ∈ L and y = st we also have t, x ∈ L and so L = H is dihedral, the
case we have already disposed of. If the order of ⟨sx⟩ is even, then y = (sx)j for j equal
to half of the order of sx, since both elements are an involution in the cyclic group F . But
we saw earlier that y = st, giving st = (sx)j and hence t = s(sx)j . This, however, with
y = (sx)j shows that H = ⟨s, x⟩, which again means that H is dihedral.
Altogether, we have shown that for non-dihedral H and for a cyclic F we must have
[H : F ] > 2. Now, by the available theory H/F embeds in Aut(F ), and as the latter is
Abelian if F is cyclic, we conclude that H/F is Abelian. But H/F is generated by four
elements of order at most 2, so that H/F ∼= Cm2 for some m such that 2 ≤ m ≤ 4. Further,
since H = ⟨st, sx, xy⟩ = ⟨st, ty, xy⟩ it follows that m ∈ {2, 3} and both sxF and tyF
have order at most 2, so that (sx)2, (ty)2 ∈ F .
From earlier calculations we recall that both k, ℓ are even and greater than 2, and as-
suming that the 2-part of k is not smaller than the 2-part of ℓ we have the following: If
c = gcd(k/2, ℓ/2), then c ∈ {1, 2}, k is a multiple of 4, and |H| = kℓ/(2c). In particular,
note that 8 ∤ ℓ, and c = 1 or 2 according as ℓ/2 is odd or even.
Under these conditions we first show that ty /∈ F . Indeed, suppose that ty ∈ F .
Observe that the order of (sx)c is ℓ/(2c) and hence odd, so that (sx)2 ∈ F implies (sx)c ∈
F also for c = 1. As ty, (sx)c ∈ F and the orders of the two elements, k/2 and ℓ/(2c), are
relatively prime, it follows that (k/2)(ℓ/(2c)) ≤ |F | = |H|/[H : F ] ≤ kℓ/(2c[H : F ]),
which yields [H : F ] ≤ 2, a contradiction.
It follows that ty /∈ F . But we know that (ty)2 ∈ F , and we may use essen-
tially the same chain of inequalities as above, with k/2 replaced by k/4 (which is the
order of (ty)2), to conclude that [H : F ] ≤ 4. We saw, however, that [H : F ] is a
power of 2 and greater than 2, so that [H : F ] = 4, and we must have equalities in
the above chain throughout. In more detail, and using the fact that F is cyclic, we have
F = ⟨(sx)c⟩⟨(ty)2⟩ = ⟨(sx)c(ty)2⟩ ∼= Cn for n = kℓ/(8c), with ty /∈ F . Observe that
the order of (sx)2 is odd in both cases for c ∈ {1, 2}, and is and relatively prime to k/4
(the order of (ty)2).
We show that F contains none of the generating involutions s, t, x, y of H , and at most
one of the involutions st, xy. For if F contained one of t, y, then this element would have
to coincide with the central involution (ty)k/4, but such an equality quickly gives t = 1 or
y = 1. If s ∈ F (and the case x ∈ F is done similarly), then s would commute with (sx)2,
which is equivalent to (sx)4 = 1. Then, ℓ/2 would have to divide 4 and since 8 ∤ ℓ we
would have ℓ = 4. But this would give H = kℓ/4 = k, so that H would be dihedral.
To address the remaining part, note that by non-orientability we know that
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 21
H = ⟨st, xy, yt⟩. From H/F = ⟨stF, xyF, ytF ⟩ and yt /∈ F while (yt)2 ∈ F , together
with [H : F ] = 4, it follows that at most one of st, xy can be contained in F .
In what follows we will without loss of generality assume that st /∈ F . As F and ⟨s, t⟩
now intersect trivially, with the help of the above finding this means that the semi-direct
product
F ⋊ ⟨s, t⟩ = ⟨(sx)c(ty)2⟩⋊ ⟨s, t⟩ ∼= Cn ⋊ V4 (5.1)
for n = kℓ/(8c) has order 4n = |H| and so H = F ⋊ ⟨s, t⟩ ∼= Cn ⋊ V4. If n = |F | is
even, then the unique non-trivial involution (ty)4 ∈ F generates a subgroup isomorphic to
C2 that is characteristic in F and hence normal in H , which means that (ty)4 is central in
H . But then H would contain the subgroup ⟨(ty)k/4⟩ × ⟨s, t⟩ ∼= C32 , contrary to the fact
that H has dihedral Sylow 2-subgroups. It follows that n is odd and so is κ = k/4 (and
we know the same about ℓ/(2c) = λ/c); also, both xy and st must lie outside F , and the
Sylow 2-subgroups of H are isomorphic to V4.
In the proof of solvability of H we have encountered the equation 2κλ− 2κ− λ = cp
for some c ∈ {1, 2}. If c = 2, then λ is exactly divisible by 2, and then oddness of κ with
2κ(λ− 1) = 2p+ λ gives a contradiction as the right-hand side is divisible by 4 while the
left-hand side is not. It follows that c = 1, and so sx ∈ F ; observe that then sy /∈ F as in
the opposite case we would have (xs)(sy) = xy ∈ F which has already been excluded.
We now let u = sx and v = ty; note that our cyclic group F , of order a product of two
odd and relatively prime numbers ℓ/2 and k/4 (the orders of u and v2), is generated e.g.
by uv2. We saw that y, ys, yt /∈ F , so that by (5.1) for c = 1 we must have y = wst for
some w ∈ F . The fact that w = yts commutes with u = sx is equivalent to ty(sx)yt =
xs = u−1, so that conjugation by v inverts u. Similarly, w = yts commutes with v2 ∈ F ,
which translates to [s, v2] = 1, and as u = sx commutes with v2 we also have [x, v2] = 1.
In somewhat more detail, let w = ua(v2)b = uav2b for uniquely determined integers
a, b such that 0 ≤ a < ℓ/2 and 0 ≤ b < k/4. Using the facts that s inverts u and commutes
with v2 and [u, v2] = 1, from y = wst it follows that v−2 = (yt)2 = (uav2bs)2 = v4b,
so that v4b+2 = 1, and for b in the above range we have 4b+ 2 = k/2 (the order of v) and
so b = (k − 4)/8. Normality of ⟨u⟩ in H (being characteristic in F ) further implies that
yuy = uj for for some j, 1 ≤ j < ℓ/2, such that j2 ≡ 1 mod ℓ/2. Since v inverts u, we
obtain u−1 = tyuyt = tujt, which implies tut = u−j . Observing now that conjugation
by st maps ua onto uaj and inverts v2, from y = uav2bst we obtain 1 = (uav2bst)2 =
ua(j+1), so that a(j + 1) ≡ 0 mod ℓ/2.
Going one step further and using properties derived above, from y = uav2bst we have
xy = xuasv2bt = u−a−1v2bt, and as conjugation by t inverts v2 and maps u onto u−j
one obtains 1 = (u−a−1v2bt)2 = u(a+1)(j−1), which gives (a + 1)(j − 1) ≡ 0 mod ℓ/2.
Subtracting the last congruence from a(j+1) ≡ 0 mod ℓ/2 yields 2a+1 ≡ j mod ℓ/2. As
2−1 = (ℓ+2)/4 mod ℓ/2 (which is odd), it follows that a ≡ (j−1)2−1 = (j−1)(ℓ+2)/4
mod ℓ/2, giving a unique value of a ∈ {0, 1, . . . , ℓ/2− 1}. Observe also that for this value
of a and for any j such that j2 ≡ 1 mod ℓ/2 one has (j + 1)a = (j2 − 1)2−1 ≡ 0 mod
ℓ/2, which is the congruence obtained earlier.
The last step will be reintroducing the notation k = 4κ and ℓ = 2λ for odd and rel-
atively prime κ and λ, and observing that 2b + 1 = κ and so 1 = tyv2buas = vκuas.
The last relation is equivalent to vκ−1ua = v−1s, and from [u, v2] = 1 (a consequence of
[s, v2] = 1 = [x, v2]) and oddness of κ it follows that 1 = [u, vκ−1ua] = [u, v−1s] and
commutation of u and v−1s is equivalent to u being inverted by conjugation by v. Sum-
ming up, the above facts well-define a group H = Hp,j of order kℓ/2 generated by four
22 Art Discrete Appl. Math. 5 (2022) #P3.08
involutions s, t, x, y and presented as follows, with u = sx and v = ty:
Hp,j = ⟨x, y, s, t |x
2, y2, s2, t2, (xy)2, (st)2, uλ, v2κ, [s, v2], [x, v2], tutuj , vκuas ⟩
(5.2)
for a non-negative integer j < ℓ/2 such that j2 ≡ 1 mod λ, with a = (j−1)(λ+1)/2. This
is the presentation appearing as a third item in Theorem 3.1. Note that here F = ⟨u, v2⟩ =
⟨uv2⟩ is cyclic, of order κλ = kℓ/8, and Hp,j = F ⋊ ⟨s, t⟩, as derived earlier.
Proof of correctness of the presentation (5.2)
Proof. Let G0 = ⟨s, x | s2, x2, (sx)λ⟩ ∼= Dℓ and let u = sx. Let us introduce a pair of
automorphisms θ and τ of G0 completely defined by letting θ(s) = s and θ(u) = u−j
for some j such that j2 ≡ 1 mod λ, and τ(u) = uj , τ(x) = x. These definitions imply,
for example, that θ(x) = su−j and τ(s) = ujx. It can be easily verified that θ and τ
commute and both are of order two. It follows that we have a well-defined split extension
of G0 by the subgroup V4 ∼= ⟨θ, τ⟩ < Aut(G0); the order of the extension is 4ℓ. By
general knowledge on split extensions the new group has an equivalent representation in
the form G1 = ⟨s, x⟩ ⋊ ⟨t, y⟩, where t, y are two commuting involutions acting on G0 by
conjugation the same way as the two earlier automorphisms do, that is, θ(z) = tzt and
τ(z) = yzy for every z ∈ G0. Note that this also implies that the subgroups G0 and ⟨t, y⟩
intersect trivially. Further, using u = sx and v = ty it can be verified that in G1 one has
the relations uvu = v; moreover, we have [s, t] = [x, y] = 1 by the definition of the two
automorphisms and their conversion to conjugations. It follows that the group G1 has a
presentation of the form
G1 = ⟨x, y, s, t |x
2, y2, s2, t2, (xy)2, (st)2, uλ, v2, tutuj , uvuv−1⟩ (5.3)
where, again, u = sx, v = ty, and j is an integer such that j2 ≡ 1 mod λ.
Next, let us consider the group G2 generated by the same involutions as G1 but with a
presentation obtained from that of G1 by omitting the relator v2 and adding the conditions
of v2 commuting with s and x:
G2 = ⟨x, y, s, t |x
2, y2, s2, t2, (xy)2, (st)2, uvuv−1, uλ, tutuj , [s, v2], [x, v2]⟩ (5.4)
The relators uvuv−1 and tutuj imply yuy = uj and txt = su−j , and these together
with tut = u−j and yxy = x show that ⟨u, x⟩ = ⟨s, x⟩ is a normal subgroup of G2.
By inspection, in the absence of any condition involving the element v the presentation of
G2/⟨s, x⟩ reduces to ⟨t, y | t2, y2⟩, which is an infinite dihedral group; hence G2 is infinite.
The subgroup N = ⟨v2⟩ is obviously normal in G2 and as G2/N ∼= G1 we conclude
that N is isomorphic to an infinite cyclic group. The subgroup κN = {(v2κ)i; i ∈ Z}
is characteristic in N and so normal in G2. Applying the Third isomorphism theorem we
obtain G2/N ∼= (G2/κN)/(N/κN) and so |G2/κN | = κ|G2/N |. Since G2/N ∼= G1,
the new group G3 = G2/κN of order 8κλ has a presentation obtained from that of (5.4)
by adding the relator v2κ:
G3 = ⟨x, y, s, t |x
2, y2, s2, t2, (xy)2, (st)2, uvuv−1, uλ, tutuj , [s, v2], [x, v2], v2κ⟩
(5.5)
The last step is to consider the element z = vκuas ∈ G3, where a = 2−1(j − 1)
mod λ as before. For the calculations that follow it is useful to observe that from the
earlier congruence a(j + 1) ≡ 0 mod λ we have aj ≡ −a mod λ, so that tuat = ua and
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 23
yuay = u−a; note also that u2a+1 = uj . We begin by showing that z is an involution.
Indeed, using [u, v2] = 1, the facts that both s and v invert u, and the properties of u listed
before one obtains (uasvκ)2 = u2asv−1vκ+1svκ = uλu−1sv−1sv = uλyu−1y = 1,
which yields z2 = 1.
We show even more; namely, that z is a central involution of G3. To show that z
commutes with s, note that the above implies that zs = vκua is an involution and so
(sz)2 = s(vκua)2s = 1, so that [s, z] = 1. With the help of the fact that conjugation
by t preserves ua and vκ (note that vκ = v−κ) we obtain [z, t] = vκuast · tvκuas =
z2 = 1. As κ is odd and so vκ inverts u, it follows that [x, z] = xvκua(sx)vκuas =
xvκua+1vκua+1x = 1. Last, using inversion of ua and preservation of vκ by conjugation
by y, as well as preservation of u by conjugation by sv, one has [z, y] = vκuasvκu−aysy =
vκuavκ−1u−asvysy = v−1svysy = 1.
We have proved that for z = vκuas the subgroup ⟨z⟩ ∼= C2 is central in G3. But this
means that the group G3/⟨z⟩, which is isomorphic to Hp,j , has order |G3|/2 = 4κλ =
kℓ/2, as claimed. This completes the proof of correctness of the presentation (5.2); the
relator uvuv−1 can be omitted since it is a consequence of the remaining ones, as shown
in the paragraph immediately preceding the presentation (5.2).
For completeness we show that none of the edge-biregular maps with automorphism
groups Hp,j presented as in (5.2) are fully regular. Indeed, in the opposite case Hp,j would
have to admit an automorphism interchanging s with x and t with y. In such a case,
however, along with s = vκua the group Hp,j would also have to admit the relation x =
v−κu−a = uavκ. This would imply xs = u2a and hence u2a+1 = 1, and as 2a + 1 ≡ j
mod λ we would have uj = 1, contrary to j2 ≡ 1 mod λ.
Finally, the Euler-Poincaré formula yields that p = 2κλ − 2κ − λ so p + 1 = (2κ −
1)(λ − 1). We know κ and λ are both odd so letting p + 1 = 2αbd′, where b ≡ 1 mod
4, and d′ is odd, then we have 2κ − 1 = b and λ − 1 = 2αd′ = d, which, so long as κ
and λ are coprime, will yield an edge-biregular map as described above. Hence, for every
factorisation p + 1 = bd such that b ≡ 1 mod 4 and gcd(b + 1, d + 1) = 1 we have such
a map of type (2(b + 1), 2(d + 1)), completing the analysis related to the third item of
Theorem 3.1.
5.2 The case when the 2-part of the Fitting subgroup is dihedral
Recall that we are investigating an edge-biregular map M = (H;x, y, s, t) of type (k, ℓ)
with both entries even, on a surface of Euler characteristic −p for some odd prime p; at the
beginning of section 5 we also made the assumption that the 2-part of k is not smaller than
the 2-part of ℓ and we had k = 4κ and ℓ = 2λ. By Proposition 5.2 we know that H is a
solvable group and so has a non-trivial Fitting subgroup F , of which we may assume that
F ̸= H . By earlier results and observations we also know that either F is cyclic, or F has
a dihedral 2-part, denoted F2. We have dealt with the first possibility in subsection 5.1 and
from now on we will assume that F2 is dihedral, which of course means that |F2| ≥ 4. Our
next result places a substantial restriction on F2 and Dk (the vertex-stabilizer in M ).
Proposition 5.3. If F2 is dihedral, then Dk is a 2-group, [Dk : F2] = 2, and k is a multiple
of 8 while ℓ/(2c) is odd.
Proof. We have assumed that the 2-part of k is not smaller than that of ℓ, and we know
that |H| = kℓ/(2c) for c = 1, 2. Analysis of equation (3.1) with these conditions yields
24 Art Discrete Appl. Math. 5 (2022) #P3.08
that a Sylow 2-subgroup of H is contained in Dk. As F2 is the Sylow 2-subgroup of F ,
normality of F in H implies that F2 is a subgroup of Dk.
Suppose F2 = Dk = ⟨t, y⟩. Then H/F2 is generated by sF2 and xF2, each of which
has order less than or equal to two. But |H|/|F2| = |H|/k = ℓ/(2c) = λ/c and since
c = gcd(2κ, λ) ≤ 2 it follows that λ/c must be an odd integer and hence the group
H/F2 does not contain any involution. Hence s, x ∈ F2 and so H = Dk, contrary to our
assumption that H is not dihedral.
To show that [Dk : F2] ≤ 2, up to the choice of labelling of orbits we may assume
F2 = ⟨y, (ty)
µ⟩ for some µ [the alternative option would be F2 = ⟨t, (ty)µ⟩]. Now, F2 is
normal in H (because it is characteristic in F ) and so yty = tyyyt = tyt ∈ F2. But y ∈ F2
also so this means (ty)2 ∈ F2, which implies |Dk : F2| ≤ 2. Hence Dk is a 2-group; the
conclusions about k and ℓ are obvious from the above.
Since F2 is normal in H and hence also in ⟨t, y⟩ ∼= Dk, we have established that there
are only three possibilities for a dihedral F2 if k ≥ 8: either F2 = ⟨t, y⟩, or F2 is one of
⟨t, (ty)2⟩, ⟨y, (ty)2⟩; in particular, k must be a power of 2. In the first case we have H/F2
generated by (at most) four involutions, but oddness of ℓ/(2c) = |H/F2| implies that the
generating involutions s, t, x, y all belong to F2 and hence H = F2 ∼= Dk; such maps
with a dihedral automorphism group have already been sorted out. In what follows we will
assume that F2 = ⟨z, (ty)2⟩ for some z ∈ {t, y}.
If k ≥ 16, the cyclic subgroup ⟨(ty)2⟩ of F2, of order k/4, is characteristic in F2 and
therefore normal in H . Note that for k = 8 and F2 ∼= C2 × C2 this need not be valid. For
now we will assume that ⟨(ty)2⟩ is normal in H also for k = 8 and we will return to the
opposite case later.
Before proceeding we make a remark about the case c = 2. By the proof of solvability
of H , for c = 2 the subgroups Dk and Dℓ intersect in a subgroup isomorphic to V4. For
k ≥ 16 the subgroup ⟨(ty)k/4⟩ ∼= C2 generated by the centre of F2 is characteristic in
F2 and hence normal in H , which means that (ty)k/4 is a central involution also in H .
Note that this is now also valid for k = 8 because of the assumption made in the previous
paragraph. If c = 2 we may also assume that ℓ ≥ 12 (as for ℓ = 4 we would have H a
dihedral group), and so for the central element of ⟨s, x⟩ ∼= Dℓ we have (sx)ℓ/4 = (ty)k/4.
Normality of ⟨(ty)2⟩ of F2 implies that s(ty)2s = (ty)2i and x(ty)2x = (ty)2j for
some integers i, j such that i2 ≡ j2 ≡ 1 mod k/4; the exponents 2 in the congruences
match the orders of s and x. It follows that sx(ty)2xs = s(ty)2js = (ty)2ij . But as
(sx)ℓ/(2c) is either the identity or a central element of H , we must have (ij)ℓ/(2c) ≡ 1
mod k/4. In view of the previous two congruences, oddness of ℓ/(2c) implies that 1 ≡
(ij)ℓ/(2c) ≡ ij, so that sx commutes with (ty)2. Since k and ℓ/(2c) are relatively prime
and k/4 is even while ℓ/(2c) is odd, it follows that the subgroup J = ⟨sx, (ty)2⟩ of H is
cyclic, of order kℓ/(8c), and generated by the product (sx)(ty)2.
From the fact that sx commutes with (ty)2 we have s(ty)2s = x(ty)2x and this element
is in ⟨(ty)2⟩ by normality of this subgroup in H . Note that we cannot have x ∈ J ; in the
opposite case x would have to be equal to (ty)k/4, the unique involution in J , but then x
would commute with sx and hence with s, giving (sx)2 = 1, contrary to ℓ ≥ 6c. Thus, the
semi-direct product K = J ⋊ ⟨x⟩ is a subgroup of H of order kℓ/(4c), and hence normal
(of index 2) in H .
The subgroup ⟨sx⟩ < J as the unique cyclic subgroup of K of order ℓ/2 is character-
istic in K and hence normal in H . Thus, y(sx)y = (sx)a and t(sx)t = (sx)b for some
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 25
positive integers a, b < ℓ/2, with both a, b odd if c = 2. By [x, y] = 1 = [s, t] we then have
ysy = s(xs)a−1 and (sy)2 = (xs)a−1; similarly, txt = (xs)b−1x and (tx)2 = (xs)b−1.
By normality of F2 in H , the element sys or xtx (depending on whether y ∈ F2 or t ∈ F2)
is equal to either the central element (ty)k/4 or an involution of the form (ty)2jz for some
j < k/4. In the first case, either sys or xtx would commute with xs, which is easily seen
to be equivalent to (sy)2 = 1 or (tx)2 = 1. In the second case, either (sy)2 or (tx)2
are a power of (ty)2 and so their order is a power of 2. But we have also established that
(sy)2 = (xs)a−1 or (tx)2 = (xs)b−1. Both (xs)a−1 and (xs)b−1 have, however, odd
order; this is obvious for c = 1 and for c = 2 it follows from oddness of a and b. The
two order parities can be matched only if (sy)2 = 1 or (tx)2 = 1. In both cases we have
established that, depending on whether y ∈ F2 or t ∈ F2, we have (sy)2 = 1 or (tx)2 = 1,
i.e., [s, y] = 1 or [t, x] = 1.
Next, we show that ty /∈ K. Indeed, if ty ∈ K, then K would contain the cyclic group
⟨ty⟩ ∼= Ck/2 and also the dihedral group ⟨s, x⟩ ∼= Dℓ. By comparing their orders with
|K| = kℓ/(4c) it follows that the two groups intersect non-trivially. If c = 2 then the two
groups would have to intersect in a group of order 4, which would have to be cyclic and
dihedral at the same time, a contradiction. It follows that c = 1 and the only non-trivial
element in their intersection is an involution. Since the only involution contained in the
cyclic group is the central one, we would have (ty)k/4 = s(sx)j for some j. But as the
central element commutes with sx, the same must hold for s(sx)j and this is easily seen
to be equivalent to (sx)2 = 1, contrary to the bound on ℓ. Thus, ty /∈ K, as claimed.
But then, from H/K = ⟨tK, yK⟩ ∼= C2 it follows that either t ∈ K or y ∈ K. This
means that K contains a dihedral subgroup of order k/2, which is in fact the unique Sylow
2-subgroup F2 of F . Hence F2 < K and, moreover, from [t, x] = 1 or [s, y] = 1 it follows
that K = ⟨sx⟩ · F2 ∼= Cℓ/2 × Dk/2. (In fact, at this stage it follows that K is the Fitting
subgroup F of H; to see this one only needs to see that F ̸= H but in the opposite case
F/F2 would be trivial (being generated by involutions) and we would be back in the case
F = Dk. However, in the light of the conclusion we will arrive at, the fact that K = F
will turn out to be irrelevant.)
To finish this part of our argument we explore the fact that x ∈ K. By the above
structural information this means that x ∈ F2, and so x would commute with sx, which is
equivalent to (sx)2 = 1, contrary to our bound on ℓ.
It thus remains to investigate the case when k = 8 and the subgroup ⟨(ty)2⟩ ∼= C2 of
F2 = ⟨z, (ty)
2⟩ ∼= C2 × C2 for some z ∈ {t, y} is not a normal subgroup of H , with
|H| = kℓ/(2c) = 4ℓ/c (note that now 8 exactly divides the order of H). For definiteness
we will assume that t ∈ F2; the case when y ∈ F2 is done in a completely analogous way
by replacing s with x and t with y in the subsequent arguments.
Thus, let F2 = {1, t, yty, (ty)2}; by our assumption y /∈ F2. If x was in F2 then clearly
x ̸= 1, t, and as x = yty implies ty has order two, the only possibility would be x = (ty)2.
This would mean that F2 = ⟨t, x⟩; by normality in H , conjugation by s preserves F2. We
cannot have [s, x] = 1 as this contradicts our bound on ℓ, and since [s, t] = 1 it follows
that sxs = xt and hence (sx)4 = 1, contrary to ℓ ≥ 6c. Therefore x /∈ F , and, as we
know, y /∈ F either, but note that xy ∈ F . Indeed, in the opposite case, by normality of
F2 and its trivial intersection with ⟨x, y⟩, the subgroup F2 ⋊ ⟨x, y⟩ of H would have order
16, contrary to 8 exactly dividing the order of H . A calculation as above shows that the
only option for xy ∈ F is xy = (ty)2 which is is equivalent to txty = 1 (hence tx has
order 4). Observe that this relation also shows that conjugation of F2 by x fixes (ty)2, and
26 Art Discrete Appl. Math. 5 (2022) #P3.08
commutativity of xy, t ∈ F2 implies xtx = yty.
Since F2 is normal in H , it is preserved by conjugation by s, which fixes t. If s fixed
(ty)2, then with x fixing (ty)2 the subgroup ⟨(ty)2⟩ would be normal in H , contrary to our
assumption made in this special case k = 8. It follows that conjugation by s induces an
automorphism of F2 fixing t and transposing yty with (ty)2; the relation s(yty)s = (ty)2
is equivalent to (syty)2t = 1. The composition of the two conjugations, namely, w 7→
(xs)w(sx) for w ∈ F2 ∼= C2 × C2, induces and automorphism in Aut(C2 × C2) ∼= S3
represented by the cycle t 7→ yty 7→ (ty)2 7→ t of length 3. In particular, conjugation
by (sx)3 centralizes t. If now ℓ/2 = 3q + r for r = ±1, then [(sx)3, t] = 1 implies
[(sx)r, t] = 1, contrary to the fact established earlier that conjugation by sx does not fix t.
It follows that r = 0 and hence ℓ/2 is a multiple of 3.
We are now approaching derivation of a presentation of H . Observe first that the rela-
tion xtx = yty derived earlier simplifies (syty)2t = 1 to (sxy)2t = 1. From y = txt and
the fact that (sx)3 centralizes F2 we obtain y(sx)3y = (xs)3, so that ⟨(sx)3⟩ is a normal
cyclic subgroup of H . If c = 2, then, by the findings in the previous paragraph, the odd in-
teger ℓ/4 is a multiple of 3 and so (sx)ℓ/4 is a central element of H as obviously commutes
with s and x, and also with t (because (sx)3 does) and hence also with y = txt. But then
⟨(sx)ℓ/4⟩ must be a subgroup of F2, otherwise its product with F2 would be isomorphic
to C32 , contrary to the Sylow 2-subgroups of H being dihedral. We cannot have (sx)
ℓ/4
equal to t or yty because the two elements do not commute with y, so that the only option
is (sx)ℓ/4 = (ty)2, but while the element on the left commutes with s the one on the right
does not. It follows that c = 1 and hence |H| = 4ℓ.
We note that normality of ⟨(sx)3⟩ cannot be extended to normality of ⟨sx⟩, as otherwise
we would have t(sx)t = (sx)d and then (xt)2 = (sx)d−1 for some d, which are elements
of orders of different parity (4 versus some (odd) divisor of ℓ/2). This shows, as an aside,
that the Fitting subgroup of H is F ∼= ⟨(sx)3⟩ × F2 ∼= Cℓ/6 × V4. It may also be useful to
note that y = txt implies that sy = stxt = t(sx)t, showing that the orders of sx and sy
are the same, namely, ℓ/2.
This way, in the case when k = 8 and F2 = ⟨t, yty⟩, with ℓ/2 = 3m for some odd
m ≥ 1, we have arrived at a presentation of H = Hp of the form
Hp = ⟨x, y, s, t |x
2, y2, s2, t2, (xy)2, (st)2, (sx)3m, (ty)4, (sxy)2t, txty⟩ (5.6)
representing the edge-biregular map M = Mp of type (k, ℓ) = (8, 6m) that appears in
item 4 of Theorem 3.1. The corresponding dual map is obtained by interchanging x with y
and s with t, while the twin map is found by interchanging x with s and y with t.
Proof of correctness of the presentation (5.6)
Proof. It remains to prove that the presentation (5.6) indeed defines a group of order
kℓ/2 = 24m. For this we begin with a ‘universal’ group U with relators as in (5.6) but
with (sx)3m omitted:
U = ⟨x, y, s, t |x2, y2, s2, t2, (xy)2, (st)2, (ty)4, (sxy)2t, txty⟩ (5.7)
We show that N = ⟨(sx)3⟩ is a normal subgroup of U . For this it is sufficient to prove
that N is invariant under conjugation by t, as from y = txt and automatic invariance of N
under conjugation by s and x it then follows that yNy = N . From the last two relators in
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 27
(5.7) we have 1 = (sxy)2t = (s(ty)2)2t, which, using (ty)4 = 1, gives s(ty)2s = yty. It
follows that conjugation by s fixes t (by the relation (st)2 = 1) and interchanges yty with
(ty)2. On the other hand, as x = tyt, one similarly obtains that conjugation by x fixes
(ty)2 and interchanges t with yty. It follows that the composition of the two conjugations,
that is, the mapping z 7→ (xs)z(sx), induces a 3-cycle t 7→ yty 7→ (ty)2 7→ t, so that
t commutes with (sx)3 and hence preserves N . (This is what we saw before but now we
needed to make sure that it was established solely from the presentation (5.7).)
By the Reidemeister-Schreier theory implemented in MAGMA in the form of its Rewrite
command one may check that N is a free group, that is, N is infinite cyclic. Also, by
MAGMA one can check that U/N ∼= S4, of order 24. Now, for an arbitrary integer
m ≥ 1 let Nm = ⟨(sx)3m⟩ be the cyclic subgroup of N of index m. Since Nm is char-
acteristic in N and so normal in U , we may use the Third isomorphism theorem to write
U/N ∼= (U/Nm)/(N/Nm) and as U/Nm ∼= H , N/Nm ∼= Cm and U/N ∼= S4 it follows
that |H| = 24m, as claimed. This proves correctness of the presentation (5.6).
We conclude by showing that none of the edge-biregular maps given by the group Hp
from (5.6) is regular. For such a map to be regular there would have to be an automorphism
of H of order 2 interchanging s with x and t with y. If this is the case then H would
also contain the relator ysyt. From the relator txty of H we have y = txt, which, when
substituted into ysyt = 1 and canceling terms, gives txsx = 1. Combining this with
tysy = 1 yields xsx = ysy, or, equivalently, (sxy)2 = 1. But this in combination with
the relator (sxy)2t of (5.6) gives t = 1, a contradiction. This completes both the analysis
related to the fourth item of Theorem 3.1 as well as our proofs of the main results from
section 3.
6 Concluding remarks
The maps Mp identified in the last part of our proof, those defined by the group Hp pre-
sented as in (5.6), deserve particular attention. Their structure is best visualized by consid-
ering the associated embedded Cayley graph Cay(Hp, X) for the group Hp and the gen-
erating set X = {x, y, s, t}. If superimposed onto the map Mp, the associated embedding
of Cay(Hp, X) displays cycles labelled alternately with y and t ‘around’ each vertex and
cycles labelled with x and s ‘around’ each face, while the 4-cycles labelled xyxy and stst
‘surround’, respectively, bold and dashed edges. The existence of the relator txty in the
presentation (5.6), equivalent to the relation x = tyt, demonstrates that all the bold edges
are loops. Applying this knowledge to the relator (sxy)2t, which (being of odd length)
signals non-orientability, one sees that the edges in the unshaded orbit partition into pairs
of double-edges, each forming a ‘central cycle’ of a Möbius strip in the embedding. The
Cayley graph Cay(Hp, X), part of which is shown in Figure 2, makes this clear. This is
also an alternative way of proving non-regularity for this map. Namely, the two edge orbits
contain (bold) loops on the one hand, and non-orientable (dashed) 2-cycles on the other,
so there will certainly not be an automorphism of the group which interchanges these two
orbits of edges.
In section 2 we pointed out that edge-biregular maps of type (k, ℓ) arise as smooth
quotients of the subgroup of the full triangle group (2.1) uniquely determined by inclusion
of the generators R0 and R2 while excluding R1. In this context it may be interesting
to review results for maps on surfaces of negative prime Euler characteristic arising from
smooth quotients of all the (up to) seven possible index-two subgroups of the full triangle
28 Art Discrete Appl. Math. 5 (2022) #P3.08
Key
x
y
s
t
Figure 2: Part of the Cayley graph Cay(Hp, X) with generating set X = {x, y, s, t}.
groups. Out of these, the subgroup most frequently referred to is ⟨R0, R1, R2⟩, the smooth
quotients of which form the class of orientably-regular maps. A complete classification of
orientably-regular maps of Euler characteristic −p for prime p can be found in [5], with ear-
lier results for orientably-regular maps with ‘large’ automorphism groups available in [1].
The bi-rotary maps of negative prime Euler characteristic, i.e., those arising from smooth
quotients of the subgroup ⟨R0, R1, R2⟩, have been classified in [6]. Since the interchange
of the involutory generators R0 and R2 induces map duality, analogous classification of
maps defined by smooth quotients of the subgroup ⟨R0, R1, R2⟩ of (2.1) follows from [6]
by dualisation.
This way, results in this paper are a further contribution to classification of highly sym-
metric maps on surfaces as above, this time arising as smooth quotients of the subgroup
⟨R0, R1, R2⟩ of index two in the full triangle group. In order to have a complete list
of highly symmetric maps on surfaces with Euler characteristic −p for prime p that can
be obtained as smooth quotients of index-two subgroups of triangle groups it is therefore
sufficient to examine, up to duality, only two more subgroups, namely, ⟨R0, R1, R2⟩ and
⟨R0, R1, R2⟩. For completeness we recall that the fully regular maps on these surfaces,
that is those arising as smooth quotients of the full triangle groups, have been classified in
[2].
ORCID iDs
Olivia Reade https://orcid.org/0000-0002-7598-9680
Jozef Širáň https://orcid.org/0000-0002-5901-7646
O. Reade and J. Širáň: Classifying edge-biregular maps of negative prime Euler characteristic 29
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[13] J. Širáň, How symmetric can maps on surfaces be?, in: S. R. Blackburn, S. Gerke and
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.09
https://doi.org/10.26493/2590-9770.1405.4ce
(Also available at http://adam-journal.eu)
The string C-group representations of the
Suzuki, Rudvalis and O’Nan sporadic groups
Dimitri Leemans* , Jessica Mulpas
Département de Mathématique, Université libre de Bruxelles, C.P.216,
Boulevard du Triomphe, 1050 Brussels, Belgium
Received 5 March 2021, accepted 2 June 2021, published online 26 July 2022
Abstract
We present new algorithms to classify all string C-group representations of a given
group G. We use these algorithms to classify all string C-group representations of the
Suzuki, Rudvalis and O’Nan sporadic groups. The new rank three algorithms also permit
us to get all string C-group representations of rank three for the Conway group Co2 and the
Fischer group Fi22.
Keywords: Abstract regular polytopes, string C-group representations, sporadic simple groups, algo-
rithms.
Math. Subj. Class.: 52B11, 20D08, 20F05
1 Introduction
Marston Conder has been one of the pioneers in generating all geometric or combinatorial
objects of a certain kind, as for instance all regular (hyper)maps of small genus, all abstract
regular or chiral polytopes with a sufficiently small number of flags, etc. Conder’s website1
contains an impressive collection of data that greatly helped researchers over the years to
get a better insight on the related topics. Most of the data available on Conder’s website are
in terms of groups and generators, the groups being automorphism groups of the objects
considered. A natural way of gaining knowledge on the structure of a group is indeed to
search for geometric and combinatorial objects on which it can act. Among those objects,
abstract regular polytopes are of great interest as they are highly symmetric combinatorial
*This work was partially supported by an ARC Advanced grant from the Fédération Wallonie-Bruxelles, Bel-
gium. The author would like to thank Martin Macaj for interesting discussions on finding a faster algorithm in the
rank three case.
E-mail addresses: Leemans.Dimitri@ulb.be (Dimitri Leemans), jessica.mulpas@outlook.com (Jessica
Mulpas)
1https://www.math.auckland.ac.nz/˜conder/
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.09
structures with many singular algebraic and geometric properties. Moreover, due to the
well-known one-to-one correspondence between abstract regular polytopes and string C-
groups, finding an abstract regular polytope with a given automorphism group G means
obtaining a sequence of involutions generating G, satisfying a certain intersection property
and such that two nonconsecutive involutions commute. This means that G is a string C-
group. Finally, string C-groups are smooth quotients of Coxeter groups, hence any Coxeter
diagram found in this process permits to conclude that the group appears as a quotient of
the corresponding (infinite) Coxeter group.
In 2006, Michael Hartley published “An atlas of small regular polytopes” [7], where
he classified all regular polytopes with automorphism groups of order at most 2000 (not
including orders 1024 and 1536). His work has been complemented later on by Conder
who computed all regular polytopes with up to 2000 flags, up to isomorphism and duality,
excluding those of rank 2 (regular polygons) and the degenerate examples that have a ′2′ in
their Schläfli symbol, using the LowIndexNormalSubgroups function of MAGMA [1]. The
complete list is available on Conder’s website.
Also in 2006, Dimitri Leemans and Laurence Vauthier published ”An atlas of poly-
topes for almost simple groups” [11], in which they gave classifications of all abstract
regular polytopes with automorphism group an almost simple group with socle of order up
to 900, 000 elements. More recently in [8], Hartley also classified, together with Alexan-
der Hulpke, all polytopes for the sporadic groups as large as the Held group (of order
4, 030, 387, 200) and Leemans and Mark Mixer classified, among others, all polytopes
for the third Conway group Co3 (of order 495, 766, 656, 000) [10]. Leemans, Mixer and
Thomas Connor then computed all regular polytopes of rank at least four for the O’Nan-
Sims group O′N [3].
All these computational data lead to many theoretical results. For more details we refer
to the recent survey [9].
The recorded computational times and memory usages for the use of previous algo-
rithms on groups among which O′N and Co3 provide a motivation to improve the algo-
rithms in order to obtain computational data for groups currently too large.
The challenges involve dealing with groups that have a relatively large smallest permu-
tation representation degree as well as groups with large Sylow 2-subgroups. The largest
sporadic group for which a classification of all string C-group representations exists was
Co3 group whose smallest permutation representation degree is on 276 points and whose
Sylow 2-subgroups are of order 210.
We use the technique described in [3] and develop it further to obtain an algorithm that
can be used on any permutation group as well as any matrix group to find polytopes of rank
at least four.
We also present a new algorithm to find polytopes of rank three and two new algorithms
to find polytopes of rank at least four that outperform the previous known algorithms.
With our new algorithms we manage to classify all string C-group representations of
the Suzuki sporadic group Suz and the Rudvalis sporadic group Ru. Although these groups
have order smaller than the one of Co3, their smallest permutation representation degrees
are respectively 1782 and 4060, and their Sylow 2-subgroups are of respective orders 213
and 214. We also finish the classification of the string C-group representations for O′N
started in [3] by computing all of its rank three representations. Finally, we classify all
string C-group representations of rank three for the Conway group Co2 and the Fischer
group Fi22.
D. Leemans and J. Mulpas: The string C-group representations of sporadic groups 3
The classifications for Suz, Ru and O′N are made available in the online version of the
Atlas [11].
The paper is organised as follows. In Section 2, we give the background needed to
understand this paper. In Section 3, we describe the new algorithm to compute rank three
string C-group representations of a group. In Section 4, we describe three new algorithms
to compute rank three string C-group representations of a group and compare the efficiency
of the algorithms. In Section 5, we describe the new results we manage to obtain with the
algorithms described in the previous two sections.
2 Abstract regular polytopes and string C-groups
An abstract n-polytope P is a ranked partially ordered set (P,≤) with four defining prop-
erties (P1), (P2), (P3) and (P4) as defined below. Here, n is the rank of P . We shall
name the elements of P faces and differentiate the elements by rank calling an element of
rank i ∈ {−1, 0, · · · , n − 1, n} an i-element. For any two faces F and G of P such that
F ≤ G, we also define the section G/F to be the collection of all faces H of P such that
F ≤ H ≤ G. Note that any section of a polytope is itself a polytope.
(P1) P has two improper faces : a least face F−1 of rank −1 and a greatest face Fn of
rank n.
(P2) Each flag (i.e. maximal totally ordered subset) of P contains n+ 2 faces (including
the two improper faces).
(P3) P is strongly connected, that is, each section of P (including P itself) is connected
(in the sense given below).
(P4) P satisfies the diamond condition, that is, for any two faces F and G of P such
that F < G and rank(G) = rank(F ) + 2, there are exactly two faces H such that
F < H < G.
A ranked partially ordered set of rank d with properties (P1) and (P2) is said to be
connected if either d ≤ 1 or d ≥ 2 and for any two proper faces F andG of P (i.e. any faces
other than F−1 and Fd) there is a sequence of proper faces F = H0, H1, · · · , Hk−1, Hk =
G such that Hi ≤ Hi+1 for i ∈ {0, 1, · · · , k − 1}.
Two flags of an n-polytope are said to be adjacent if they differ by only one face.
In particular, two flags are said to be i-adjacent if they differ only by their i-face. As a
consequence of the diamond condition, any flag Φ of a polytope has exactly one i-adjacent
(for any i ∈ {0, 1, ..., n− 1}) flag that we shall denote by Φi. Note that (Φi)i = Φ for all i
and that (Φj)i = (Φi)j for all i and j such that | i− j |≥ 2
The automorphism group of an n-polytope P is denoted by Γ(P). An n-polytope is
said to be regular if its automorphism group Γ(P) has exactly one orbit on the flags of P ,
or equivalently, if for any flag Φ = {F−1, F0, F1, · · · , Fn} and each i ∈ {0, 1, ..., n − 1},
there is a unique (involutory) automorphism ρi of P such that ρi(Φ) = Φ
i. In this case,
one usually chooses a flag as a reference flag and calls it base flag.
Finally, an n-polytope is called equivelar is for any i ∈ {1, 2, ..., n − 1}, there is an
integer pi such that any section G/F of P defined by an (i− 2)-face F and an (i+1)-face
G is a pi-gon. If this is the case then we say that P has Schläfli type {p1, p2, ..., pn−1}.
Let G be a group and S := {ρ0, ρ1, ..., ρn−1} be a set of elements of G such that
G = ⟨S⟩. We define the following two properties.
4 Art Discrete Appl. Math. 5 (2022) #P3.09
(SP) the string property, that is (ρiρj)
2 = 1G for all i, j ∈ {0, 1, ..., n−1} with | i−j |≥
2;
(IP) the intersection property, that is ⟨ρi | i ∈ I⟩ ∩ ⟨ρj | j ∈ J⟩ = ⟨ρk | k ∈ I ∩ J⟩ for
any I, J ⊆ {0, 1, ..., n− 1}.
A pair (G,S) as above that satisfies property (SP) is called a string group generated by
involutions (or sggi in short). A string C-group is an sggi that satisfies property (IP).
The rank of (G,S) is the size of S. For any subset I ⊆ {0, . . . , n − 1}, we denote
⟨ρj : j ∈ {0, . . . , n − 1} \ I⟩ by GI . If I = {i}, we denote G{i} by Gi. Similarly, if
I = {i, j}, we denote G{i,j} by Gij .
As shown in [12], the automorphism group Γ(P) of an abstract regular polytope P ,
together with the involutions that map a base flag to its adjacent flags is a string C-group
representation (see [12, Propositions 2B8, 2B10 and 2B11]). More precisely, if one fixes
a base flag Φ of P , Γ(P) = ⟨ρ0, ρ1, ..., ρn−1⟩ with ρi the unique involution such that
ρi(Φ) = Φ
i then the pair (Γ(P), {ρ0, ρ1, ..., ρn−1}) is a string C-group representation.
The Schläfli type of a string C-group representation (G, {ρ0, . . . , ρn−1}) is the ordered set
{p1, p2, ..., pn−1} where pi is the order of the element ρi−1ρi for i = 1, . . . , n− 1.
Conversely, as shown in [12, Theorem 2E11], a regular n-polytope can be constructed
uniquely from a string C-group representation (G,S) with S := {ρ0, ..., ρn−1}. Let Gi :=
⟨ρj | j ̸= i⟩ for any i ∈ {0, 1, ..., n − 1}. We also set G−1 = Gn := G. For i ∈
{−1, 0, 1, ..., n − 1, n}, we take the set of i-faces of P to be the set of all right cosets
Giϕ of Gi in G. We define a partial order on P as follows : Giϕ ≤ Gjψ if and only if
−1 ≤ i ≤ j ≤ n and Giϕ ∩Gjψ ̸= ∅.
We say that two string C-group representations (G,S) and (G,S′) ofG are isomorphic
if there exists an automorphism ofG that maps S onto S′. The dual of a string C-group rep-
resentation (G, {ρ0, ρ1, ..., ρn−1}) is the string C-group representation (G, {ρn−1, ρn−2,
..., ρ0}).
Due to the one-to-one correspondence between string C-group representations and au-
tomorphism groups of abstract regular polytopes, finding all abstract regular polytopes
with a fixed automorphism group amounts to considering all ways of presenting a partic-
ular group as a string C-group and verifying whether they yield non-isomorphic abstract
regular polytopes. Our algorithms classify string C-group representations of a group G up
to isomorphism and duality.
3 The rank three case
Every rank three string C-group representation of a group G is a pair (G, {ρ0, ρ1, ρ2})
where ⟨ρ0, ρ1⟩ is a dihedral group and ρ2 commutes with ρ0.
The dihedral subgroups of G can be generated from the conjugacy classes of elements
of G using the following simple observation.
Lemma 3.1. Let G be a group. The group G has a dihedral subgroup D if and only if
there exists an element τ of G and an involution ρ ̸= τ of G such that D := ⟨τ, ρ⟩ and
τρ = τ−1.
Proof. If there exists a dihedral subgroup D in G, take two involutions that generate D,
say ρ and ρ′. Then τ = ρρ′ is such that D := ⟨τ, ρ⟩ and τρ = τ−1.
On the other hand, suppose there is a subgroup D of G such that D := ⟨τ, ρ⟩ and
τρ = τ−1 for some element τ of G and some involution ρ of G distinct from τ . Then
D. Leemans and J. Mulpas: The string C-group representations of sporadic groups 5
ρ′ := ρτ is an involution (as ρ′ρ′ = ρτρτ = ρ−1τρτ = τ−1τ = 1G and ρ
′ ̸= 1G since
ρ ̸= τ ) and ρρ′ = ρρτ = τ . Therefore D is a dihedral subgroup of G.
So an easy way to construct all conjugacy classes of dihedral subgroups of a group G
is the following:
1. Let D := ∅.
2. Compute a set C containing one representative of each conjugacy class of elements
of G.
3. For each τ ∈ C, compute N(τ) := NG(⟨τ⟩).
4. For each involution ρ ∈ N(τ), if τρ = τ−1 then add ⟨τ, ρ⟩ to D provided it is not
conjugate in G to an element already in D.
At the end of the above process, D contains one representative of each conjugacy class
of dihedral subgroups of G.
In order to generate all rank three string C-group representations of G, we now need to
do the following.
1. Let S := ∅.
2. For each D ∈ D
3. For each pair ρ0, ρ1 of generating involutions of D, if it is not conjugate in G to a
pair already in S , add this pair to S
At the end of the above process, S contains all the pairs ρ0, ρ1 that we try now to extend to
a triple of involutions.
Here we can either try to extend the ordered pair [ρ0, ρ1] to the right or to the left
(which is equivalent to extending the reversed pair [ρ1, ρ0] to the right). If we try to extend
it to the right, that is by adding a ρ2, we know that ρ2 ∈ CG(ρ0). So we can try every
involution in CG(ρ0), and if G is simple, we also know that ρ2 does not commute with
ρ1, for otherwise the string C-group (G, {ρ0, ρ1, ρ2}) is degenerate and G is therefore not
simple (e.g. ⟨ρ1, ρ2⟩ being a non-trivial normal subgroup of G).
This algorithm turns out to be extremely efficient as we will show in Section 5. It
permitted us to compute all rank three string C-group representations of Ru, Suz, O′N, Co2
and Fi22.
4 The ranks four and above
We describe three techniques to find string C-group representations of rank at least four.
4.1 Using centralizers of involutions
This algorithm uses the following observations.
Lemma 4.1. LetG be a group. Let (G, {ρ0, . . . , ρn−1}) be a string C-group representation
ofG. The subgroupG1 is a subgroup of the centraliser of ρ0, in particularG1 = ⟨ρ0⟩×G01
where G01 ≤ CG(ρ0).
6 Art Discrete Appl. Math. 5 (2022) #P3.09
Proof. This is a direct consequence of Proposition 2B12 in [12] and of the definition of
string C-groups.
This lemma implies that we can take for ρ0 a representative of a conjugacy class of
involutions of G, compute its centraliser CG(ρ0) and then the subgroup G1 must be a
subgroup of CG(ρ0).
We then proceed to construct G1. In order to do so, we produce all string C-group
representations of subgroups H of CG(ρ0) that have ρ0 as first generator. As CG(ρ0)
is substantially smaller than G, one may now apply the former algorithms (from [10])
to CG(ρ0) in order to classify all the string C-group representations of all its subgroups.
With each string C-group representation ⟨ρ0, ρ2, ..., ρn−1⟩ of G1, we proceed by adding an
appropriate involution ρ1 of G to it, giving a string C-group representation for G. In order
to restrict the number of involutions ρ1 to consider, we use the following observation.
Lemma 4.2. If (G, {ρ0, . . . , ρn−1}) is a string C-group representation of the group G,
then ρ1 ∈ CG(ρ3) ∩ . . . ∩ CG(ρn−1).
Proof. Since ρ1 has to commute with every generator ρi for i ≥ 3 by property (SP), we
have that ρ1 ∈ CG(ρ3) ∩ . . . ∩ CG(ρn−1).
We then check that the resulting pair (G, {ρ0, ρ1, ..., ρn−1}) is a string C-group repre-
sentation of G using the following proposition.
Proposition 4.3 ([12, Proposition 2E16(a)]). Let (G, {ρ0, ρ1, ..., ρn−1}) be an sggi. If G0
and Gn−1 are string C-groups and if Gn−1 ∩ G0 = G0,n−1 then (G, {ρ0, ρ1, ..., ρn−1})
is a string C-group.
Finally, let us note that although the former algorithms only allowed to be implemented
for groups of permutations, this new algorithm also works for matrix groups. This gives
access to groups much larger in size, as long as the centralizers of involutions of G are
small enough to be treated as permutation groups.
We give in Table 1 the pseudo-code of our new algorithm.
4.2 Higher ranks from the new rank three algorithm
If in the process of generating the rank three string C-group representations with the al-
gorithm described in Section 3, we keep also the triples that do not generate the group
G, we can then try to extend these triples to 4-tuples and so on, getting string C-group
representations of all possible ranks.
There are two possible approaches here. We could either decide to keep triples of
involutions ρ0, ρ1, ρ2 and try to extend them to rank four and so on, or keep the triples
of the shape ρ0, ρ1, ρ3, namely, keeping the triples where the third involution commutes
with the first two. We have designed algorithms to try both approaches. We call the first
approach the linear approach and the second one the central approach.
4.3 The linear approach
We can easily give an algorithm based on the rank three method presented in Section 3 to
produce all string C-group representations of all ranks for a given group. Indeed, if, while
computing the rank three representations we keep track of all triples ρ0, ρ1, ρ2 that satisfy
D. Leemans and J. Mulpas: The string C-group representations of sporadic groups 7
Input: G the group for which we want to compute all
string C-group representations.
Output: L a sequence containing the pairwise non-isomorphic
string C-group representations.
1. Compute the conjugacy classes CC(G) of elements of G.
2. Initialise L.
3. For each conjugacy class c of elements of order 2 in
CC(G):
4. Let r0 be a representative of that class.
5. Build the centralizer H of r0 in G.
6. If G is a matrix group,
7. find a permutation representation P (H) of H;
8. reduce the degree of P (H);
9. If G is already a permutation group then
10. only reduce its degree and call it P (H).
11. Using the existing procedures to do so (see [11]),
12. compute the string C-group representations with
generators {r0, r2, . . . , rn−1} for subgroups of CG(r0),
forcing r0 as first generator.
13. For each such representation, try to complete it by
inserting an involution r1 between r0 and r2,
using the fact that it has to be in the centralisers
of the ri’s for i = 3, . . . , n− 1.
14. Compute G0 = ⟨r1, ..., rn−1⟩, Gn−1 = ⟨r0, ..., rn−2⟩ and
G0,n−1 = ⟨r1, ..., rn−2⟩.
15. Check the intersection property for G0 and Gn−1 and
16. Check that G0 ∩Gn−1 = G0,n−1.
17. Let GP = ⟨r0, r1, ..., rn−1⟩ and let
∼
GP = ⟨rn−1, ..., r1, r0⟩ be
its dual.
18. If GP and
∼
GP are non-isomorphic to all the elements
of L,
19. add GP to L.
20. At the end, L contains one representative of each
isomorphism class of string C-group representation
of G.
Table 1: The pseudo-code of the new algorithm for ranks 4 and above.
8 Art Discrete Appl. Math. 5 (2022) #P3.09
the intersection property but do not generate the full group G, we can then try to extend
these triple to four-tuples and so on in the following way. Suppose P contains the triples
ρ0, ρ1, ρ2 that satisfy the intersection property but do not generate G.
• While P is not empty do the following.
• Let Q be an empty set.
• Let r be the number of involutions in a tuple of P .
• For each tuple ρ0, . . . , ρr−1 of P , try to extend it on the left by looking for invo-
lutions ρ−1 ∈ CG(ρ1) ∩ . . . ∩ CG(ρr−1) such that o(ρ−1ρ0) > 2 and such that
⟨ρ−1, . . . , ρr−1⟩ = G and {ρ−1, . . . , ρr−1} satisfies the intersection property. All
such ρ−1 give a string C-group representation of rank r + 1. In this process, when-
ever a ρ−1 is found such that ⟨ρ−1, . . . , ρr−1⟩ < G and {ρ−1, . . . , ρr−1} satisfies
the intersection property, add it in Q provided it is not isomorphic to any element of
Q yet.
• For each tuple ρ0, . . . , ρr−1 of P , try to extend it on the right by looking for in-
volutions ρr ∈ CG(ρ0) ∩ . . . ,∩CG(ρr−2) such that o(ρr−1ρr) > 2 and such that
⟨ρ0, . . . , ρr⟩ = G and {ρ0, . . . , ρr} satisfies the intersection property. All such ρr
give a string C-group representation of rank r + 1. In this process, whenever a ρr
is found such that ⟨ρ0, . . . , ρr⟩ < G and {ρ0, . . . , ρr} satisfies the intersection prop-
erty, add it in Q provided it is not isomorphic to any element of Q yet.
• At the end of this for loop, we have found all string C-group representations of rank
r + 1 of G and in Q we have r + 1 tuples that we can try to extend further.
• Let P := Q and continue the while loop.
4.4 The central approach
We can also give an algorithm based on the rank three one to produce all string C-group
representations of all ranks for a given group but using the observations of Section 4.1.
Indeed, if, while computing the rank three representations we keep track of all ordered
triples ρ0, ρ1, ρ2 that satisfy the intersection property but do not generate the full group
G, and such that ρ2 commutes with both ρ0 and ρ1, we can then try to extend these triple
to four-tuples and so on in the following way. Suppose P contains these triples ρ0, ρ1, ρ2
described above.
• While P is not empty do the following.
• Let Q be an empty set.
• Let r be the number of involutions in a tuple of P .
• For each tuple ρ0, . . . , ρr−1 of P , try to extend it on the left by looking for involutions
ρ−1 ∈ CG(ρ1)∩. . . ,∩CG(ρr−2) such that o(ρ−1ρ0) > 2, o(ρ−1ρr−1) > 2 and such
that ⟨ρ−1, . . . , ρr−1⟩ = G and {ρ−1, . . . , ρr−1} satisfies the intersection property.
All such ρ−1 give a string C-group representation (G, {ρr−1, ρ−1, ρ0, . . . , ρr−2}) of
rank r+1. In this process, whenever a ρ−1 is found such that ⟨ρ−1, . . . , ρr−1⟩ < G,
o(ρ−1ρr−1) = 2 and {ρ−1, . . . , ρr−1} satisfies the intersection property, add the
D. Leemans and J. Mulpas: The string C-group representations of sporadic groups 9
ordered tuple {ρ−1, ρ0, . . . , ρr−2, ρr−1} in Q, provided it is not isomorphic to any
element of Q yet.
• For each tuple ρ0, . . . , ρr−1 of P , try to extend it on the right by looking for invo-
lutions ρr ∈ CG(ρ0) ∩ . . . ,∩CG(ρr−3) such that o(ρr−1ρr) > 2, o(ρr−2ρr) > 2
and such that ⟨ρ0, . . . , ρr⟩ = G and {ρ0, . . . , ρr} satisfies the intersection property.
All such ρr give a string C-group representation (G, {ρ0, . . . , ρr−2, ρr, ρr−1}) of
rank r + 1. In this process, whenever a ρr is found such that ⟨ρ0, . . . , ρr⟩ < G and
{ρ0, . . . , ρr} satisfies the intersection property, add the ordered tuple {ρ0, . . . , ρr−2,
ρr, ρr−1} in Q, provided it is not isomorphic to any element of Q yet, o(ρr−1ρr) = 2
and o(ρr−2ρr) > 2.
• At the end of this for loop, we have found all string C-group representations of rank
r + 1 of G and in Q we have r + 1 tuples that we can try to extend further.
• Let P := Q and continue the while loop.
4.5 Computational times
Table 2 gives, for a given group G, the Time2 it takes (in seconds) to compute all string
C-group representations with the central approach, the linear approach, the rank three al-
gorithm and the algorithm described in Section 4.1 for the higher ranks, the old algorithm
(when this time is small enough and the old algorithm was capable of getting the whole
result), the number of string C-group representations of rank greater than 3 and the number
of string C-group representations of rank three. The time in the ”High+Rank3” column is
given as a sum of two numbers. The first number is the time it takes to compute string
C-group representations of rank at least four with the algorithm of Section 4.1. The second
number is the time it takes to compute the rank three ones with the new algorithm described
in Section 3. We leave a ??? when the computer was unable to finish the computation.
5 New results
Two sporadic groups smaller than Co3, namely Suz and Ru, had apparently never been
investigated. The two main reasons were that these groups have a higher permutation
representation degree than Co3 and that they have larger Sylow 2-subgroups (of respective
sizes 213 and 214 while the Sylow 2-subgroups of Co3 have order 2
10). We analysed them
with our new set of programs.
Our new algorithm also permitted to complete the classification of string C-group rep-
resentations for the O’Nan group, and to obtain all string C-group representations of rank
three for the Conway group Co2 and the Fischer group Fi22.
We now summarize our findings.
5.1 The Rudvalis group
The Rudvalis group was discovered by Arunas Rudvalis in 1973 [14]. It has order
145, 926, 144, 000 = 214 ·33 ·53 ·7 ·13 ·29 and smallest permutation representation degree
2The timings presented in this section were obtained using MAGMA [1] running on a computer with 8 cores
running at 3.9Ghz and 512Gb of RAM at 2.9Ghz. Note that MAGMA does not do parallel computing, it is using
one core at a time.
10 Art Discrete Appl. Math. 5 (2022) #P3.09
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D. Leemans and J. Mulpas: The string C-group representations of sporadic groups 11
4060. It has two conjugacy classes of involutions. It has 21594 string C-group representa-
tions of rank three, 227 of rank four and none of higher rank.
5.2 The Suzuki sporadic group
The Suzuki sporadic group was discovered by Michio Suzuki in 1968 [15]. It has order
448, 345, 497, 600 = 213 · 37 · 52 · 7 · 11 · 13 and smallest permutation representation
degree 1782. It has two conjugacy classes of involutions. It has 7119 string C-group
representations of rank three, 257 of rank four, 13 of rank five and none of higher rank.
5.3 The O’Nan group
In 1973, Michael O’Nan provided in [13] strong evidence for the existence of a new spo-
radic group now called O′N. Later in the seventies, Sims constructed this group with help
of a computer (see [6] for a survey of the story of O′N) but his work seems to be unpub-
lished. The group has order 4, 608, 155, 059, 20 = 29 · 34 · 5 · 73 · 11 · 19 · 31 and smallest
permutation representation degree 122, 760. This makes it a difficult group to deal with
even though its 2-Sylows are not that large compared to Suz and Ru.
All its string C-group representations of rank at least four were determined by Connor,
Leemans and Mixer [3]. In [2], Connor and Leemans computed the number of regular maps
having the O’Nan group as automorphism group, therefore bounding the number of string
C-group representations of rank three. However, they were unable to compute all string
C-group representations of rank three. Thanks to our new algorithm, we finally fill in this
gap in the classification of string C-group representations of the O’Nan group. The O’Nan
group has 6536 string C-group representations of rank three, 16 of rank four and none of
higher rank.
5.4 The Conway group Co2
The Conway group Co2 was discovered by John Horton Conway in 1968 as a subgroup of
a group he called .0, among with the other two simple sporadic groups Co1 and Co3 [4].
As pointed out by Conway in his paper, the simplicity of these groups was proven by John
Thompson. The group has order 42, 305, 421, 312, 000 = 218 ·36 ·53 ·7·11·23 and smallest
permutation representation degree 2300. It has three conjugacy classes of involutions. It
has 60370 string C-group representations of rank three.
5.5 The Fischer group Fi22
The Fischer group Fi22 was discovered by Bernd Fischer in 1971 [5] while studying groups
generated by 3-transpositions. The group has order 64, 561, 751, 654, 400 = 217 ·39 ·52 ·7 ·
11 · 13 and smallest permutation representation degree 3510. It has three conjugacy classes
of involutions. It has 25052 string C-group representations of rank three.
ORCID iDs
Dimitri Leemans https://orcid.org/0000-0002-4439-502X
Jessica Mulpas https://orcid.org/0000-0002-6128-2561
12 Art Discrete Appl. Math. 5 (2022) #P3.09
References
[1] W. Bosma, J. Cannon and C. Playoust, The Magma Algebra System I: the user language, J.
Symbolic Comput. 3-4 (1997), 235–265, doi:10.1006/jsco.1996.0125.
[2] T. Connor and D. Leemans, Algorithmic enumeration of regular maps, Ars Math. Contemp. 10
(2016), 211–222, doi:10.26493/1855-3974.544.1b4.
[3] T. Connor, D. Leemans and M. Mixer, Abstract regular polytopes for the O’Nan group, Inter-
nat. J. Algebra Comput. 24 (2014), 59–68, doi:10.1142/s0218196714500052.
[4] J. H. Conway, A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple
groups, Proc. Nat. Acad. Sci. U.S.A. 61 (1968), 398–400, doi:10.1073/pnas.61.2.398.
[5] B. Fischer, Finite groups generated by 3-transpositions. I, Invent. Math. 13 (1971), 232–246,
doi:10.1007/bf01404633.
[6] D. Gorenstein, The classification of finite simple groups, I. Simple groups and local analysis,
J. Am. Math. Soc. 1 (1979), 43–199, doi:10.1090/s0273-0979-1979-14551-8.
[7] M. I. Hartley, An atlas of small regular polytopes, Periodica Math. Hungar. 53 (2006), 149–
156, doi:10.1007/s10998-006-0028-x.
[8] M. I. Hartley and A. Hulpke, Polytopes derived from sporadic simple groups, Contrib. Discrete
Math. 5 (2010), 106–118, doi:10.11575/cdm.v5i2.61945.
[9] D. Leemans, String C-group representations of almost simple groups: A survey, in: Polytopes
and discrete geometry, Amer. Math. Soc., [Providence], RI, volume 764 of Contemp. Math.,
pp. 157–178, 2021, doi:10.1090/conm/764/15335.
[10] D. Leemans and M. Mixer, Algorithms for classifying regular polytopes with a fixed automor-
phism group, Contrib. Discrete Math. 7 (2012), 105–118, doi:10.11575/cdm.v7i2.62152.
[11] D. Leemans and L. Vauthier, An atlas of abstract regular polytopes for small groups, Aequa-
tiones Math. 72 (2006), 313–320, doi:10.1007/s00010-006-2843-9.
[12] P. McMullen and E. Schulte, Abstract regular polytopes, volume 92, Cambridge University
Press, 2002.
[13] M. O’Nan, Some evidence for the existence of a new simple group, Proc. London Math. Soc.
(3) 32 (1976), 421–479, doi:10.1112/plms/s3-32.3.421.
[14] A. Rudvalis, A new simple group of order 214 · 33 · 53 · 7 · 13 · 29, Notices Am. Math. Soc.
(1973), A–95.
[15] M. Suzuki, A simple group of order 448, 345, 497, 600, in: Theory of Finite Groups (Sympo-
sium, Harvard Univ., Cambridge, Mass., 1968), Benjamin, New York, 1969 pp. 113–119.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.10
https://doi.org/10.26493/2590-9770.1411.6ee
(Also available at http://adam-journal.eu)
On Roli’s cube
Barry Monson*
Department of Mathematics and Statistics, University of New Brunswick
Fredericton, New Brunswick, Canada
Received 6 April 2021, accepted 4 June 2021, published online 26 July 2022
Abstract
First described in 2014, Roli’s cube R is a chiral 4-polytope, faithfully realized in
Euclidean 4-space (a situation earlier thought to be impossible). Here we describe R in a
new way, determine its minimal regular cover, and reveal connections to the Möbius-Kantor
configuration.
Keywords: Regular and chiral polytopes, realizations of polytopes.
Math. Subj. Class.: 51M20, 52B15
1 Introduction
Actually Roli’s cube R isn’t a cube, although it does share the 1-skeleton of a 4-cube. First
described by Javier (Roli) Bracho, Isabel Hubard and Daniel Pellicer in [3], R is a chiral 4-
polytope of type {8, 3, 3}, faithfully realized in E4 (a situation earlier thought impossible).
Of course, Roli didn’t himself name R; but the eponym is pleasing to his colleagues and
has taken hold.
Chiral polytopes with realizations of ‘full rank’ had (incorrectly) been shown not to
exist by Peter McMullen in [11, Theorem 11.2]. Mind you, these objects do seem to be
elusive. Pellicer has proved in [15] that chiral polytopes of full rank can exist only in ranks
4 or 5.
Roli’s cube R was constructed in [3] as a colourful polytope, starting from a hemi-4-
cube in projective 3-space. (For more on this, see Section 3.) The construction given here in
Section 6 is a bit different, though certainly closely related. In Section 7 we can then easily
manufacture the minimal regular cover T for R, and give both a presentation and faithful
*This work was generously supported at the Universidad Nacional Autónoma de México (Morelia) by PAPIIT-
UNAM grant #IN100518. Besides greeting Marston Conder, I also want to thank Daniel Pellicer for generously
welcoming me to the Centro de Ciencias Matemáticas at UNAM (Morelia). Finally, thanks are also due the
referee, whose useful advice clearly indicated a careful reading of the manuscript.
E-mail address: bmonson@unb.ca (Barry Monson)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.10
representation for its automorphism group. Along the way, we encounter the Möbius-
Kantor Configuration 83, the regular complex polygon 3{3}3 and the 24-cell {3, 4, 3}.
In what follows, we can make our way with concrete examples, so we won’t need much
of the general theory of abstract regular or chiral polytopes and their realizations. We refer
the reader to [12, 13] and [16] for more.
This is a good place to offer a thank you to Marston Conder: first, for his wonderfully
fresh insights into the world of symmetry; and second, for his humour, which so often has
us weeping with laughter!
2 The 4-cube: convex, abstract and colourful
The most familiar of the regular convex polytopes in Euclidean space E4 is surely the 4-
cube P = {4, 3, 3}. A projection of P into E3 is displayed in Figure 1.
v
w
−v
Figure 1: A 2-dimensional look at a 3-dimensional projection of the 4-cube.
Let us equip E4 with its usual basis b1, . . . , b4 and inner product. Then we may take
the vertices of P to be the 16 sign change vectors
e = (ε1, . . . , ε4) ∈ {±1}4. (2.1)
At any such vertex there is an edge (of length 2) running in each of the 4 coordinate di-
rections, so that P has 32 = 16·42 edges. Similarly we count the 24 squares {4} as faces
of dimension 2. Finally, P has 8 facets; these faces of dimension 3 are ordinary cubes
{4, 3}. They lie in four pairs of supporting hyperplanes orthogonal to the coordinate axes.
It is enjoyable to hunt for these faces in Figure 2, where the 8 parallel edges in each of the
coordinate directions have colours black, red, blue and green, respectively.
Let us turn to the symmetry group G for P . Each symmetry γ is determined by its
action on the vertices, which clearly can be permuted with sign changes in all possible
ways. Thus G has order 24 · 4! = 384, and we may think of it as being comprised of all
4× 4 signed permutation matrices.
In fact, G can be generated by reflections ρ0, ρ1, ρ2, ρ3 in hyperplanes. Here ρ0 negates
the first coordinate x1 (reflection in the coordinate hyperplane orthogonal to b1); and, for
B. Monson: On Roli’s cube 3
x
1
x
2
x
3
x
4
−v
v = (1,1,1,1)
w
Figure 2: The most symmetric 2-dimensional projection of the 4-cube.
1 ≤ j ≤ 3, ρj transposes coordinates xj , xj+1 (reflection in the hyperplane orthogonal to
bj − bj+1).
Note that the reflection in the j-th coordinate hyperplane is ρ
ρ1···ρj−1
0 for 1 ≤ j ≤ 4.
(We use the notation γη := η−1γη.) The product of these 4 special reflections, in any
order, is the central element ζ : t 7→ −t. It is easy to check as well that
ζ = (ρ0ρ1ρ2ρ3)
4 . (2.2)
The Petrie symmetry π = ρ0ρ1ρ2ρ3 therefore has period 8.
For purposes of calculation, we note that G ≃ C42 ⋊ S4 is a semidirect product. Under
this isomorphism, each γ ∈ G factors uniquely as γ = eµ, where µ ∈ Sn is a permutation
of {1, . . . , 4} (labelling the coordinates); and e is a sign change vector, as in (2.1). Note
that
eµ = (ε1, ε2, ε3, ε4)
µ = (ε(1)µ−1 , ε(2)µ−1 , ε(3)µ−1 , ε(4)µ−1) .
Now really γ is a signed permutation matrix. But it is convenient to abuse notation, keeping
in mind that each e corresponds to a diagonal matrix of signs and each µ to a permutation
matrix. Thus we might write
π = ρ0ρ1ρ2ρ3 = (−1, 1, 1, 1) · (4, 3, 2, 1) =


0 0 0 −1
1 0 0 0
0 1 0 0
0 0 1 0

 . (2.3)
Next we use the group G = ⟨ρ0, ρ1, ρ2, ρ3⟩ to remanufacture the cube. In this (ge-
ometric) version of Wythoff’s construction [7, §2.4] we choose a base vertex v fixed by
the subgroup G0 := ⟨ρ1, ρ2, ρ3⟩ (which permutes the coordinates in all ways). Thus,
v = c(1, 1, 1, 1) for some c ∈ R. To avoid a trivial construction we take c ̸= 0, so, up to
similarity, we may use c = 1. Then the orbit of v under G is just the set of 16 points in
4 Art Discrete Appl. Math. 5 (2022) #P3.10
(2.1); and their convex hull returns P to us. Since G0 is the full stabilizer of v in G, the
vertices correspond to right cosets G0γ.
The beauty of Wythoff’s construction is that all faces of P can be constructed in a
similar way by induction on dimension (see [4, 12] or [13, Section 1B]). For example, the
vertices v = (1, 1, 1, 1) and vρ0 = (−1, 1, 1, 1) of the base edge of P are just the orbit of
v under the subgroup G1 := ⟨ρ0, ρ2, ρ3⟩; and edges of P correspond to right cosets of the
new subgroup G1. Furthermore, a more careful look reveals that a vertex is incident with
an edge just when the corresponding cosets have non-trivial intersection.
Pursuing this, we see that the face lattice of P can be recontructed as a coset geometry
based on subgroups
G0, G1, G2, G3, where Gj := ⟨ρ0, . . . , ρj−1, ρj+1, . . . , ρ3⟩. (2.4)
From this point of view, P becomes an abstract regular 4-polytope, a partially ordered set
whose automorphism group is G. Notice that the distinguished subgroups in (2.4) provide
the proper faces in a flag in P , namely a mutually incident vertex, edge, square and 3-cube.
The crucial structural property of G is that it should be a string C-group with respect to
the generators ρj . A string C-group is a quotient of a Coxeter group with linear diagram
under which an ‘intersection condition’ on subgroups generated by subsets of generators,
such as those in (2.4), is preserved [13, Section 2E].
For the 4-cube P , G is actually isomorphic to the Coxeter group B4 with diagram
• 4 • 3 • 3 • (2.5)
Comparing the geometric and abstract points of view, we say that the convex 4-cube is a
realization of its face lattice (the abstract 4-cube).
When we think of a polytope from the abstract point of view, we often use the term rank
instead of ‘dimension’. An abstract polytope Q is said to be regular if its automorphism
group is transitive on flags (maximal chains in Q). Intuitively, regular polytopes have
maximal symmetry (by reflections). Next up are chiral polytopes, with exactly two flag
orbits and such that adjacent flags are always in different orbits (so maximal symmetry by
rotations, but without reflections).
We will soon encounter less familiar abstract regular or chiral polytopes, with their
realizations. For a first example, suppose that we map (by central projection) the faces of P
onto the 3-sphere S3 centred at the origin. We can then reinterpret P as a regular spherical
polytope (or tessellation), with the same symmetry group G. Now recall that the centre of
G is the subgroup ⟨ζ⟩ of order 2. The quotient group G/⟨ζ⟩ has order 192 and is still a
string C-group. The corresponding regular polytope is the hemi-4-cube H = {4, 3, 3}4,
here realized in projective space P3 [13, Section 6C]; see Figure 3. By (2.2), the product
of the four generators of G/⟨ζ⟩ has order 4; this is recorded as the subscript in the Schläfli
symbol for H.
Now we can outline the construction of Roli’s cube given in [3].
3 Colourful polyopes
The image in Figure 1 or on the left in Figure 2 can just as well be understood as a graph
G, namely the 1-skeleton of the 4-cube P . In fact, we can recreate the abstract (or combi-
natorial) structure of P from just the edge colouring of G: for 0 ≤ j ≤ 4, the j-faces of P
B. Monson: On Roli’s cube 5
can be identified with the components of those subgraphs obtained by keeping just edges
with some selection of the j colours (over all such choices). We therefore say that P is a
colourful polytope.
Such polytopes were introduced in [1]. In general, one begins with a finite, connected d-
valent graph G admitting a (proper) edge colouring, say by the symbols 1, . . . , d. Thus each
of the colours provides a 1-factor for G. The graph G determines an (abstract) colourful
polytope PG as follows. For 0 ≤ j ≤ d, a typical j-face (C, v) is identified with the set
of all vertices of G connected to a given vertex v by a path using only colours from some
subset C of size j taken from {1, . . . , d}. The j-face (C, v) is less than or equal the k-
face (D,w) just when C ⊆ D and w can be reached from v by a D-coloured path. (This
means that j ≤ k; and we can just as well replace w by v. The minimal face of rank
−1 in PG is formal.) Notice that the 1-skeleton of the d-polytope PG is just G itself. The
polytope is simple: each of its vertex-figures is isomorphic to a (d − 1)-simplex. From
[1, Theorem 4.1], the automorphism group of PG is isomorphic to the group of colour-
respecting graph automorphisms of G. (Such automorphisms are allowed to permute the
1-factors.)
It is easy to see that the hemi-4-cube H is also colourful. Its 1-skeleton is the complete
bipartite graph K4,4 found in Figure 3. We obtain this graph from Figure 1 or Figure 2 by
identifying antipodal pairs of points, like v and −v.
v
y w x
(a) (b) (c)
α
vvw
x
y w
x
y
Figure 3: The graph K4,4 in (a) is the 1-skeleton of the hemi-4-cube {4, 3, 3}4 in (b), (c).
If we lift H, as it is now, to S3, we regain the coloured 4-cube P . Now keep K4,4
embedded in P3, as in Figure 3. But, following [3], observe that K4,4 admits the auto-
morphism α which cyclically permutes, say, the first three vertices y, w, x in the top block,
leaving the rest fixed. Clearly, α is a non-colour-respecting automorphism of K4,4, so its
effect is to recolour 12 of the edges in the embedded graph. On the abstract level nothing
has changed for the resulting colourful polytope; it is still the hemi-4-cube H. But faces
of ranks 2 and 3 are now differently embedded in P3. For example, the red-blue 2-face on
v, which is planar in Figure 3(b), becomes a skew quadrangle in Figure 3(c) and also ac-
quires an orientation. According to Definition 4.1 below, these skew quadrangles are Petrie
polygons for the standard realization of H in Figure 3(b).
The newly coloured geometric object, which we might label HR, is a chiral realization
of the abstract regular polytope H. Comforted by the fact that P3 is orientable, we could
just as well apply α−1 to obtain the left-handed version HL. These two enantiomorphs are
6 Art Discrete Appl. Math. 5 (2022) #P3.10
oppositely embedded in P3, though both remain isomorphic to H as partially ordered sets.
If we lift either enantiomorph to S3, we obtain a chiral 4-polytope faithfully realized in E4
[3, Theorem 2]. This is Roli’s cube R.
Next we set the stage for a slightly different construction of R, without the use of P3.
4 Petrie polygons of the 4-cube
Let us consider the progress of the base vertex v = (1, 1, 1, 1) for P as we apply successive
powers of π in (2.3). We get a centrally symmetric 8-cycle of vertices
v → (1, 1, 1,−1) → (1, 1,−1,−1)
→ (1,−1,−1,−1) → −v = (−1,−1,−1,−1) → . . . .
Starting from v in Figure 2 we therefore proceed in coordinate directions 4, 3, 2, 1 (indi-
cated by different colours), then repeat again. This traces out the peripheral octagon C,
which in fact is a Petrie polygon for P .
Definition 4.1 (See [6, p. 223] or [13, p. 163]). A Petrie polygon of a 3-polytope is an
edge-path such that any 2 consecutive edges, but no 3, belong to a 2-face. We then say that
a Petrie polygon of a 4-polytope Q is an edge-path such that any 3 consecutive edges, but
no 4, belong to (a Petrie polygon of) a facet of Q.
For the cube P , the parenthetical condition is actually superfluous; compare [9].
Clearly, we can begin a Petrie polygon at any vertex, taking any of the 4! orderings of
the colours. But this counts each octagon in 16 ways. We conclude that P has 24 Petrie
polygons. What we really use here is the fact that G is transitive on vertices, and that at
any fixed vertex, G permutes the edges in all possible ways. We see that G acts transitively
on Petrie polygons.
But the (global) stabilizer of the polygon C (constructed above with the help of π and
v) is the dihedral group K of order 16 generated by µ0 = ρ0ρ2ρ3ρ2 = (−1, 1, 1, 1) · (2, 4),
and µ1 = µ0π = ρ2ρ3ρ2ρ1ρ2ρ3 = (1, 1, 1, 1) · (1, 4)(2, 3). (Such calculations are routine
using either signed permutation matrices or the decomposition in C42 ⋊S4. Note that any 4
consecutive vertices of C form a basis of E4.) We confirm that P has 24 = 384/16 Petrie
polygons.
Now we move to the rotation subgroup
G+ = ⟨ρ0ρ1, ρ1ρ2, ρ2ρ3⟩.
It has order 192 and consists of the signed permutation matrices of determinant +1. Note
that K < G+. Thus, under the action of G+, there are two orbits of Petrie polygons of 12
each. Let’s label these two chiral classes R and L for right- and left-handed, taking C in
class R.
The two chiral classes must be swapped by any non-rotation, such as any ρj . To distin-
guish them, we could take the determinant of the matrix whose rows are any 4 consecutive
vertices on a Petrie polygon. The two chiral classes R and L then have determinants +8,
respectively, −8. Or starting from a common vertex, the edge-colour sequence along a
polygon in one class is an odd permutation of the colour sequence for a polygon in the
other class.
The inner octagram C∗ in Figure 2 is another Petrie polygon. Start at the vertex
w = (v)ρ1ρ0ρ1 = (1,−1, 1, 1) which is adjacent to v along a red edge; then proceed
B. Monson: On Roli’s cube 7
in directions 4, 1, 2, 3 and repeat. However, the remaining Petrie polygons appear in less
symmetrical fashion in Figure 2.
Note that µ0 actually acts on the diagram in Figure 2 as a reflection in a vertical line,
whereas π rotates the octagon C through one edge-step and the octagram C∗ by three. On
the other hand, µ2 = ρ1ρ2ρ0ρ1 = (1,−1, 1, 1) · (1, 3) is an element of G+ which swaps C
and C∗. Thus there are 6 such unordered pairs like C, C∗ in class R and another 6 pairs in
class L.
Remark 4.2. It can be shown that Figure 2 is the most symmetric orthogonal projection of
P to a plane [6, Section 13.3]. Since all edges after projection have a common length, we
may say that this projection is isometric.
The Petrie symmetry π is one instance of a Coxeter element in the group G = B4,
namely the product of the four reflections attached to a simple system of roots. All such
Coxeter elements are conjugate [10, Sections 1.3 and 3.16]. Each of them has invariant
planes which give rise to the sort of orthogonal projection displayed in Figure 2. A pro-
cedure for finding these planes is detailed in [10, Section 3.17]. For π, the two planes are
spanned by the rows of
[
1√
2
1
2 0
−1
2
0 12
1√
2
1
2
]
and
[
1√
2
−1
2 0
1
2
0 12
−1√
2
1
2
]
.
These planes are orthogonal complements; and π acts on them by rotations through 45◦
and 135◦, respectively. Figure 2 results from projecting P onto the first plane. □
5 The map M and the Möbius-Kantor Configuration 83
Look again at the companion Petrie polygons C, C∗ in Figure 2. Now working around the
rim clockwise from v delete the edges coloured blue, red, black, green, and repeat. We are
left with the trivalent graph L displayed in Figure 4.
In fact, L is the generalized Petersen graph {8} + { 83}, studied in detail by Coxeter
in [5, Section 5]. The graph is transitive on 2-arcs but not on 3-arcs, so its automorphism
group has order 96 = 16 · 3 · 22−1 [2, Chapter 18]. We return to this group later.
We have labelled alternate vertices of L by the residues 0, 1, 2, 3, 4, 5, 6, 7 (mod 8).
These will represent the points in a Möbius-Kantor configuration 83. The remaining (un-
labelled) vertices of L represent the 8 lines in the configuration. Thus we have lines 013
(represented by the ‘north-west’ vertex (−1,−1, 1, 1)), 124, 235, and so on, including line
671 represented by v.
Notice that we can interpret the configuration as being comprised of two quadrangles
with vertices 0, 2, 4, 6 and 1, 3, 5, 7, each inscribed in the other: vertex 0 lies on edge 13,
vertex 1 lies on edge 24, and so on.
So far this configuration 83 is purely abstract. In fact, it can be realized as a point-line
configuration in a projective (or affine) plane over any field in which
z2 − z + 1 = 0
has a root, certainly over C. However, 83 cannot be realized in the real plane.
Coxeter made other observations in [5], including the fact, used above, that the graph
L is a sub-1-skeleton of the 4-cube. Altogether L contains 3 complementary pairs of Petrie
8 Art Discrete Appl. Math. 5 (2022) #P3.10
1
3
5
7
2
0
4
6
v
Figure 4: The Levi graph L = {8}+ { 83} for the configuration 83.
polygons, which we can briefly describe by their alternate vertices:
0246(= C∗) 0541 1256
1357(= C) 2367 0743
Hence, the configuration can be regarded as a pair of mutually inscribed quadrangles in
three ways.
Observe that each edge of L lies on exactly two of the 6 octagons. For example, the top
edge with vertices labelled 1 and v lies on octagons 1357 and 1256. (It does not matter that
two such octagons then share a second edge opposite the first.) Furthermore, each vertex
lies on the three octagons determined by choices of two edges. We can thereby construct a
3-polytope M of type {8, 3}, with 6 octagonal faces, whose 1-skeleton is L. In short, M
is realized by substructures of the 4-cube P .
Moving sideways, we can reinterpret M in a more familiar topological way as a map
on a compact orientable surface of genus 2. Recall that M is covered by the tessellation
{8, 3} of the hyperbolic plane, as indicated in Figure 5.
Now return to E4 where the combinatorial structure of M is handed to us as faithfully
realized. Drawing on [8, Section 8.1], we have that the rotation group Γ(M)+ for M is
generated by two special Euclidean symmetries:
σ1 = π = ρ0ρ1ρ2ρ3 = (−1, 1, 1, 1) · (4, 3, 2, 1) (preserving the base octagon C); and
σ2 = ρ3ρ2ρ1ρ3 = (1, 1, 1, 1) · (1, 2, 4) (preserving the base vertex v on C).
The order of Γ(M)+ must then be twice the number of edges in M, namely 48. Let us
assemble these and further observations in
Theorem 5.1. (a) The 3-polytope M is abstractly regular of type {8, 3}, here realized
in E4 in a geometrically chiral way.
(b) The rotation subgroup Γ(M)+ = ⟨σ1, σ2⟩ has order 48 and presentation
⟨σ1, σ2 |σ81 = σ32 = (σ1σ2)2 = (σ−31 σ2)2 = 1⟩. (5.1)
B. Monson: On Roli’s cube 9
1
3 5
7
6
3
2
52
4
5
0
7
4
6
0
v
Figure 5: Part of the tessellation {8, 3} of the hyperbolic plane.
(c) The full automorphism group Γ(M) has order 96 and presentation
⟨τ0, τ1, τ2 | τ20 = τ21 = τ22 = (τ0τ1)8 = (τ1τ2)3 = (τ0τ2)2 = ((τ1τ0)3τ1τ2)2 = 1⟩.
(5.2)
Proof. We begin with (b), where it is easy to check that the relations in (5.1) do hold for the
matrix group ⟨σ1, σ2⟩. By a straightforward coset enumeration [8, Chapter 2], we conclude
from the presentation in (5.1) that the subgroup ⟨σ1⟩ has the 6 coset representatives
1, σ2, σ
2
2 , σ2σ
−1
1 , σ
2
2σ1, σ2σ
−1
1 σ2.
(We abuse notation by passing freely between the matrix group and abstract group.) This
finishes (b).
We next note that ⟨σ1⟩∩⟨σ2⟩ = {1}, since σj1 fixes v only for j ≡ 0 (mod 8). Now we
are justified in invoking [16, Theorem 1(c)], whereby the 3-polytope M is regular (rather
than just chiral) if and only if the mapping σ1 7→ σ−11 , σ2 7→ σ21σ2 induces an involutory
automorphism τ of Γ(M)+. But the new relations induced by applying the mapping to
(5.1) are easily verified formally, or even by matrices. For instance, since σ1σ2 = σ
−1
2 σ
−1
1 ,
we have
(σ21σ2)
3 = (σ1σ
−1
2 σ
−1
1 )
3 = σ1σ
−3
2 σ
−1
1 = 1.
Thus M is abstractly regular and Γ(M) has order 96. The presentation in (5.2) follows
at once by extending Γ(M)+ by ⟨τ⟩, then letting τ0 := τ, τ1 := τσ1, τ0 := τσ1σ2. (In
Theorem 7.1 we use the same labels for the first three generators of the linear group T .)
10 Art Discrete Appl. Math. 5 (2022) #P3.10
It remains to check that our realization is geometrically chiral [3, Theorem 2]. This
means that τ is not represented by a symmetry of M as realized in E4. From the combi-
natorial structure, τ would have to swap vertices 1 and v while preserving the two Petrie
polygons on that edge. Thus τ would have to act just like µ0, that is, just like reflection in
a vertical line in Figure 2. But µ0 does not preserve the set of 8 edges deleted to give L in
Figure 4.
Remark 5.2. It is helpful to note that the centre of Γ(M)+ is generated by σ41 . Referring
to [8, Section 6.6], we find that Γ(M)+ is isomorphic to the group ⟨−3, 4|2⟩, which in turn
is an extension by C2 of the binary tetrahedral group ⟨3, 3, 2⟩. Indeed, a = σ−11 σ2σ−11 , b =
σ2σ
4
1 satisfy a
3 = b3 = (ab)2 (= ζ). Thus, ⟨3, 3, 2⟩ ◁ Γ(M)+. □
6 Roli’s cube – a chiral polytope R of type {8, 3, 3}
Under the action of G+ we expect to find 4 = 192/48 copies of M. To understand this
better, recall that there are 12 Petrie polygons in one chiral class, say R. As with C and C∗,
each polygon D is paired with a unique polygon D∗ (with the disjoint set of 8 vertices).
For each D there are then two ways to remove 8 edges (connecting D and D∗) so as to get
a copy of L and hence a copy of M. Since M has six 2-faces like C, we once more find
12 · 2/6 = 4 copies of M.
Each Petrie polygon lies on 2 copies of M, again from the two ways to remove 8 edges.
For example, C lies on both M and (M)µ0. (The same is true for C∗.)
The pointwise stabilizer in G+ of the base edge joining v = (1, 1, 1, 1) and (v)µ0 =
(−1, 1, 1, 1) must consist of pure, unsigned even permutations of {2, 3, 4}. Therefore it is
generated by
σ3 := ρ2ρ3 = (1, 1, 1, 1) · (2, 4, 3).
It is easy to check that G+ = ⟨σ1, σ2, σ3⟩.
Since three consecutive edges of a Petrie polygon lie on two adjacent square faces in a
cubical facet of P , it must be that every vertex of R has the same vertex-figure as P , thus
of tetrahedral type {3, 3}.
We have enumerated and (implicitly) assembled the faces of a 4-polytope R, faithfully
realized in E4 and symmetric under the action of G+. Let’s take stock of its proper faces:
rank stabilizer in G+ order number of faces type
0 ⟨σ2, σ3⟩ 12 16 vertex of cube P
1 ⟨σ1σ2, σ3⟩ 6 32 edge of P
2 ⟨σ1, σ2σ3⟩ 16 12 Petrie polygons of P in one class R
3 ⟨σ1, σ2⟩ 48 4 copy of M
It is not hard to see that our 4-polytope R is isomorphic to Roli’s cube, as constructed
in [3] and as described in Section 3.
B. Monson: On Roli’s cube 11
Theorem 6.1. (a) The 4-polytope R is abstractly chiral of type {8, 3, 3}. Its symmetry
group Γ(R) ≃ G+ has order 192 and the presentation
⟨σ1, σ2, σ3|σ81 = σ32 = σ33 = (σ1σ2)2 = (σ2σ3)2 = (σ1σ2σ3)2 = 1 (6.1)
(σ−31 σ2)
2 = 1 (6.2)
(σ−11 σ3)
4 = 1 ⟩. (6.3)
(b) R is faithfully realized as a geometrically chiral polytope in E4.
Proof. The relations in (6.1) are standard for chiral 4-polytopes [16, Theorem 1]; and we
have seen that the relation in (6.2) is a special feature of the facet M. Enumerating cosets
of the subgroup ⟨σ1, σ2⟩, which still has order 48, we find at most the 8 cosets represented
by
1, σ3, σ
2
3 , σ
2
3σ1, σ
2
3σ
2
1 , σ
2
3σ
2
1σ2, σ
2
3σ
2
1σ
2
2 , σ
2
3σ
2
1σ3.
Thus the group defined by (6.1) and (6.2) has order at most 384. But G+, where these
relations do hold, has order 192. We require an independent relation. In Section 7, we will
see why (6.3) is just what we need.
To show that R is abstractly chiral we must demonstrate that the mapping σ1 7→
σ−11 , σ2 7→ σ21σ2, σ3 7→ σ3 does not extend to an automorphism of G+. This is easy,
since
(σ1σ3)
4 = ζ whereas (σ−11 σ3)
4 = 1. (6.4)
Clearly, R is realized in a geometrically chiral way in E4; we have already seen this for
its facet M.
Our concrete geometrical arguments should suffice to convince the reader that we really
have described here a chiral 4-polytope identical to the original Roli’s cube. A skeptic can
nail home the proof by applying [16, Theorem 1] to the group G+, as generated above.
7 Realizing the minimal regular cover of R
The rotation group G+ for the cube has order 192 and ‘standard’ generators ρ0ρ1, ρ1ρ2, ρ2ρ3.
But for our purposes we use either of two alternate sets of generators. We already have
σ1 = (ρ0ρ1)(ρ2ρ3), σ2 = (ρ3ρ2)(ρ1ρ2)(ρ2ρ3), σ3 = ρ2ρ3. (7.1)
Now we also want
σ1 = σ
−1
1 , σ2 = σ
2
1σ2, σ3 = σ3. (7.2)
Recalling our shorthand for such matrices, we have
σ1 = (−1, 1, 1, 1) · (4, 3, 2, 1); σ2 = (1, 1, 1, 1) · (1, 2, 4); σ3 = (1, 1, 1, 1) · (2, 4, 3),
and
σ1 = (1, 1, 1,−1) · (1, 2, 3, 4); σ2 = (−1,−1, 1, 1)(1, 3, 2); σ3 = (1, 1, 1, 1) · (2, 4, 3).
12 Art Discrete Appl. Math. 5 (2022) #P3.10
We have seen that the group G+ = ⟨σ1, σ2, σ3⟩ (with these specified generators) is the
rotation (and full automorphism) group of the chiral polytope R of type {8, 3, 3}. From
[17, Section 3] we have that the (differently generated) group G+ = ⟨σ1, σ2, σ3⟩ is the au-
tomorphism group for the enantiomorphic chiral polytope R . By generating the common
group in these two ways we effectively exhibit right- and left-handed versions of the same
polytope.
Our geometrical realization of R began with the base vertex v = (1, 1, 1, 1) (which
also served as base vertex for the 4-cube P). It is crucial here that v does span the subspace
fixed by σ2 and σ3. By instead taking G+ with base vertex v = (−1, 1, 1, 1) fixed by σ2,
and σ3⟩, we have a faithful geometric realization of R, still in E4, of course.
We will soon have good reason to mix G+ and G+ in a geometric way. Each group acts
irreducibly on E4. Construct the block matrices, κj = (σj , σj), j = 1, 2, 3, now acting on
E
8 and preserving two orthogonal subspaces of dimension 4. Obviously we may extend
our notation for signed permutation matrices to the cubical group B8 acting on E
8. Thus,
taking the second copy of E4 to have basis b5, b6, b7, b8, we may combine our descriptions
of σj , σj to get
κ1 = (−1, 1, 1, 1, 1, 1, 1,−1) · (4, 3, 2, 1)(5, 6, 7, 8),
κ2 = (1, 1, 1, 1,−1,−1, 1, 1) · (1, 2, 4)(5, 7, 6),
κ3 = (1, 1, 1, 1, 1, 1, 1, 1) · (2, 4, 3)(6, 8, 7).
Now let T+ = ⟨κ1, κ2, κ3⟩. In slot-wise fashion, κ1, κ2, κ3 satisfy relations like those
in (6.1) and (6.2). From the proof of Theorem 6.1, we conclude that T+ has order 384. We
even get a presentation for it.
Recall that the centre of G+ is generated by ζ = σ41 = σ
4
1. Thus the centre of T
+ has
order 4, with non-trivial elements
(ζ, 1) = (κ1κ3)
4, (1, ζ) = (κ−11 κ3)
4, and (ζ, ζ) = κ41. (7.3)
(This is at the heart of the proof that R is abstractly chiral.) Looking at (6.4), we see that
T+/⟨(1, ζ)⟩ ≃ Γ(R),
and thus see the reason for the special relation in (6.3). Similarly, T+/⟨(ζ, 1)⟩ ≃ Γ(R).
Finally, we have
T+/⟨(ζ, ζ)⟩ ≃ Γ(P)+, (7.4)
the rotation group of the 4-cube (isomorphic to G+ generated in the customary way).
Now T+ is clearly isomorphic to the mix G+♢G+ described in [14, Theorem 7.2].
Guided by that result, we seek an isometry τ0 of E
8 which swaps the two orthogonal sub-
spaces, while conjugating each (σj , σj) to (σj , σj). It is easy to check that
τ0 = (1, 1, 1, 1, 1, 1, 1, 1) · (1, 5)(2, 6)(3, 7)(4, 8)
does the job. We find that T+ is the rotation subgroup of a string C-group T = ⟨τ0, τ1, τ2, τ3⟩,
where τ1 = τ0κ1, τ2 = τ0κ1κ2, τ3 = τ0κ1κ2κ3. The corresponding directly regular 4-
polytope has type {8, 3, 3} and must be the minimal regular cover of each of the chiral
polytopes R and R.
For a little background here, we note that T is an instance of the group H appearing in
the proof of [14, Theorem 7.2]. There it was shown that H is isomorphic to the ‘connection
B. Monson: On Roli’s cube 13
group’ Mon(R) for the chiral polytope R. From [14, Proposition 3.16], this is just what
we need to discuss the regular covers of R. We consolidate all this in
Theorem 7.1. (a) The group T = ⟨τ0, τ1, τ2, τ3⟩ is a string C-group of order 768 and
with the presentation
⟨τ0, τ1, τ2, τ3 | τ2j = (τ0τ1)8 = (τ1τ2)3 = (τ3τ3)3 = 1, 0 ≤ j ≤ 3,
(τ0τ2)
2 = (τ0τ3)
2 = (τ1τ3)
2 = ((τ1τ0)
3τ1τ2)
2 = 1⟩.
(b) The corresponding regular 4-polytope T has type {8, 3, 3} and is faithfully realized
in E8, with base vertex (v, v) = (1, 1, 1, 1,−1, 1, 1, 1). The polytope T is the min-
imal regular cover for Roli’s cube R and its enantiomorph R. It is also a double
cover of the 4-cube P .
(c) T ≃ {M , {3, 3}} is the universal regular polytope with facets M and tetrahedal
vertex-figures.
Proof. The centre of T is generated by (ζ, ζ). It is easy to check that T/⟨(ζ, ζ)⟩ ≃ G,
the full symmetry group of the cube; compare (7.4). In other words, the mapping τj 7→
ρj , 0 ≤ j ≤ 3, induces an epimorphism φ : T → G. Since τ1, τ2, τ3 and ρ1, ρ2, ρ3 both
satisfy the defining relations for Γ({3, 3}) ≃ S4, φ is one-to-one on ⟨τ1, τ2, τ3⟩. By the
quotient criterion in [13, 2E17], T really is a string C-group. The remaining details are
routine. For background on (c) we refer to [13, 4A].
Much as in the proof, the assignment κj 7→ σj , (j = 1, 2, 3), induces an epimorphism
φR : T
+ → G+. On the abstract level, this in turn induces a covering φ̃R : T → R, in
other words, a rank- and adjacency-preserving surjection of polytopes as partially ordered
sets. The corresponding covering of geometric polytopes is induced by the projection
E
8 → E4
(x, y) 7→ x
The projection (x, y) 7→ y likewise induces the geometrical covering φ̃L : T → R. Both
φ̃R and φ̃L are 3-coverings, meaning here that each acts isomorphically on facets M and
vertex-figures {3, 3} [13, page 43]. Notice that each face of R or R has two preimages in
T .
The polytope T is also a double cover of the 4-cube P . But there is no natural way to
embed P in E8 to illustrate the geometric covering, since κ41 = −1 on any subspace of E8,
whereas (ρ0ρ1)
4 = 1 for P .
8 The Möbius-Kantor configuration again, and the 24-cell
We noted earlier that 83 can be ‘realized’ as a point-line configuration in C
2. We will
show this here by first endowing E4 with a complex structure. Thus, we want a suitable
orthogonal transformation J on E4 such that J2 = ζ. Keeping the addition, we then define
(a+ ıb)u = au+ b(uJ), for a, b ∈ R, u ∈ E4.
14 Art Discrete Appl. Math. 5 (2022) #P3.10
Thus ıu = uJ ; and over C, E4 has dimension 2. Our choice for the matrix J is motivated
by an orthogonal projection different from that in Figures 2 and 4.
The vectors representing the vertices labelled 0, . . . , 7 in Figure 4 are either opposite or
perpendicular. Thus, these eight points are the vertices of a cross-polytope O = {3, 3, 4},
one of two inscribed in P . (Its vertices are those of P which have an odd number of entries
+1.)
04
2
6
1
3 5
7
Figure 6: Another projection of the cross-polytope O.
In [7, Figure 4.2A], Coxeter gives another projection of O which nicely displays certain
2-faces of O. We show this in Figure 6, where each vertex of either of the two concentric
squares forms an equilateral triangle with one edge of the other square. These 8 triangles
correspond to the unlabelled nodes in Figure 4, and also to the lines of the configuration
83. (Each of these real triangles lies on a unique complex line in C
2.) We may take the
vertices in Figure 6 to be (±1,±1) and (±r, 0), (0,±r), where r =
√
3− 1.
But what plane Λ in E4 actually gives such a projection? Starting with an unknown
basis a1, b1 for Λ, we can force a lot. For example, the vector 02 from vertex 0 to vertex
2 in Figure 6 is the projection of (0, 2,−2, 0) and is obtained from 07, the projection of
(2, 2, 0, 0), by a rotation through 60◦. From such details in the geometry, we soon find that
Λ is uniquely determined and get a basis satisfying a1 · a1 = b1 · b1 and a1 · b1 = 0. But
any such basis can still be rescaled or rotated within Λ. Tweaking these finer details, we
find it convenient to take a1, b1 to be the first two rows of the matrix
L =
1
2
√
3


√
3 −1 1 −(2 +
√
3)
−1 2 +
√
3
√
3 −1
−1 −
√
3 2 +
√
3 1
2 +
√
3 1 1
√
3

 . (8.1)
The last two rows a2, b2 of L give a basis for the orthogonal complement Λ
⊥.
B. Monson: On Roli’s cube 15
Since we want J to induce 90◦ rotations in both Λ and Λ⊥, we have
J =
1√
3


0 1 −1 −1
−1 0 −1 1
1 1 0 1
1 −1 −1 0

 . (8.2)
Notice that a1J = b1 and a2J = b2, so {a1, a2} is a C-basis for E4; and the plane Λ in
Figure 6 is just z2 = 0 in the resulting complex coordinates. The points in the configuration
83 now have these complex coordinates:
Label (z1, z2)
0 (r, 1− ı)
1 (−1 + ı, r)
2 (rı,−1− ı)
3 (−1− ı,−rı)
4 (−r,−1 + ı)
5 (1− ı,−r)
6 (−rı, 1 + ı)
7 (1 + ı, rı)
(Recall that r =
√
3−1.) The first coordinates do give the points displayed in Figure 6.
The second coordinates describe the projection onto Λ⊥ (z1 = 0). There labels on the inner
and outer squares are suitably swapped.
The line in the configuration 83 which contains points 1, 6, 7 is typical. It has the
equation
r(1− ı)z1 + 2z2 = 2r(1 + ı) .
After consulting [7, Sections 10.6 and 11.2], we observe that the eight points are also
the vertices of the regular complex polygon 3{3}3. Its symmetry group (of unitary trans-
formations on C2) is the group 3[3]3 with the presentation
⟨γ1, γ2 | γ31 = 1, γ1γ2γ1 = γ2γ1γ2⟩ . (8.3)
In fact, this group of order 24 is isomorphic to the binary tetrahedral group ⟨3, 3, 2⟩. But
in our context, we may identify it with the centralizer in G of the structure matrix J . A bit
of computation shows that this subgroup of G is generated by
γ1 = ρ1ρ2ρ3ρ2 = (1, 4, 2) and γ2 = ρ2ρ0ρ1ρ0 = (−1, 1,−1, 1) · (1, 2, 3),
which do satisfy the relations in (8.3).
Remark 8.1. The orthogonal projection which underlies Figure 6 has another explanation
[7, Section 4.2]. The group G (≃ B4) can be embedded with index 3 in an isometry group
H = ⟨β0, β1, β2, β3⟩ which is isomorphic to the Coxeter group F4 with diagram
• 3 • 4 • 3 • . (8.4)
16 Art Discrete Appl. Math. 5 (2022) #P3.10
To do this we may take β1 to be reflection in the hyperplane orthogonal to (1,−1,−1, 1),
then set β0 := ρ0, β2 := ρ3, β3 := ρ2. Indeed, H has order 1152 and ρ1 = β
β1
2 . Notice
that the base vertex v is also fixed by β1, β2, β3. Applying Wythoff’s construction anew, we
obtain the 24-cell Q = {3, 4, 3}, whose symmetry group is just H ≃ F4. The 24 vertices of
this self-dual polytope include the original 16 vertices of the 4-cube P , along with another
8, namely all permutations of (±2, 0, 0, 0). We see that Q has 3 inscribed cross-polytopes,
one of which is the polytope O whose projection appears in Figure 6. (For more on such
regular compounds, see [6].)
As we did for the cube, we can use the procedure in [10, Section 3.17] to find invariant
planes for Coxeter elements in H . After considerable tinkering, we find that Λ,Λ⊥ are the
invariant planes for the Coxeter element
π̃ = (β0β1β2β3)
β2β1β0β1β2β3
in H . We thereby obtain an instance of the most symmetrical projection of the 24-cell Q
onto a plane. In Figure 7, we have altered Figure 6 by showing as well the images of the
remaining 16 vertices of Q. (The edges of O are not edges of Q, so have been removed.)
The vertices of Q appear in two concentric rings of 12. Indeed, Coxeter elements in H ,
such as π̃, have period 12. (The peculiar choice of π̃ is an artefact of our pleasant way of
first labelling the vertices of the Möbius-Kantor configuration.) □
1
3 5
7
0
2
4
6
Figure 7: The dodecagonal projection of Q ={3,4,3}, with vertices of O labelled.
ORCID iDs
Barry Monson https://orcid.org/0000-0001-9901-9681
B. Monson: On Roli’s cube 17
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[2] N. Biggs, Algebraic Graph Theory, Cambridge Mathematical Library, Cambridge University
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[4] H. S. M. Coxeter, Wythoff’s Construction for Uniform Polytopes, Proc. London Math. Soc. (2)
38 (1935), 327–339, doi:10.1112/plms/s2-38.1.327.
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413–455, doi:10.1090/S0002-9904-1950-09407-5.
[6] H. S. M. Coxeter, Regular Polytopes, Dover Publications, Inc., New York, 3rd edition, 1973.
[7] H. S. M. Coxeter, Regular Complex Polytopes, Cambridge University Press, Cambridge, 2nd
edition, 1991.
[8] H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups,
Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14, Springer-Verlag, New York-
Heidelberg, 3rd edition, 1972, doi:10.1007/978-3-662-21946-1.
[9] H. S. M. Coxeter and A. I. Weiss, Twisted honeycombs {3, 5, 3}t and their groups, Geom.
Dedicata 17 (1984), 169–179, doi:10.1007/BF00151504.
[10] J. E. Humphreys, Reflection Groups and Coxeter Groups, volume 29 of Cambridge Stud-
ies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990, doi:10.1017/
CBO9780511623646.
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(2021), 651––665, doi:10.1007/s13366-020-00545-0.
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.11
https://doi.org/10.26493/2590-9770.1406.ec5
(Also available at http://adam-journal.eu)
The interlacing number for alternating
semiregular polytopes
Barry Monson
University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3
Egon Schulte*
Northeastern University, Boston, Massachussetts, USA, 02115
Received 7 March 2021, accepted 20 September 2021, published online 5 August 2022
Abstract
In the classical setting, a convex polytope is said to be semiregular if its facets are
regular and its symmetry group is transitive on vertices. This paper continues our study of
‘alternating’ semiregular abstract polytopes. These structures have abstract regular facets,
still with combinatorial automorphism group transitive on vertices and with two kinds of
regular facets occurring in an alternating fashion. Here we investigate a parameter called
the interlacing number k for a compatible pair of regular n-polytopes P and Q: in what
circumstances can we have k copies each of P and Q occuring in alternating fashion as the
facets of a semiregular (n+ 1)-polytope S?
Keywords: Abstract semiregular polytopes, tail-triangle C-groups.
Math. Subj. Class.: 51M20, 52B15
1 Introduction
The cuboctahedron is an example of what we call an alternating semiregular 3-polytope:
its facets are squares and equilateral triangles, two of each occuring in alternate fashion
around each vertex. Taking these words as instructions for assembling a convex model,
then up to similarity there can be only one end result S (the cuboctahedron1). This sort of
alternating behaviour also appears in the familiar tiling T of Euclidean 3-space by regular
*Supported by the Simons Foundation Award No. 420718
E-mail addresses: bmonson@unb.ca (Barry Monson), e.schulte@northeastern.edu (Egon Schulte)
1Polytopes like this are often called quasi-regular in the literature. See [1, pages 18 and 69], for example,
where the cuboctahedron is denoted by {3
4
}. We generalize such notation later. For much more on geometric
realizations of quasi-regular polytopes see [10].
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.11
octahedra and tetrahedra. Indeed, T is an infinite alternating semiregular 4-polytope. Our
main concern in this paper will be abstract semiregular polytopes like this, with two kinds
of regular facets P and Q occurring in an ‘alternating’ fashion.
We began our study of such polytopes in [14]. Here our task is to investigate a parameter
called the interlacing number k for a compatible pair of regular n-polytopes P and Q: in
what circumstances can we have k copies each of P and Q occuring in alternating fashion
as the facets of a semiregular (n+ 1)-polytope S?
In fact, when k = ∞ such a polytope S actually does exist. In particular, there is a
universal alternating semiregular polytope UP,Q with infinite interlacing number. We have
described its properties with considerable care in [16].
For finite k the situation is less settled. Sometimes a polytope S with the required
structure does exist. In this case, as we show in Theorem 3.3, there exists a universal
polytope UkP,Q of this kind. Its universal covering property is described in Theorem 4.9.
(These results help explain, for example, why the cuboctahedron, as constructed above
from such simple instructions, must be as symmetric as it is!)
But in general there may be obstructions to assembling P and Q in the desired way [15].
In Section 5 we survey some interesting examples. Then in Sections 6 and 7 we describe
some families of compatible polytopes P and Q, together with interlacing number k, such
that UkP,Q does exist.
It is our pleasure to dedicate this paper to Marston Conder. We thank him, first of all,
for his insights into the world of graphs, maps, polytopes and their groups; and second, for
his great good humour!
In addition, we here thank the referee for various helpful suggestions.
2 Abstract polytopes
An abstract n-polytope P is a partially ordered set rather like the face lattice of a convex
n-polytope. In general, however, P need not be a lattice or finite or at all familiar. See [11,
Chapter 2] for a detailed look at these objects or [16, Section 2] for a briefer discussion.
The elements F of P are called its faces.
First we recall that P has a (unique) minimal face F−1 and maximal face Fn such that
F−1 ⩽ F ⩽ Fn, for all F ∈ P . Every maximal chain, or flag, in P has exactly n (proper)
faces in addition to the improper faces F−1, Fn. We let F(P) be the set of all flags Φ in P .
It follows that P has a rank function rk, where rk(F )+1 is the number of faces strictly
below F in any flag containing F . An element F ∈ P with rk(F ) = j is called a j-face;
and faces of rank 0, 1, n−2 or n−1 are called vertices, edges, ridges or facets, respectively.
If F ⩽ G are incident faces in P , the section G/F is
G/F := {H ∈ P |F ⩽ H ⩽ G}.
We next recall the diamond property, that whenever F < G with rk(F ) = j − 1 and
rk(G) = j+1, there must be exactly two j-faces H with F < H < G. For 0 ⩽ j ⩽ n− 1
and any flag Φ ∈ F(P), there thus exists a unique j-adjacent flag Φj , differing from Φ in
just the face of rank j . With this notion of adjacency, F(P) becomes the flag graph for P .
Finally we assume that P is strongly flag–connected, that is, the flag graph for each section
(including P itself) is connected.
Suppose F ⩽ G are incident faces in P , with rk(F ) = j ⩽ k = rk(G). It is easy to
prove that the section G/F is itself a (k − j − 1)-polytope. It is often useful to treat a face
B. Monson and E. Schulte: The interlacing number for alternating semiregular polytopes 3
F as if it were the full section F/F−1 below it. Likewise, if F is a vertex of P , then the
section Fn/F is called the vertex-figure at F . More generally, if F is a j-face, then F is
said to have co-rank n− j − 1, that being the rank of its co-face Fn/F .
The automorphism group Γ (P) of an n-polytope P consists of all order-preserving
bijections on P . We say P is regular if Γ (P) is transitive on the flag set F(P). In this
case we may choose any one flag Φ ∈ F(P) as base flag, then define ρj to be the (unique)
automorphism mapping Φ to Φj , for each j in the index set N := {0, . . . , n − 1}. From
[11, Section 2B] we recall that Γ (P) is then a string C-group. This means first that Γ (P) is
generated by {ρj : j ∈ N}. Second, these involutory generators satisfy the commutativity
relations typical of a Coxeter group with string diagram, namely
(ρjρk)
pjk = 1, for 0 ⩽ j ⩽ k ⩽ n− 1, (2.1)
where pjj = 1 and pjk = 2 whenever |j − k| > 1. Finally, Γ (P) is a C-group, meaning
that it satisfies the intersection condition
⟨ρk : k ∈ I⟩ ∩ ⟨ρk : k ∈ J⟩ = ⟨ρk : k ∈ I ∩ J⟩, for any I, J ⊆ N. (2.2)
The fact that one can reconstruct a regular polytope in a canonical way from any string
C-group Γ is at the heart of the theory [11, Section 2E].
The periods pj := pj−1,j in (2.1) satisfy 2 ⩽ pj ⩽ ∞ and are assembled into the
Schläfli symbol {p1, . . . , pn−1} for the regular polytope P . As a familiar example, we recall
that every 2-polytope or polygon {p} is automatically abstractly regular; its automorphism
group is the dihedral group Dp of order 2p.
There are many ways to relax symmetry and thereby broaden the class of groups Γ (P)
[9, page 77]. In [14], for instance, we considered the kind of polytopes described in
Definition 2.1. An abstract polytope S is semiregular if it has regular facets and its auto-
morphism group Γ (S) is transitive on vertices. The semiregular polytope S is alternating
if its facets (all regular) are of two kinds P and Q, which further occur in alternating fashion
around any face in S of co-rank 2. (We allow P ≃ Q.)
Clearly every regular polytope is semiregular. The classical Archimedean solids, prisms
and antiprisms are convex semiregular polyhedra.
Remark 2.2. Suppose S is any polytope with only regular facets. Then all ridges in S (i.e.
facets of facets of S) are isomorphic to a common regular polytope K [13, Lemma 5.16].
We shall say that regular polytopes are compatible if they have the same facets K up to
isomorphism.
One can weaken a little the defining property for a semiregular polytope S to be alter-
nating. Fix one face F of co-rank 2. For some k ⩾ 1, F is surrounded by k copies of P
alternating with k copies of Q, where P,Q are compatible regular polytopes. The section
S/F is then a 2k-gon.
Now, in a polytope S of rank n+ 1, the (n− 2, n− 1)-incidence graph (whose nodes
are the faces of S of ranks n − 2 or n − 1) is connected. From this fact it follows easily
that copies of the same P and Q must occur alternately at each other face of co-rank 2 in
S . However, k can vary from one such face to another: □
Example 2.3. Construct a 3-polytope M by gluing two regular octahedra face-to-face,
removing the triangular barrier between. M has vertices of valency 4 or 6. Now construct
4 Art Discrete Appl. Math. 5 (2022) #P3.11
the 4-polytope S = 2M, as described in [18, Section 3]. Then S is vertex-transitive, with
each vertex-figure isomorphic to M. Moreover, all facets of S are isomorphic to 2{3}, that
is, to the cube, which of course is regular. Therefore, S is a semiregular 4-polytope, but with
4 facets surrounding some edges and 6 facets around certain others. Topologically, this S
is a “cubical” 3-manifold with vertex-links determined by M, and it admits an embedding
into the 3-skeleton of the 9-dimensional cube.
Evidently, we could here replace M by any simplicial convex 3-polytope or indeed by
any polytopal map with triangular facets. □
As in [14], we confine our attention to those semiregular polytopes for which the param-
eter k is constant over all faces F of co-rank 2 in S . In such cases we call k the interlacing
number for S . This certainly happens when S has rank 3 (by vertex transitivity) and also,
for instance, if S is hereditary [12]. In a hereditary polytope, every automorphism of every
facet extends to an automorphism of the whole polytope.
First of all, let us dispose of the somewhat degenerate class of examples which have
k = 1. Recall that a polytope is said to be (combinatorially) flat if each of its vertices is
incident with each of its facets [11, Section 4E]. For example, any (not necessarily regular)
n-polytope P has a trivial extension {P, 2} [19, page 377]. This flat (n+ 1)-polytope has
just two facets, each a copy of P .
Lemma 2.4. Suppose P is a regular n-polytope. Then S = {P, 2} is an alternating
semiregular (n+ 1)-polytope with interlacing number k = 1. In fact, S is regular.
Conversely, if each face of co-rank 2 in the polytope S is incident with just two facets
of S , then S has in total exactly two facets, which are furthermore isomorphic. In other
words, S must be a trivial extension.
Proof. The straightforward argument for the converse has nothing to do with the automor-
phism group Γ (S).
We are now justified in assuming from here on that k ⩾ 2. In fact, we will focus even
more narrowly on semiregular polytopes which can be constructed using the combinatorial
version of Wythoff’s construction described in [14, Section 4] and outlined below. Suppose
then that Γ = ⟨α0, . . . , αn−2, αn−1, βn−1⟩ is a group generated by involutions which
satisfy the commutativity relations implicit in the tail-triangle diagram
t t
t
t
t t❤ ✑
✑
✑
✑
◗
◗
◗
◗
αn−3
∗
αn−2
∗
∗
α0
∗
α1
. . . . . .
βn−1
αn−1
k (2.3)
The label ‘k’ indicates that αn−1βn−1 has period k, for some k = 2, . . . ,∞. However,
all other periods of products of two ‘adjacent’ generators are unspecified for the moment
(and indicated by a ∗). The label ‘2’ is possible and indicates the actual absence of the
corresponding branch in the diagram. The group Γ is called a tail-triangle group. (Antici-
pating Theorem 2.5, we also say that the group Γ has interlacing number k.) We allow the
degenerate (base) case n = 1, in which Γ = ⟨α0, β0⟩ is just the dihedral group Dk.
Suppose also that Γ is a C-group, now satisfying the intersection condition (2.2) over
a suitable index set N for the n+ 1 generators of Γ . Then Γ is a tail-triangle C-group. It
B. Monson and E. Schulte: The interlacing number for alternating semiregular polytopes 5
follows that the subgroups
ΓPn := ⟨α0, . . . , αn−2, αn−1⟩
and
ΓQn := ⟨α0, . . . , αn−2, βn−1⟩
are string C-groups, indeed automorphism groups for regular n-polytopes P and Q, respec-
tively.
As described in Definition 4.2 of [14], the ringed node in (2.3) initiates Wythoff’s con-
struction for an (n+ 1)-polytope S = S(Γ ) as a coset geometry over Γ . For later use, we
record some details here. First of all, for 0 ⩽ j ⩽ n− 2, we let
Γ−j := ⟨α0, . . . , αj−1⟩, Γ
+
j := ⟨αj+1, . . . , αn−1, βn−1⟩ (2.4)
and
Γj := ⟨α0, . . . , αj−1, αj+1, . . . , αn−1, βn−1⟩ = Γ
−
j × Γ
+
j . (2.5)
The direct product follows easily from (2.2) and the structure of the diagram in (2.3). When
j = n− 1, we also interpret (2.4) and (2.5) as giving Γn−1 = Γ
−
n−1, with Γ
+
n−1 = {1}.
The j-faces of S with j ⩽ n − 1 can be identified with all right cosets of Γj in Γ .
Likewise, the n-faces of S are all right cosets of either ΓPn or Γ
Q
n . For the improper faces
of S we may take two distinct copies of Γ . Faces of distinct rank are now incident if and
only if the corresponding cosets have non-empty intersection. We find that the polytope S
has two flag orbits under the action of Γ , with base flags
Φ := [Γ0, . . . , Γn−2, Γn−1, Γ
P
n ] and Ψ := [Γ0, . . . , Γn−2, Γn−1, Γ
Q
n ]. (2.6)
(As usual, we have suppressed the two improper faces.)
We combine the key results of [14, Section 4] about the structure of S into the following
Theorem 2.5. Suppose Γ is a tail-triangle C-group corresponding to the diagram (2.3)
and let S = S(Γ ) be the resulting (n+ 1)-polytope. Then
(a) S is an alternating semiregular polytope. Its facets are isomorphic to P or Q, with
k of each of these occurring alternately around each face R of co-rank 2. Each
2-section S/R is therefore a 2k-gon; and S has interlacing number k.
(b) The face-wise Γ -stabilizer of any ridge of S is trivial.
(c) Each vertex-figure of S is isomorphic to the alternating semiregular n-polytope Ŝ
defined by deleting the node labelled α0 in the diagram (2.3), then ringing the node
labelled α1.
(d) S is a regular polytope if and only if Γ admits a group automorphism induced by the
diagram symmetry which swaps αn−1 and βn−1 in (2.3), while fixing the remaining
αj’s. In this case P ≃ Q, say with Schläfli type {p1, . . . , pn−1}, and S is regular of
type {p1, . . . , pn−1, 2k}; moreover, Γ (S) ≃ Γ ⋊ C2.
(e) If S is not regular, then S is a 2-orbit polytope and Γ (S) ≃ Γ . In particular, this is
so if the facets P and Q are non-isomorphic.
6 Art Discrete Appl. Math. 5 (2022) #P3.11
Recall that an abstract polytope is called a 2-orbit polytope if its automorphism group
has precisely two flag orbits. (See [5]; for general structural results about the groups of
2-orbit polytopes, see also [6].)
Typically when working with C-groups it is the verification of the intersection condition
(2.2) which causes pain. For relief, we often can make use of an inductive approach, using,
for instance, [14, Lemma 4.12]. Here is a refinement of that result.
Lemma 2.6. Suppose that Γ = ⟨α0, . . . , αn−2, αn−1, βn−1⟩ is a tail-triangle group with
n ⩾ 2, and suppose that its subgroups Γ 1n := Γ
P
n = ⟨α0, . . . , αn−2, αn−1⟩, Γ
2
n := Γ
Q
n =
⟨α0, . . . , αn−2, βn−1⟩ and Γ0 := ⟨α1, . . . , αn−2, αn−1, βn−1⟩ are C-groups. Then Γ is a
tail-triangle C-group if and only if
(a) Γ 1n ∩ Γ
2
n = ⟨α0, . . . , αn−2⟩; and
(b) Γ0 ∩ Γ
1
n = ⟨α1, . . . , αn−1⟩ and Γ0 ∩ Γ
2
n = ⟨α1, . . . , αn−2, βn−1⟩.
Proof. In our original formulation of this in [14, Lemma 4.12], we required a stronger
hypothesis in place of (b), namely
(b+) Γ+j ∩ Γ
1
n = ⟨αj+1, . . . , αn−1⟩ and Γ
+
j ∩ Γ
2
n = ⟨αj+1, . . . , αn−2, βn−1⟩,
for 0 ⩽ j ⩽ n− 2.
Recall from (2.4) that Γ+j := ⟨αj+1, . . . , αn−2, αn−1, βn−1⟩. Notice that Γ0 = Γ
+
0 , so
that now just j = 0 appears in condition (b). To see that this suffices, suppose in (b+) that
1 ⩽ j ⩽ n− 2. Then Γ+j ⊆ Γ0, so that
Γ+j ∩ Γ
1
n = (Γ
+
j ∩ Γ0) ∩ Γ
1
n
= Γ+j ∩ (Γ0 ∩ Γ
1
n)
= Γ+j ∩ ⟨α1, . . . , αn−1⟩ (by (b))
= ⟨αj+1, . . . , αn−1⟩.
The last equality follows from the assumption that Γ0 is a C-group. The calculation for
Γ+j ∩ Γ
2
n is similar.
We now have a tool for manufacturing lots of examples. For example, we could at least
take Γ to be a Coxeter group with a diagram like that displayed in (2.3). (In general, Γ
might satisfy independent relations not implied by the structure of the diagram.) Anyway,
by [7, Theorem 5.5], any Coxeter group satisfies the intersection condition (2.2), so if Γ is
a tail-triangle Coxeter group, then it is a tail-triangle C-group. However, the corresponding
regular n-polytopes P and Q now have a special structure. They are universal for their
Schläfli type [11, Section 3D]. In finite cases, such polytopes are isomorphic to classical
convex (that is, spherical) regular polytopes. What happens, however, if the building blocks
P and Q are selected from the many other kinds of abstract regular polytopes? We are led
to a fundamental
B. Monson and E. Schulte: The interlacing number for alternating semiregular polytopes 7
Assembly Problem: suppose P and Q are regular n-polytopes with isomorphic facets K;
and fix k ⩾ 2. Does there exist an alternating semiregular (n + 1)-polytope S with facets
P and Q, and interlacing number k (as described in Theorem 2.5)?
We will use ⟨PQ⟩
k to denote the class of all alternating semiregular (n + 1)-polytopes
with the above data, as constructed from a tail-triangle C-group like that in (2.3). We will
recall in Section 5 some situations where this class is empty – somehow there is an obstruc-
tion to our alternating assembly of facets. The basic structural problem is summarized in
the following diagram, in terms of the relevant automorphism groups:
Γ (K)
 _

  // Γ (Q)

Γ (P) // Γ ?
The compatibility of P and Q means that natural maps on generators induce the indicated
inclusions of Γ (K) into Γ (P) and Γ (Q). We then ask whether there exists a tail-triangle
C-group Γ making a commutative diagram in which the dotted maps are also injective.
In other words, the class ⟨PQ⟩
k is non-empty precisely when there is a solution Γ to the
diagram. We will prove in such cases (Section 3), that the class then contains a univer-
sal polytope, which we might denote {PQ}
k (in conformity with certain classical notations
[1, p. 18]). But for consistency with our earlier papers, we will instead write UkP,Q for this
universal polytope.
Remark 2.7. Chapter 4 of [11] concerns the analogous amalgamation problem for regular
polytopes: given regular n-polytopes R, T , does there exist a regular (n+1)-polytope with
facets isomorphic to R and vertex-figures isomorphic to T ? For this to be so, the vertex-
figures of R must be isomorphic to facets of T ; however, even given this compatibility
condition, the answer can still be ‘no’. If the class ⟨R, T ⟩ of such regular (n+1)-polytopes
is in fact non-empty, then there must be a ‘universal polytope’, denoted {R, T }, in the class.
The trivial extensions mentioned in Lemma 2.4 can be reinterpreted as universal polytopes.
3 The universal semiregular polytope Uk
P,Q
In the extreme case k = ∞, the interlacing number is never an obstruction to alternating
copies of compatible regular polytopes P and Q. In [16] we cover this situation in depth.
On an intuitive level, we explain there how to construct an alternating semiregular (n+1)-
polytope UP,Q in a ‘free’ manner: start with a copy of P; ‘attach’ copies of Q to each
facet of the P; then attach new copies of P to each ‘exposed’ facet of a Q, and so on [16,
Theorem 4.10]. To make sense of this we exploit a group Υ which we next describe more
carefully.
As always, then, P and Q will be regular n-polytopes with isomorphic facets K. Let
Υ be the free product of their automorphism groups Γ (P) and Γ (Q) amalgamated along
Γ (K). That is, identify corresponding standard generators for the facet subgroups (copies
of Γ (K)) in each of Γ (P) and Γ (Q). The result [14, Theorem 5.5] is that Υ is a tail-triangle
C-group with a diagram like this:
8 Art Discrete Appl. Math. 5 (2022) #P3.11
[h!] t t
t
t
t t❤ ∗ ✑
✑
✑
✑
◗
◗
◗
◗
ρn−3 ρn−2
∗
∗
ρ0
∗
ρ1
. . . . . .
τn−1
ρn−1
∞
P
Q
(3.1)
(For later clarity we use new labels ρ0, . . . , ρn−2, ρn−1, τn−1 for the generators.)
From properties of such amalgamated free products, it follows that
Γ (P) ≃ ⟨ρ0, . . . , ρn−2, ρn−1⟩, Γ (Q) ≃ ⟨ρ0, . . . , ρn−2, τn−1⟩ and Γ (K) ≃ ⟨ρ0, . . . , ρn−2⟩
are faithfully embedded as subgroups of Υ. Thus we may take Γ (P) and Γ (Q) simply to
be those subgroups of Υ. With this understanding, a presentation for Υ on the given gen-
erators is obtained just by taking the union of the various defining relations for Γ (P) and
Γ (Q). This means, for example, that a Coxeter group with a tail-triangle diagram like that
in (3.1), and so with k = ∞, can be viewed as an amalgamated group Υ of the above kind.
(As we remarked earlier, the regular polytopes P and Q in this situation are universal for
their Schläfli types.)
We recall from Theorem 2.5 that the tail-triangle C-group Υ gives an alternating semireg-
ular (n+ 1)-polytope UP,Q, constructed as a sort of coset geometry. In [16, Theorem 5.7]
we show that UP,Q is ‘universal’: it covers any (n + 1)-polytope whose facets alternate
between P and Q. Below we will redescribe UP,Q as U
∞
P,Q; but before we review how
a polytope can have some sort of universal property, we must choose a sensible class of
morphisms.
Definition 3.1 ([11, Section 2D]). Let A and B be polytopes of the same rank. A covering
is a rank and adjacency preserving homomorphism (rap-map) η : A → B. We then say that
A is a cover of B and write A ↘ B. If also η restricts to an isomorphism on any section of
rank at most m in A, then we say that η is an m-covering.
Remark 3.2. It is a useful exercise to check that the flag-connectedness of B forces the
rap-map η to be surjective. It follows that B is isomorphic, in a natural way, to a quotient
of A [13, Lemmas 2.5 and 2.10]. Let us say that the quotient B is induced by the rap-map
η from A to B. The polytope A might have other sorts of quotients which we definitely
want to avoid here [13, Example 2.13].
Most of our covers will originate from group maps. Suppose, for example, that
ϕ : Θ → Γ is an epimorphism of tail-triangle C-groups of the same rank, meaning that ϕ
maps the distinguished generators of Θ to those of Γ , each set indexed by the correspond-
ing diagram. If we refer to Section 2 for the details of the construction of the alternating
semiregular polytopes A and S , from Θ and Γ , respectively, we find that j-faces in the
polytopes are right cosets of naturally defined subgroups (see equation (2.5) for Γ ); and
incidence is given by non-empty intersection. It is quite clear then that application of ϕ
induces a covering map η : A → S . A similar reckoning holds in the regular case for
epimorphisms of string C-groups.
Theorem 3.3. Suppose P and Q are compatible regular n-polytopes and k ⩾ 1 is an
integer for which the class ⟨PQ⟩
k ̸= ∅. Then there exists a universal polytope UkP,Q ∈ ⟨
P
Q⟩
k
such that for each polytope S in the class there is a covering
η : UkP,Q → S,
B. Monson and E. Schulte: The interlacing number for alternating semiregular polytopes 9
which acts as an isomorphism on each facet.
Proof. For k = 1 we observed in Lemma 2.4 that ⟨PQ⟩
1 is non-empty just when P ≃ Q and
contains (up to isomorphism) only the trivial extension of P . Thus U1P,P ≃ {P, 2}.
Now assume k ⩾ 2. The case n = 0 is of little interest, since then P ≃ Q are single
‘vertices’ with empty facets. When n = 1, P ≃ Q are segments, and up to isomorphism
⟨PQ⟩
k contains just the (2k)-gon, so UkP,P ≃ {2k}.
Suppose that n ⩾ 1. Let S ∈ ⟨PQ⟩
k be constructed from a group
Γ = ⟨α0, . . . , αn−2, αn−1, βn−1⟩ associated to a diagram like that in (2.3). We explicitly
assume that P,Q are realized as facets of S , so that Γ (P), Γ (Q) are embedded as sub-
groups of Γ . But, as we remarked earlier, such embeddings also hold for Υ = Γ (UP,Q),
the free product with amalgamation described above (with diagram (3.1)). Trivially, the
subgroups of Υ and Γ corresponding to Γ (P), Γ (Q) are isomorphic. By the universal
property of a free product with amalgamation, we therefore have an epimorphism
ϕ : Υ → Γ
ρj , τn−1 7→ αj , βn−1,
which is bijective on Γ (P), Γ (Q).
Let N be the normal closure of (ρn−1τn−1)
k in Υ and set Υk := Υ/N . Thus
N ⩽ kerϕ and there is an induced map
ϕ̃ : Υk → Γ.
We abuse notation by letting ρj , τn−1 refer also to their images Nρj , Nτn−1 in Υ
k.
We want to show that Υk is a tail-triangle C-group. We will use Lemma 2.6 but simplify
the notation by letting
A := ⟨ρ0, . . . , ρn−2, ρn−1⟩ (≃ Γ (P)), B := ⟨ρ0, . . . , ρn−2, τn−1⟩ (≃ Γ (Q)), and
Cj := ⟨ρn−1−j , . . . , ρn−2, ρn−1, τn−1⟩, for j = 0, . . . , n− 1,
be certain crucial subgroups of Υk. (Here Cn−1 = Υ
k and C0 = ⟨ρn−1, τn−1⟩.)
Since ϕ̃ acts bijectively on A and B, these two subgroups are embedded faithfully in
Υk. Certainly the ρj , τn−1 survive as involutions in Υ
k, which does have a diagram like
that in (2.3); and C0 ≃ Dk, since ϕ̃ must also be bijective on C0.
We will show that Cj is a tail-triangle C-group by induction on j = 0, . . . ,
n − 1. This is trivially the case for the dihedral group C0, so assume Cj is a tail-triangle
C-group for some j ⩾ 0. We apply Lemma 2.6 to Cj+1, so must show that Aj :=
⟨ρn−2−j , . . . , ρn−2, ρn−1⟩ and Bj := ⟨ρn−2−j , . . . , ρn−2, τn−1⟩ are C-groups and that
Aj ∩ Bj ⊆ ⟨ρn−2−j , . . . , ρn−2⟩; Aj ∩ Cj ⊆ ⟨ρn−j−1, . . . , ρn−2, ρn−1⟩; and finally
Bj ∩ Cj ⊆ ⟨ρn−j−1, . . . , ρn−2, τn−1⟩.
Now since Γ is a tail-triangle C-group, the corresponding inclusions certainly hold
there. But since ϕ̃ is a bijection on A and on B, we can pull-back each such inclusion
to Υk. (See [11, Theorem 2E17] for this ‘quotient criterion’ argument in regular cases.)
It also follows that Aj and Bj are C-groups with respect to the given generators, since
A ≃ Γ (P) and B ≃ Γ (Q) are themselves string C-groups. By our induction hypothesis,
Cj is a C-group. Thus Cj+1 is a tail-triangle C-group by Lemma 2.6.
We have shown that Υk is a tail-triangle C-group. Let UkP,Q be the corresponding
alternating semiregular (n + 1)-polytope. From Remark 3.2 above, we have the desired
10 Art Discrete Appl. Math. 5 (2022) #P3.11
covering η : UkP,Q → S . Furthermore, the construction of U
k
P,Q is quite independent of any
special features of S , apart from the interlacing number k. Thus UkP,Q covers all polytopes
in the class. This completes the proof.
Remark 3.4. Theorem 3.3 implies, of course, that if it exists at all, UkP,Q will be unique to
isomorphism. The covering map η is also uniquely specified by a choice of type P flag in
UkP,Q and its image (also of type P) in S . Notice that U
∞
P,Q = UP,Q.
We now see that the Assembly Problem reduces to determining whether or not the
universal polytope UkP,Q even exists. Therefore, let us rethink how we can construct
its automorphism group. Again we are guided by the tail-triangle diagram (2.3), whose
nodes (for a moment) will index the generators α0, . . . , αn−1, βn−1 of a free group of rank
n + 1. We then impose all relations implied by the diagram, including (αn−1βn−1)
k =
1, along with any relations special to Γ (P) = ⟨α0, . . . , αn−2, αn−1⟩ or to Γ (Q) =
⟨α0, . . . , αn−2, βn−1⟩. (Of course, the ambiguous periods ∗ are now also specified.) The
resulting quotient is Γ (UkP,Q), apart from the extension required for the regular case de-
scribed in Theorem 2.5(d). Informally speaking, the group of UkP,Q has no defining rela-
tions which ‘cross’ between Γ (P) and Γ (Q). Note that there could be all sorts of other
‘non-universal’ tail-triangle C-groups which do support such relations. In Theorem 4.9
below, we prove that UkP,Q must cover each of the corresponding polytopes, indeed other,
more general (n + 1)-polytopes which share the local structure described by P,Q and k.
Example 3.5. In Section 5 we recall some situations where even the relations specific to
Γ (P) and to Γ (Q) are somehow made contrary by imposing a particular finite value of k.
For example, take k = 2, P the hemicube {4, 3}3 and Q the toroidal map {4, 4}(s,0). (See
[15, Section 3] for details.) Then
〈
{4,3}3
{4,4}(s,0)
〉2
= ∅
for odd s. On the other hand, U2{4,3}3, {4,4}(s,0) exists and is finite when s is even.
Next we observe that the universal polytope must exist when Q, say, is a trivial exten-
sion of its facet.
Corollary 3.6. Suppose P is a regular n-polytope with facets isomorphic to K; and let
Q be the trivial extension {K, 2}. Then for any even k ⩾ 2 the universal polytope UkP,Q,
exists.
Proof. To apply Theorem 3.3 we need only exhibit a polytope S in the class ⟨PQ⟩
k. But by
a dual version of Theorem 5.1 in [17] we do have a regular extension {P, k} of P , so long
as k ⩾ 4 is even. (The case k = 2 is covered by Lemma 2.4.) If we merely redraw the
corresponding linear diagram as
B. Monson and E. Schulte: The interlacing number for alternating semiregular polytopes 11
t t
t
t
t t∗❤ ∗
∗
✑
✑
✑
✑
. . . . . . k
P
Q
(3.2)
we see that we have a tail-triangle diagram of the required type. (Compare [16, Exam-
ple 5.10].)
What positive results we obtained in [15] suggested the following, seemingly reason-
able
Conjecture 1. If k divides m and the universal polytope UkP,Q exists (equivalently, the
class ⟨PQ⟩
k is non-empty), then also UmP,Q exists.
At first glance it seems that we can use the machinery in the proof of Theorem 3.3 to
prove this conjecture. However, we must be cautious and make do with
Proposition 3.7. Suppose P and Q are compatible regular n-polytopes; and suppose for
k ⩾ 1 that UkP,Q exists, corresponding to the tail-triangle C-group
Υk = ⟨ρ0, . . . , ρn−2, ρn−1, τn−1⟩, as in the proof of Theorem 3.3. Then if k|m, the group
Υm = ⟨ρ̃0, . . . , ρ̃n−2, ρ̃n−1, τ̃n−1⟩ is also a tail-triangle C-group which covers Υ
k. How-
ever, the supposed period m of ρ̃n−1τ̃n−1 conceivably could collapse to some l, with k|l
and l|m.
Proof. We use the notation in the proof of Theorem 3.3. Since k|m, we clearly have an
epimorphism ψ : Υm → Υk, which is bijective when restricted to the subgroups isomor-
phic to Γ (P), Γ (Q). We may then apply Lemma 2.6, again by induction on the iterated
vertex-figure subgroups of Υm. This begins with the dihedral group ⟨ρ̃n−1, τ̃n−1⟩, which
could, in principle, be some Dl, where k|l|m.
Remark 3.8. Of course, we don’t actually have an example of the sort of collapse sug-
gested in Proposition 3.7.
4 The universal covering property of Uk
P,Q
In order to understand covers of most (that is, less symmetrical) polytopes, we need some
new tools. Suppose A is a polytope of rank n+ 1. For 0 ⩽ j ⩽ n, let
rj : F(A) → F(A)
Λ 7→ Λj
Thus rj maps a flag Λ of A to its j-adjacent flag Λ
j in A. Note that each rj is a fixed-
point-free involution on the flag set F(A).
Definition 4.1. The connection group of A is
Mon(A) = ⟨r0, . . . , rn⟩.
12 Art Discrete Appl. Math. 5 (2022) #P3.11
Remark 4.2. The connection group was used in [21] as a tool in the theory of maps. One
can interpret relations in the group as describing how an abstract set of flags might be
connected together to constitute a polytope. We use the notation Mon(A) as a reminder
that this group is often called the ‘monodromy group’ in the literature. Etymologically,
‘monodromy’ does make sense; however, in complex analysis, ‘monodromy group’ has
come to mean something a bit at odds with the intent here. The connection group appears
as the ‘cartographic group’ in [4, page 246]; see also [8].
Observe that Mon(A) is a subgroup of the symmetric group on the flag set. It is always
a string group generated by involutions (sggi), meaning that relations parallel to those dis-
played in (2.1) must hold on the specified generators. However, the intersection condition
(2.2) can fail if the rank n+ 1 ⩾ 4. See [13, Example 6.8], for instance.
If g = rj1 · · · rjl ∈ Mon(A) and Λ is any flag of A, we write Λ
g or even Λj1...jl for
the image of Λ under g. Notice that j1 . . . jl could be any string in the symbols {0, . . . , n}.
The following simple result is crucial to our calculations.
Lemma 4.3. If γ is any automorphism of A, Φ is any flag of A and g ∈ Mon(A), then
(Λγ)g = (Λg)γ. (4.1)
Proof. Since γ preserves adjacency of flags, the actions of γ and each rj on F(A) com-
mute.
Many covering properties can be rephrased in terms of connection groups. For this, it
is, in turn, crucial to understand the flag stabilizer in Mon(A). (Notice that when A is a
polytope, Mon(A) acts transitively on the flag set.) For more details we refer to [13], in
particular to Propositions 3.11 and 3.13, which we rephrase here as
Proposition 4.4. Suppose that A and B are (n+ 1)-polytopes.
(a) Any covering η : A → B induces an epimorphism
η : Mon(A) → Mon(B)
rj 7→ sj
(of sggi’s). Furthermore, for a given flag Λ of A with image Λ′ = (Λ)η in B, the
stabilizer of Λ in Mon(A) is mapped by η into the stabilizer of Λ′ in Mon(B).
(b) Suppose that η : Mon(A) → Mon(B) is an epimorphism of sggi’s and that there are
flags Λ of A and Λ′ of B such that η maps the stabilizer of Λ in Mon(A) into the
stabilizer of Λ′ in Mon(B). Then there is a covering η : A → B which maps Λ to Λ′.
When S is the alternating semiregular (n + 1)-polytope constructed from the tail-
triangle C-group Γ = ⟨α0, . . . , αn−2, αn−1, βn−1⟩, with diagram (2.3), we can compare
more closely the actions of Γ and Mon(S) on flags of S . Recall that S has two flag orbits
under the action of Γ , containing in turn the base flags Φ and Ψ described in (2.6).
Lemma 4.5. (a) Φβn−1 = Φ
n,n−1,n and Ψαn−1 = Ψ
n,n−1,n.
(b) Φαn−1 = Φ
n−1 and Ψβn−1 = Ψ
n−1.
B. Monson and E. Schulte: The interlacing number for alternating semiregular polytopes 13
(c) Φαj = Φ
j and Ψαj = Ψ
j , for 0 ⩽ j ⩽ n− 2.
(d) The relation (rn−1rn)
2k = 1 holds in Mon(S).
Proof. From the description of the faces of S in Section 2, as well as the base flags in (2.6),
we recall that Φ has n-face ΓPn . Thus Φ
n has n-face ΓQn ; so Φ
n,n−1 has (n − 1)-face
Γn−1βn−1; so finally Φ
n,n−1,n has n-face ΓPn βn−1. Since βn−1 ∈ Γj , for 0 ⩽ j ⩽ n− 2,
we have Φβn−1 = Φ
n,n−1,n. The remaining calculations for parts (a), (b), (c) are similar.
(For a figure aiding in these calculations see also [14, Equation (9)].) We then have
(Φ)αn−1βn−1 = (Φ
n−1)βn−1 = (Φβn−1)
n−1 = Φn,n−1,n,n−1.
Since (αn−1βn−1)
k = 1, we get Φ(rnrn−1)
2k
= Φ. Similarly, Ψ(rn−1rn)
2k
= Ψ. Applying
Lemma 4.3 one last time, we obtain the relation in (d).
We now focus on the case that S = UP,Q (= U
∞
P,Q) is universal, with ρj , τn−1 in-
stead of αj , βn−1. Let M := Mon(UP,Q) = ⟨r0, . . . , rn⟩ and M
k := Mon(UkP,Q) =
⟨s0, . . . , sn⟩. As in the proof of Theorem 3.3, the natural map π : Υ → Υ
k induces a cov-
ering η : UP,Q → U
k
P,Q. We recall from Section 2 that a typical j-face of UP,Q is some
right coset Υjµ, where µ ∈ Υ. (For facets we have Υ
P
n µ or Υ
Q
n µ.) The image of such a
j-face in UkP,Q is just
(Υjµ)π = {Nσ : σ ∈ Υj}(Nµ) = {Nσµ : σ ∈ Υj},
where again N = kerπ is the normal closure of (ρn−1τn−1)
k in Υ.
The map η in turn induces an epimorphism
η : M →Mk
rj 7→ sj , 0 ⩽ j ⩽ n
(see Proposition 4.4(a) above).
Lemma 4.6. If two flags of type P (or two of type Q) in UP,Q have the same image under η,
then these flags lie in the same N -orbit. No two flags of different types lie in the same N -
orbit.
Proof. The base flag Φ of type P is like that described in equation (2.6), now with group Υ
in place of Γ . If (Φµ)η = (Φγ)η for µ, γ ∈ Υ, then we have equal cosetsNΥjµ = NΥjγ,
for 0 ⩽ j ⩽ n− 1. (Also NΥPn µ = NΥ
P
n γ.) This means that
Nµγ−1 ∈ (Υ0)π ∩ . . . ∩ (Υn−1)π ∩ (Υ
P
n )π = {N},
by the interesection condition in Υk. Thus µ ∈ Nγ. The case of two flags of type Q is
similar.
If two flags of different types had the same image under η, then we would soon get
NUgPn = NΥ
Q
n . This already contradicts the fact that facets in U
k
P,Q alternate. But to nail
home the argument, note that τn−1 ∈ Υ
Q
n , so that then NΥ
P
n = Υ. This would mean the
face (Υn−1)η lies on only one facet in U
k
P,Q, a contradiction.
14 Art Discrete Appl. Math. 5 (2022) #P3.11
We now need two important subgroups of the connection group M . First of all, we
let L be the normal closure in M of the element (rn−1rn)
2k. Second, we recall from [16,
Theorem 5.6], that the stabilizer K in M of the base flag Φ of type P in UP,Q is generated
by elements of the form wt, where t ∈M , w ∈Mn = ⟨r0, . . . , rn−1⟩ and either
(a) w fixes Φ and rn appears an even number of times in a word for t; or
(b) w fixes Ψ and rn appears an odd number of times in a word for t.
But remember that for any flag Φ′ of type P in UP,Q there exists γ ∈ Υ such that Φ
′ =
(Φ)γ. By Lemma 4.3, K therefore fixes all flags of type P . Since the base flag Ψ of type
Q is n-adjacent to Φ, all flags of type Q are stabilized by Krn = rnKrn.
Now let us take Φ̃ := (Φ)η and Ψ̃ := (Ψ)η as n-adjacent base flags in UkP,Q.
Lemma 4.7. If for g ∈M we have Φ̃(g)η = Φ̃, then g ∈ KL. Furthermore,
ker η ⊆ KL ∩KrnL.
Proof. Since η : UPQ → U
k
P,Q is a rap-map, we have
(Φg)η = (Φη)(g)η = Φ̃ = (Φ)η.
By Lemma 4.6, there exists α ∈ N such that Φg = (Φ)α. But such an α is a product of
various conjugates in Υ of the word (ρn−1τn−1)
k. We may apply the rewriting rules in
Lemma 4.5, with αj , βn−1 replaced by ρj , τn−1. Thus (Φ)α = Φ
h, where now h ∈M is a
product of various conjugates of (rn−1rnrn−1rn)
k = (rn−1rn)
2k. In other words, h ∈ L
and Φgh
−1
= Φ. Thus gh−1 ∈ K.
If g ∈ ker η, then we also have Ψ̃(g)η = Ψ̃, so that we similarly get g ∈ KrnL.
Remark 4.8. For any flag Λ of UPQ and g ∈M we have (Λ
g)η = (Λη)(g)η .
Theorem 4.9. Suppose that P and Q are compatible regular n-polytopes and that k is
a fixed positive integer such that UkPQ exists. Let B be an (n + 1)-polytope whose facets
are, in alternating fashion, various quotients of P and Q, all induced by rap-maps. More
precisely, suppose that for each face F of co-rank 2 in B, there is a positive integral divisor
kF of k such that F is surrounded by kF quotients of P alternating with kF quotients of Q
(so that B/F ≃ {2kF }). Then there exists a covering
κ : UkP,Q → B.
Proof. By [16, Corollary 5.9], there is a covering λ : UP,Q → B, so there is an epimor-
phism λ :M → Mon(B), as in
M
η

λ // Mon(B)
Mk
κ ?
::
B. Monson and E. Schulte: The interlacing number for alternating semiregular polytopes 15
We seek a map κ (of sggi’s) making the diagram commute. But ker η ⊆ KL ∩KrnL
from Lemma 4.7, so we want to show that KL ∩ KrnL ⊆ kerλ. First of all, from the
assumption concerning kF at face F in B, we have ((rn−1rn)
2k)λ = 1 in Mon(B). Thus
L ⊆ kerλ.
Next consider the generators wt of type (a) or (b) for K. Since, as we observed, wt
fixes all flags of type P in UP,Q, and since the corresponding facets of B are quotients of
P by rap-maps, (wt)λ fixes all flags of this kind in B. Likewise, (Krn)λ fixes the flags in
B whose facets arise as quotients of Q.
We conclude that ker η ⊆ kerλ, so the epimorphism κ exists and respects specified
generators of sggi’s. Furthermore, if g̃ = (g)η fixes the flag Φ̃ in UkP,Q, then again by
Lemma 4.7, g ∈ KL. Once more, on geometrical grounds, (g̃)κ = (g)λ fixes the flag (Φ)λ
in B. By Proposition 4.4(b), we have a covering κ : UkP,Q → B (sending Φ̃ to (Φ)λ).
5 Examples with finite interlacing number k
To make some sense of the Assembly Problem, it is useful to understand cases in which
collapse occurs and others when it does not. In Table 1 we summarize several pertinent ex-
amples from [15]. By Theorem 3.3 we are justified in examining only universal polytopes.
P Q k UkP,Q Comments
exists (Γ = Γ (UkP,Q) )
{4, 3}3 {4, 2} or {4, 4}(2,0) any odd k ⩾ 3 no
(hemicube)
{4, 3} {4, 4}(s,0) 2 yes, finite only for s = 2,
for any s ⩾ 2 in which case |Γ | = 384
{4, 3}3 {4, 4}(s,0) 2 yes, for even s ⩾ 2 finite; |Γ | = 24s
3
2 no, for odd s ⩾ 1
{6, 3}(1,1) {6, 3} 2 no Q forced to collapse
to a finite toroid.
See Remark 5.1.
{6, 3}(1,1) {6, 3}(b,b) 2 yes, just when b = 1, 2
{6, 3}(b,0) 2 no
{6, 3}(1,1) {6, 3}(2,0) 4 no only known collapse
for even k ⩾ 4
{4, 3n−3, 3}n {4, 3
n−3, 4}(s,0n−2) 2 no, for odd s ⩾ 3 collapse in all
(hemi-n-cube) an n-toroid higher ranks
has FAP also has FAP 2 yes [15, Theorem 5.3] explicitly
w.r.t. facet K w.r.t. facet K describes an alt. semireg.
poly. S, so implying U2P,Q exists.
See Remark 5.2.
Table 1: Assembly problems when k <∞.
Remark 5.1. Since the geometry has some strange features, we say more about the case
that both P and Q have Schläfli type {6, 3}. Here {6, 3} really indicates the familiar tiling
of the Euclidean plane by regular hexagons. Its automorphism group Γ = ⟨α0, α1, α2⟩ is
the Coxeter group [6, 3] (or G̃2) with diagram
t
α0
t
α1
t
α2
36
16 Art Discrete Appl. Math. 5 (2022) #P3.11
(For consistency with notation introduced in [15], we are deviating here from our earlier
convention that the letter Γ should denote a tail-triangle group.) The translation subgroup
of Γ is generated by x = (α0α1)
2α2α1 = (α0α1α2)
2 and y = xα1 = (α1α0)
2α1α2.
These translations, along with mirrors for the generating reflections, are indicated in Fig-
ure 5.1. The fundamental region enclosed by these mirrors has been shaded in. It is con-
venient to use the redundant reflection α3 := α
α0
1 . Thus the mirrors for α1, α2, α3 enclose
an equilateral triangle and furthermore generate the Coxeter group Ã2, having index 2 in
Γ . Notice that xy = (α3α1α2)
2.
1
xy
α
α
α
0
2
Figure 1: The plane tiling {6, 3}.
For each b ⩾ 2, the group of the finite regular toroid {6, 3}(b,0) is obtained by setting
xb = 1; this group has order 12b2 [2, Section 8.4]. Likewise, for c ⩾ 1, the toroid
{6, 3}(c,c) is obtained by setting (xy)
c = 1, or equivalently (α1α2α3)
2c = 1, giving an
automorphism group of order 36c2.
Here our concern is really with alternating semiregular polytopes with facets P =
{6, 3}(1,1) and Q = {6, 3}, so in this instance Q is the universal cover of P . For interlacing
number k = 2, we first ask whether the polytope U2P,Q even exists, which means that we
must study the group Λ defined by the diagram and subsidiary relation indicated in (5.1).
t
α0



❅
❅
❅
t
α1
t
t
α2
β2
3
3
6
Λ: + (α1α2α3)
2 = 1 (5.1)
To force P to be the toroid {6, 3}(1,1) we have imposed the relation (α1α2α3)
2 = 1.
Remarkably, we find [15, Section 4] that this implies that
(α1β2α3)
4 = 1 (5.2)
must hold in the subgroup ⟨α0, α1, β2⟩ of Λ. In other words, Q cannot survive as the
infinite, universal polyhedron {6, 3}. Through its interlacing with k = 2 copies of P =
{6, 3}(1,1), Q must collapse at least to the toroid Q̃ = {6, 3}(2,2). Any other choice for
Q (still of type {6, 3}) must result in some sort of loss of structure for Q or even for P .
For example, if Q = {6, 3}(2,0), then with the aid of GAP [3], we find that Λ collapses
B. Monson and E. Schulte: The interlacing number for alternating semiregular polytopes 17
to a group of order 48. In this case, Q survives but P collapses to the non-polytopal map
{6, 3}(1,0).
If Q = P = {6, 3}(1,1) (and still k = 2), U
2
P,Q is the universal regular 4-polytope
{{6, 3}(1,1), {3, 4}} with automorphism group S3 ⋊ [3, 4], of order 288 (see [11, Theo-
rem 11C13]).
Now increase the interlacing number to k = 4, again with the ‘worrisome’ facets
P = {6, 3}(1,1) and Q = {6, 3}(2,0). We get our first (and so far, only!) instance of
collapse for even k > 2. The group Λ now has order 17280; and Γ (P) and Γ (Q) survive
unscathed. But the intersection condition (2.2) fails. In particular, the vertex-figure sub-
group ⟨α1, α2, β2⟩ has order 8640, so that the corresponding coset geometry could only
have 2 vertices.
Remark 5.2. This construction yields alternating semiregular polytopes which are unex-
pectedly ‘small’, at least compared to the enormous group orders which frequently arise.
Informally speaking, a regular polytope is said to have the flat amalgamation property
(FAP), with respect to its facets, if each facet is the fundamental region, in the entire poly-
tope, for the subgroup generated by the reflections in its own facets (see [11, Section 4E]).
We can formalize the FAP in terms of groups. Suppose n ⩾ 2 and let R be a regular
n-polytope with facet L and automorphism group Γ (R) = ⟨γ0, . . . , γn−1⟩. Define
NR := ⟨ϕ
−1γn−1ϕ : ϕ ∈ Γ (R)⟩ = ⟨ϕ
−1γn−1ϕ : ϕ ∈ ⟨γ0, . . . , γn−2⟩⟩,
which is, in fact, the normal closure of γn−1 in Γ (R). Then
Γ (R) = NR ·⟨γ0, . . . , γn−2⟩ = ⟨γ0, . . . , γn−2⟩·NR,
a product of subgroups. The polytope R is said to have the flat amalgamation property
(with respect to its facet L), if this product of subgroups is semi-direct, that is, if
Γ (R) ≃ NR ⋊ ⟨γ0, . . . , γn−2⟩ ≃ NR ⋊ Γ (L).
While clearly not every polytope has the FAP, the property nevertheless is not uncommon;
for examples of interesting classes of polytopes with the FAP, see [20], where the FAP was
referred to as the DAP (for “degenerate amalgamation property”).
In [15, Section 5] we describe a construction of small alternating semiregular polytopes,
with k = 2, which employs the FAP for P and Q, along with a kind of mixing of their
groups. We leave the details to [15, Theorem 5.3] and extract what we want here as
Proposition 5.3. Suppose P and Q are compatible regular n-polytopes with facets iso-
morphic to K, and suppose that P and Q have the FAP with respect to their facet K. Then
there exists an alternating semiregular (n + 1)-polytope S with two copies of P and Q
occurring alternately around each face of co-rank 2. (Thus the interlacing number k is
equal to 2.) The automorphism group of S is isomorphic to (NP ×NQ)⋊Γ (K) if P ̸≃ Q,
or ((NP ×NQ)⋊ Γ (K))⋊ C2 if P ≃ Q. In particular, S is finite if and only if P and Q
are finite.
Note that this Proposition does not assert that the constructed polytope S is universal,
although it does follow (Theorem 3.3 again) that U2P,Q exists.
18 Art Discrete Appl. Math. 5 (2022) #P3.11
Example 5.4. Among the regular toroidal maps precisely {4, 4}(2t,0), {4, 4}(2t,2t),
{3, 6}(t,t), and {3, 6}(3t,0), with t ⩾ 1, have the FAP with respect to their facets (see [20,
page 319-320]). For instance, take P = {4, 4}(2s,0) and Q = {4, 4}(2t,0), with s, t ⩾ 1,
and consider the alternating semiregular 4-polytope S obtained from Theorem 5.3 via the
diagram
{4, 4}(2s,0)
{4, 4}(2t,0)
❤t t
α0 4
α1 4
4
β2
α2
✑
✑
✑
✑✑
◗
◗
◗
◗◗
t
t
(5.3)
The 4-polytope S then has s2t2 vertices and 4(s2+t2) facets, namely 4t2 facets isomorphic
to P and 4s2 facets isomorphic to Q. The polytope is regular if and only if s = t.
6 Prescribing the interlacing number k
Recall Conjecture 1, which asserts that if UkP,Q exists and k divides m, then also U
m
P,Q
should exist. Although certain examples in Section 5 urge caution here, it is still true that
in many cases a direct construction of an alternating semiregular polytope with interlac-
ing number m from a given alternating semiregular polytope with interlacing number k
is available. The new tail-triangle C-group is obtained as a ‘mix’ of the old tail-triangle
C-group and the dihedral group Dm.
Call a regular polytope R vertex bipartite if the edge graph (1-skeleton) of R is a
bipartite graph. Then R is vertex bipartite if and only if each circuit of the edge graph
of R has even length. The faces of a vertex bipartite regular polytope are also vertex
bipartite. Call a regular polytope R dually vertex bipartite if and only if its dual R∗ is
vertex bipartite. For example, the toroidal maps {4, 4}(2t,0) and {4, 4}(t,t), with t ⩾ 1, are
all vertex bipartite and dually vertex bipartite.
Now assume, as before, that P and Q are regular n-polytopes, with a common facet
K, such that an alternating semiregular (n + 1)-polytope S with facets P and Q and with
interlacing number k does exist. Let Γ = ⟨α0, . . . , αn−2, αn−1, βn−1⟩ be the correspond-
ing tail-triangle C-group, so that ⟨αn−1, βn−1⟩ = Dk. Now suppose k divides m, and
let Dm = ⟨δ0, δ1⟩, where δ0, δ1 are standard involutory generators for Dm. Consider the
subgroup ∆ of the direct product Γ × Dm generated by
α′i :=
{
(αi, 1) if 0 ⩽ i ⩽ n− 2,
(αi, δ0) if i = n− 1
(6.1)
and
β′n−1 := (βn−1, δ1). (6.2)
Then ∆ is a tail-triangle group with ⟨α′0, . . . , α
′
n−2⟩ ≃ ⟨α0, . . . , αn−2⟩ and ⟨αn−1, βn−1⟩ =
Dm. However, in general the subgroups
∆Pn := ⟨α
′
0, . . . , α
′
n−2, α
′
n−1⟩, ∆
Q
n := ⟨α
′
0, . . . , α
′
n−2, β
′
n−1⟩
will not be isomorphic to Γ (P) or Γ (Q), respectively. In fact, these subgroups are just
the automorphism groups of the polytopes P ′ and Q′, respectively, which are the duals, of
B. Monson and E. Schulte: The interlacing number for alternating semiregular polytopes 19
the mix of the duals P∗ or Q∗ with a 1-polytope (see [11, Theorem 7A7]). It is known
from [11, Theorem 7A8] that P ′ ≃ P and Q′ ≃ Q if all circuits of the edge graph of P∗
or Q∗ are even, that is, if P and Q are dually vertex bipartite. (Note that this implies that
αn−2αn−1 and αn−2βn−1 have even periods.) We see that dual vertex bipartiteness of P
and Q is a necessary condition for ∆ to be a tail-triangle group of the desired kind.
Theorem 6.1. Suppose P and Q are dually vertex bipartite, compatible regular n-polytopes,
and k divides m. Suppose there exists an alternating semiregular (n + 1)-polytope with
facets isomorphic to P or Q, and with interlacing number k, which is derived from a tail-
triangle C-group Γ . Then the subgroup ∆ of Γ×Dm, with generators as in (6.1) and (6.2),
is a tail-triangle C-group. The corresponding alternating semiregular (n+1)-polytope still
has facets isomorphic to P or Q, and interlacing number m.
Proof. It remains to establish that ∆ is a C-group. Our proof again appeals to Lemma 2.6
and is similar to the proof of Theorem 3.3. Let π denote the restriction to ∆ of the projection
of Γ × Dm onto the first factor Γ . As P and Q are dually vertex bipartite, π defines
isomorphisms πP : ∆
P
n 7→ Γ
P
n (≃ Γ (P) ) and πQ : ∆
Q
n 7→ Γ
Q
n (≃ Γ (Q)). In particular,
∆Pn and ∆
Q
n are C-groups.
For condition (a) of Lemma 2.6 we must show that ∆Pn ∩ ∆
Q
n = ⟨α
′
0, . . . , α
′
n−2⟩.
However, this follows easily from the fact that πP and πQ are isomorphisms, and that Γ is
a C-group, with ΓPn ∩ Γ
Q
n = ⟨α0, . . . , αn−2⟩.
For condition (b) of Lemma 2.6 we argue inductively, using the fact that the vertex-
figures of dually vertex bipartite regular polytopes are also dually vertex bipartite.
The case n = 2 can be handled directly. The only intersections not yet considered
are ∆P2 ∩ ⟨α
′
1, β
′
1⟩ and ∆
Q
2 ∩ ⟨α
′
1, β
′
1⟩. As the proofs are similar, we only discuss the
first. Consider the two componentwise projections from the direct product Γ × Dm and
again use the fact that Γ is a C-group. This shows that ∆P2 ∩ ⟨α
′
1, β
′
1⟩ must necessarily
lie in ⟨α1⟩ × ⟨δ0⟩ ≃ C2 × C2. However, recalling our earlier remarks on dual vertex
bipartiteness, α0α1 has even order when n = 2. Thus, the element (1, δ0) definitely does
not lie in ∆P2 = ⟨α
′
0, α
′
1⟩ = ⟨(α0, 1), (α1, δ0)⟩, so the intersection must just be ⟨α
′
1⟩, as
required. We conclude that ∆ is a C-group when n = 2.
Now let n > 2 and suppose inductively that the subgroup ∆0 associated with the
vertex-figures P0 of P and Q0 of Q is a C-group. By Lemma 2.6 we must show that
∆0 ∩∆
P
n = ⟨α
′
1, . . . , α
′
n−1⟩, ∆0 ∩∆
Q
n = ⟨α
′
1, . . . , α
′
n−2, β
′
n−1⟩.
The proofs are similar, so we only verify the first of these conditions. Recall that the
projection mapping πP from before is an isomorphism, and that Γ0∩Γ
P
n = ⟨α1, . . . , αn−1⟩
since Γ is a C-group. Thus the desired property lifts from Γ to ∆ itself, and the proof is
complete.
7 Alternating semiregular polytopes with toroidal facets
When k is finite, it generally seems difficult to even decide the existence of UkP,Q for
given compatible regular n-polytopes P and Q, let alone pin down the corresponding group
explicitly (the group Υk in the notation of Section 3). However, in some cases this can be
done.
Our approach in this final section is similar to that in [15, Section 3]. We apply a
‘twisting operation’ of the sort discussed in detail in [11, Chapter 8] to a carefully cho-
sen Coxeter group. We will then show that certain toroidal maps of type {4, 4} can be
20 Art Discrete Appl. Math. 5 (2022) #P3.11
assembled with arbitrary interlacing number k ⩾ 2; and we can describe the group for the
universal polytope in a very satisfactory way.
To be precise let us take P = {4, 4}(r,0) and Q = {4, 4}(s,0) with any r, s ⩾ 2 and any
k ⩾ 2 (allowing k = ∞). Thus the tail-triangle diagram of Υk has the form
{4, 4}(r,0)
{4, 4}(s,0)
❤t t
α0 α1
β2
α2
✡
✡
✡
✡
✡
❏
❏
❏
❏
❏
t
t
k (7.1)
(In Section 3 we had ρj , τ2 in place of αj , β2.) As before, when k = 2, we regard the
vertical branch as missing. The key idea is to investigate how the generator α1 acts by
conjugation on α0, α2, β2 and their conjugates under α1. In a sense, this is a “reverse
twisting” method.
Suppose for a moment that S is any alternating semiregular 4-polytope with facets
P = {4, 4}(r,0) and Q = {4, 4}(s,0) and with interlacing number k, associated with a
tail-triangle C-group Γ = ⟨α0, α1, α2, β2⟩ and a diagram as in (7.1). The subgroup
W (S) := ⟨α0, α2, β2, α1α0α1, α1α2α1, α1β2α1⟩
is invariant under conjugation by α1, and either Γ = W (S) (if α1 ∈ W (S)) or Γ =
W (S) ⋊ C2 (if α1 ̸∈ W (S)). Either way, α1 permutes the generators of W (S) and acts
like the central symmetry on the following diagram for W (S):
t✡
✡
✡
✡
✡
✡
❏
❏
❏
❏
❏
❏
t
t
k
t
t
❏
❏
❏
❏
❏
❏
✡
✡
✡
✡
✡
✡
tkqα0 α1α0α1
α1β2α1
α1α2α1
α2
β2
s
r s
r
∗
∗
↶
α1
(7.2)
It is straightforward to check that the orders of the pairwise products of generators ofW (S),
if finite, are just the marks on the branches connecting the corresponding nodes, with or-
der 2 again representing an omitted branch. For example, diametrically opposite nodes
represent commuting generators, as both P and Q are of type {4, 4}. The vertical branches
are marked with the interlacing number k, as (α2β2)
k = 1 in Γ . The marks r and s on the
left and right are just the length of the 2-holes of P and Q, respectively, in accordance with
B. Monson and E. Schulte: The interlacing number for alternating semiregular polytopes 21
(α0α1α2α1)
r = 1 and (α0α1β2α1)
s = 1 in Γ [11, p. 195]. Finally, the two horizontal
branches in (7.2) are marked ∗, since we do not immediately know the actual order of the
product α1α2α1β2. There are examples of groups Γ where this product has finite order,
and others where it has infinite order. Note that α1α2α1β2 lies in the subgroup ⟨α1, α2, β2⟩
of Γ supported on the triangular subdiagram of (7.1).
Now let W =W (r, s, k) denote the Coxeter group with diagram
t✡
✡
✡
✡
✡
✡
❏
❏
❏
❏
❏
❏
t
t
k
t
t
❏
❏
❏
❏
❏
❏
✡
✡
✡
✡
✡
✡
t
k
qρ1 ρ4
ρ2
ρ6
ρ3
ρ5
s
r s
r
∞
∞
↶
σ
(7.3)
(Here, of course, ρj does not denote a generator of the group Υ
k from Section 3.) Our
previous considerations just say thatW (S) is a quotient ofW . Thus, if S is any alternating
semiregular 4-polytope with facets P and Q, interlacing number k, and tail-triangle C-
group Γ , then either Γ itself is a quotient of W , or Γ is the semi-direct product of a
quotient of W by C2.
We now stand the argument on its head and construct a polytope of the desired kind
from a tail-triangle group derived from W itself. This will necessarily be the universal
polytope of its kind.
Suppose the parameters r, s and k specifying the facets and interlacing number are
given, and consider the Coxeter group W = W (r, s, k) defined by the diagram in (7.3).
Then the central diagram symmetry σ shown in (7.3) induces a group automorphism of W
permuting the generators ρ1, . . . , ρ6, and we can form the semi-direct product of W by C2.
Abusing notation, we set Γ :=W ⋊ ⟨σ⟩ and choose the following generators for Γ :
α0 := ρ1, α1 := σ, α2 := ρ3, β2 := ρ5. (7.4)
Then it is straightforward to check that Γ is a tail-triangle group whose generators satisfy
the relations implicit in diagram (7.1). For example, (α0α1)
4 = 1, since
(α0α1)
2 = α0(α1α0α1) = ρ1ρ4.
Similarly, (α1α2)
4 = 1 and (α1β2)
4 = 1. Further, (α2β2)
k = (ρ3ρ5)
k = 1. The facet
subgroups of Γ are given by
⟨α0, α1, α2⟩ = ⟨ρ1, ρ3, ρ4, ρ6, σ⟩ ≃ (Dr × Dr)⋊ C2
and
⟨α0, α1, β2⟩ = ⟨ρ1, ρ2, ρ4, ρ5, σ⟩ ≃ (Ds × Ds)⋊ C2,
and have the required form. The vertex-figure subgroup of Γ is
⟨α1, α2, β2⟩ = ⟨ρ2, ρ3, ρ5, ρ6⟩⋊ C2,
22 Art Discrete Appl. Math. 5 (2022) #P3.11
where ⟨ρ2, ρ3, ρ5, ρ6⟩ is a Coxeter group with a rectangular diagram and with branch marks
k and ∞ alternating. (Such a subgroup of W is a Coxeter group [7, Theorem 5.5].)
Finally, Γ inherits the intersection property with respect to its generators α0, α1, α2, β2
from the intersection property of W with respect to its generators ρ1, . . . , ρ6. (Recall that
any Coxeter group has the intersection property with respect to its standard generators
[7, Theorem 5.5].) For example, ⟨α0, α1, α2⟩ ∩ ⟨α1, α2, β2⟩ coincides with
⟨ρ1, ρ3, ρ4, ρ6⟩·⟨σ⟩ ∩ ⟨ρ2, ρ3, ρ5, ρ6⟩·⟨σ⟩ = ⟨ρ3, ρ6⟩·⟨σ⟩ = ⟨α1, α2⟩,
as required.
Thus Γ = W ⋊ C2 is a tail-triangle group associated with an alternating semiregular
4-polytope with facets P := {4, 4}(r,0) and Q := {4, 4}(s,0) and interlacing number k. But
now it is clear that this must be the universal polytope UkP,Q. In fact, by our earlier consid-
erations, any tail-triangle group associated with an alternating semiregular 4-polytope with
these facets and interlacing number must necessarily sit below W or W ⋊C2, while on the
other hand the present tail-triangle group is just W ⋉ C2 itself.
In summary, we have proved the following theorem.
Theorem 7.1. Let r, s, k ⩾ 2, P := {4, 4}(r,0) and Q := {4, 4}(s,0), and let W (r, s, k)
denote the Coxeter group defined by diagram (7.3). Then the universal alternating semireg-
ular 4-polytope UkP,Q exists, and its tail-triangle C-group Υ
k is given by
Υk ∼=W (r, s, k)⋊ C2.
In particular, UkP,Q is infinite, as is its vertex-figure.
If the mark ∞ in diagram (7.3) is replaced by a finite integer t ⩾ 2, then the resulting
new Coxeter group can similarly be employed to produce an alternating semiregular 4-
polytope with facets {4, 4}(r,0) and {4, 4}(s,0) and interlacing number k. This polytope is
a proper quotient of UkP,Q and is determined by the extra relation
(α1α2α1β2)
t = 1
imposed on the generators of Γ (and affecting the vertex-figure group). Thus UkP,Q has
infinitely many quotients, each an alternating semiregular 4-polytope with the same facets
and interlacing number.
We now look at the case k = 2 in more detail. Let r, s, t ⩾ 2, allowing t = ∞.
Consider the Coxeter group Wt(r, s) with Coxeter diagram
t✡
✡
✡
✡
✡
✡
❏
❏
❏
❏
❏
❏
t t
t
❏
❏
❏
❏
❏
❏
✡
✡
✡
✡
✡
✡
tqρ1 ρ4
ρ2
ρ6
ρ3
ρ5
s
r s
r
t
t
↶
σ
(7.5)
B. Monson and E. Schulte: The interlacing number for alternating semiregular polytopes 23
Then the same choice of generators as in (7.4) gives a tail-triangle C-group Γ of the form
{4, 4}(r,0)
{4, 4}(s,0)
❤t t
α0 α1
β2
α2
✡
✡
✡
✡
✡
❏
❏
❏
❏
❏
t
t
{4, 4}(t,0) (7.6)
Now the vertex-figure subgroup of Γ is isomorphic to the automorphism group of the
toroidal map {4, 4}(t,0) (if t is finite) or the square tessellation {4, 4} (if t = ∞). By
Theorem 2.5(c), the vertex-figure of the corresponding alternating semiregular 4-polytope
itself is isomorphic to the medial polyhedron of {4, 4}(t,0) or {4, 4}, which is {4, 4}(t,t) or
{4, 4}, respectively. If t = ∞, then Wt(r, s) = W (r, s, 2) and so we recover the universal
polytope U2P,Q. Thus we have the following theorem.
Theorem 7.2. Let r, s, t ⩾ 2, P := {4, 4}(r,0) and Q := {4, 4}(s,0), and let Wt(r, s) de-
note the Coxeter group defined by diagram (7.5). Then U2P,Q has infinitely many quotients
which are alternating semiregular 4-polytopes with the same facets P , Q and the same
interlacing number 2. In particular, there exists such a polytope with tail-triangle C-group
isomorphic to Wt(r, s)⋉ C2 and with vertex-figures {4, 4}(t,t) (if t is finite); this polytope
is finite if and only if Wt(r, s) is finite (that is, (r, s, t) = (2, 2, t) with 2 ⩽ t < ∞ or
(r, s, t) = (2, 3, t) with t = 3, 4, 5, up to permutation of r, s, t).
Note that the diagram for Γ in (7.6), with the ring around the leftmost node removed,
can be interpreted in three different ways as a diagram for a tail-triangle group with 3-
generator subgroups corresponding to toroidal maps. Each is determined by ringing one of
the outer nodes of the diagram and yields a polytope as in Theorem 7.2.
Here we end, for now, our survey of the interlacing number for compatible regular
n-polytopes P,Q. For some fairly large families of such polytopes, assembly into an alter-
nating semiregular (n+ 1)-polytope S is possible. In other, rather elusive cases, assembly
fails for certain interlacing numbers. Our conjectures from [15], including Conjecture 1 in
Section 3, remain beyond reach. Intuitively, they should ‘generally’ hold. But it would also
be very nice to kill them off with counterexamples!
ORCID iDs
Barry Monson https://orcid.org/0000-0001-9901-9681
Egon Schulte https://orcid.org/0000-0001-9725-3589
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.12
https://doi.org/10.26493/2590-9770.1442.cb1
(Also available at http://adam-journal.eu)
On hamiltonian cycles in Cayley graphs
of order pqrs
Dave Witte Morris*
Department of Mathematics and Computer Science, University of Lethbridge,
4401 University Drive, Lethbridge, Alberta, T1K 3M4, Canada
Received 18 August 2021, accepted 29 June 2022, published online 5 August 2022
Abstract
Let G be a finite group. We show that if |G| = pqrs , where p, q, r, and s are distinct
odd primes, then every connected Cayley graph on G has a hamiltonian cycle.
Keywords: Cayley graph, hamiltonian cycle.
Math. Subj. Class.: 05C25, 05C45
1 Introduction
Definition 1.1 (cf. [4, page 34]). If A is a subset of a finite group G, then the corresponding
Cayley graph Cay(G;A) is the undirected graph whose vertices are the elements of G, and
such that vertices g and h are adjacent if and only if g−1h ∈ A ∪ A−1, where A−1 =
{ a−1 | a ∈ A }.
It is easy to see (and well known) that Cay(G;A) is connected if and only if A is a
generating set of G. Several papers show that all connected Cayley graphs of certain orders
are hamiltonian:
Theorem 1.2 (see [6, 7, 9] and references therein). Let A be a generating set of a finite
group G. If |G| has any of the following forms (where p, q, and r are distinct primes, and
k is a positive integer), then Cay(G;A) has a hamiltonian cycle:
1. kp, where k ≤ 47,
2. kpq, where k ≤ 7,
3. pqr,
4. kp2, where ≤ 4,
5. kp3, where k ≤ 2.
6. pk.
*I thank an anonymous referee for carefully reading the original version of the paper, and pointing out some
serious deficiencies in the exposition.
E-mail address: dmorris@deductivepress.ca (Dave Witte Morris)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.12
The purpose of this note is to add pqrs to the list when it is odd:
Theorem 1.3. If p, q, r, and s are distinct odd primes, then every connected Cayley graph
of order pqrs has a hamiltonian cycle.
Remark 1.4.
1. To remove Theorem 1.3’s assumption that the primes are odd, it would suffice to
show that every connected Cayley graph of order 2pqr is hamiltonian (where p, q,
and r are distinct odd primes). An important first step in this direction was taken by
F. Maghsoudi [7], who handled the case where one of the primes is 3.
2. Theorem 1.3 implies that if p, q, and r are distinct primes, then every connected
Cayley graph of order 3pqr is hamiltonian. Namely:
• If r = 2, then 3pqr = 6pq, so the main theorem of [7] applies.
• If r = 3, then 3pqr = 9pq, so [11, Corollary 1.5] applies.
• If {p, q, r} ∩ {2, 3} ≠ ∅, then the theorem applies (because 3, p, q, and r are
distinct odd primes).
3. Unfortunately, current methods do not seem to be sufficient to remove the assumption
that p, q, r, and s are distinct. For example, it is not known that all Cayley graphs of
order 9p2 or 3p3 are hamiltonian (cf. [6]).
2 Preliminaries
We use the following fairly standard notation.
Notation 2.1. Let A be a subset of a finite group G.
1. e is the identity element of G,
2. |g| is the order of an element g of G.
3. G′ = [G,G] is the commutator subgroup of G.
4. Z(G) = { z ∈ G | zg = gz for all g ∈ G } is the centre of G.
5. A sequence (ai)
m
i=1 = (a1, a2, . . . , am) of elements of A∪A
−1 represents the walk
in Cay(G;A) that visits the vertices
e, a1, a1a2, . . . , a1a2 · · · am.
Also, we use ak and a−k to represent the sequences
(a, a, . . . , a) and (a−1, a−1, . . . , a−1)
of length k.
Most cases of Theorem 1.3 are easy consequences of three known results that are col-
lected in the following theorem.
Theorem 2.2 ([1, page 257], Durnberger [2], Morris [11]). Let A be a generating set of a
nontrivial finite group G, and assume that |G| is odd. If either
D. W. Morris: On hamiltonian cycles in Cayley graphs of order pqrs 3
(0) |G′| = 1, or
(1) |G′| is prime, or
(2) |G′| is the product of 2 distinct primes,
then Cay(G;A) has a hamiltonian cycle.
Remark 2.3. The assumption that G has odd order is not necessary in parts (0) and (1)
of Theorem 2.2. Steps toward removing this assumption from part (2) were taken in [3]
and [8].
For ease of reference, we record a few other known facts:
Lemma 2.4 (“Factor Group Lemma” [10, §2.2]). Suppose
• A is a subset of the group G,
• N is a cyclic, normal subgroup of G,
• C = (ai)
m
i=1 is a hamiltonian cycle in Cay(G/N ;A), and
• the voltage V(C) =
∏m
i=1 ai generates N .
Then (a1, a2, . . . , am)
|N | is a hamiltonian cycle in Cay(G;A).
Lemma 2.5 ([6, Lemma 2.27]). Let A generate a finite group G and let b ∈ A, such that
the subgroup generated by b is normal in G. If Cay
(
G/⟨b⟩;A
)
has a hamiltonian cycle,
and ⟨b⟩ ∩ Z(G) = {e}, then Cay(G;A) has a hamiltonian cycle.
Lemma 2.6 (cf. [5, Theorem 9.4.3, p. 146] and [7, Lemma 2.16]). Assume G is a finite
group of square-free order. Then
(1) G′ and G/G′ are cyclic,
(2) G′ ∩ Z(G) = {e}, and
(3) if a ∈ G, such that ⟨a,G′⟩ = G, then
(a) a does not commute with any nontrivial element of G′, and
(b) |a| = |G/G′|.
Proposition 2.7 (Maghsoudi [7, Proposition 3.1]). Assume G is a finite group, such that
|G| is a product of four distinct primes. If A is an irredundant generating set of G, such
that |A| ≥ 4, then Cay(G;A) has a hamiltonian cycle.
3 Proof of Theorem 1.3
Let A be an irredundant generating set of a finite group G, such that |G| = pqrs. We wish
to show that Cay(G;A) has a hamiltonian cycle. Although some of the details are new, the
line of argument here is quite standard. (See the proofs in [7], for example.)
Let
G = G/G′.
4 Art Discrete Appl. Math. 5 (2022) #P3.12
Note that G is solvable by Lemma 2.6(1), so G′ ̸= G. Therefore |G′| has at most 3 prime
factors. We may assume it has exactly 3, for otherwise Theorem 2.2 applies. Hence, we
may assume |G′| = qrs (so |G| = p). Since G′ is cyclic (see Lemma 2.6(1)), we have
G′ = Cq × Cr × Cs,
where Cn denotes a (multiplicative) cyclic group of order n. Then each element γ of G
′
can be written uniquely in the form
γ = γq γr γs, where γq ∈ Cq , γr ∈ Cr, and γs ∈ Cs.
Let a ∈ A, such that ⟨a⟩ = G. Then |a| = p by Lemma 2.6(3)(b). Let b be another
element of A, so we may write b = akγ, where γ ∈ G′ (and 0 ≤ k < p).
Case 1. Assume A ∩ G′ ̸= ∅. Then we may assume b ∈ G′. Since G′ is cyclic (see
Lemma 2.6(1)), and it is a standard exercise in undergraduate abstract algebra to show
that every subgroup of a cyclic normal subgroup is normal, we know that ⟨b⟩ ◁ G. Let
Ĝ = G/⟨b⟩. Since |G| = pqrs has only 4 prime factors, we know that |Ĝ| has at most 3
prime factors, so we see from part (1), (2), or (3) of Theorem 1.2 that Cay(Ĝ;A) has a
hamiltonian cycle. Since
⟨b⟩ ∩ Z(G) ⊆ G′ ∩ Z(G) = {e}
(see Lemma 2.6(2)), we can now conclude from Lemma 2.5 that Cay(G;S) has a hamilto-
nian cycle, as desired.
Case 2. Assume |A| = 2 (and A ∩ G′ = ∅). This means that A = {a, b}, so ⟨a, b⟩ = G.
This implies ⟨γ⟩ = G′, so γq , γr, and γs are nontrivial. Also, we have k ≥ 1, since
A ∩G′ = ∅. Let
C = (b, a−(k−1), b, ap−k−1),
so C is a (well known) hamiltonian cycle in Cay(G;A). The voltage of C is
V(C) = ba−(k−1)bap−k−1 = akγ ·a−(k−1) ·akγ ·ap−k−1 = ak ·γqγrγs ·a·γqγrγs ·a
−k−1.
Since |G| is odd, we know that a does not invert any nontrivial element of G′, so the two oc-
currences of γq in this product do not cancel each other, and similarly for the occurrences of
γr and γs. Therefore, this voltage projects nontrivially to Cq , Cr, and Cs, so it generates G
′.
Hence, the Factor Group Lemma 2.4 provides a hamiltonian cycle in Cay(G;A).
Case 3. Assume |A| ≥ 3 (and A∩G′ = ∅). Let c be a third element of A. We may assume
|A| < 4 (for otherwise Proposition 2.7 applies), so A = {a, b, c}. Write c = aℓγ′ (with
γ′ ∈ G′). Since ⟨a, b, c⟩ = G, we must have ⟨γ, γ′⟩ = G′, but the fact that A is irredundant
implies ⟨γ⟩ ̸= G′ and ⟨γ′⟩ ̸= G′. If |γ| = q, then we must have |γ′| = rs. However, this
implies ⟨b, c⟩ = G, which contradicts the fact that the generating set A is irredundant. So
|γ| and |γ′| must each have precisely two prime factors. Therefore, we may assume without
loss of generality that |γ| = qr and |γ′| = rs. Then
γ = γq γr and γ
′ = γ′r γ
′
s,
and each of γq , γr, γ
′
r, and γ
′
s is nontrivial.
D. W. Morris: On hamiltonian cycles in Cayley graphs of order pqrs 5
We may assume k, ℓ ≤ (p − 1)/2 (by replacing some generators with their inverses,
if necessary). We may also assume, without loss of generality, that ℓ ≤ k. (If k ̸= ℓ, this
implies ℓ ≤ (p− 3)/2.) Also note that k, ℓ ≥ 1, since A ∩G′ = ∅.
Subcase 3.1. Assume p = 3. This implies a = b = c, so we have the following two
hamiltonian cycles in Cay(G;A):
C1 = (a, b, c) and C2 = (a, c, b).
Their voltages are
V(C1) = abc = a · aγ · aγ
′ = a2 γq γr a γ
′
r γ
′
s
and
V(C2) = acb = a · aγ
′ · aγ = a2 γ′r γ
′
s a γq γr.
In each of these voltages, there is nothing that could cancel the factor γq or the factor γ
′
s.
So both voltages project nontrivially to Cq and Cs.
Therefore, we may assume that both voltages project trivially to Cr, for otherwise we
have a voltage that projects nontrivially to all three factors, and therefore generates G′, so
the Factor Group Lemma 2.4 applies. Hence
a2γraγ
′
r = V(C1)r = e = V(C2)r = a
2γ′raγr,
so, letting x = γ′r γ
−1
r , we have ax = xa, which means that x commutes with a. However,
we also know from Lemma 2.6(3)(a) that a does not commute with any nontrivial element
of G′. Therefore, we must have x = e, which means γ′r = γr. Also, since a does not invert
any nontrivial element of G′ (since a has odd order), we know that γraγr ̸= a. Therefore
V(C1)r = a
2 · γraγ
′
r = a
2 · γraγr ̸= a
2 · a = e.
This voltage therefore projects nontrivially to all three factors of G′, so it generates G′.
Hence, the Factor Group Lemma 2.4 applies.
Subcase 3.2. Assume a, b, and c are not all distinct. We may assume a = c (which means
ℓ = 1). We may also assume p ≥ 5, for otherwise Subcase 3.1 applies. Therefore
p− k ≥ p− (p− 1)/2 = (p+ 1)/2 ≥ (5 + 1)/2 = 3,
so we have the following two hamiltonian cycles in Cay(G;A):
C1 = (b, a
−(k−1), b, c, ap−k−2) and C2 = (b, a
−(k−1), b, a, c, ap−k−3).
Their voltages are
V(C1) = ba
−(k−1)bcap−k−2
= akγ · a−(k−1) · akγ · aγ′ · ap−k−2
= ak γqγr a γqγr a γ
′
rγ
′
s a
−k−2
6 Art Discrete Appl. Math. 5 (2022) #P3.12
and
V(C2) = ba
−(k−1)bacap−k−3
= akγ · a−(k−1) · akγ · a · aγ′ · ap−k−3
= ak γqγr a γqγr a
2 γ′rγ
′
s a
−k−3.
Since a does not invert γq (because a has odd order), we see that the two occurrences of γq
in these voltages cannot cancel each other. Hence, both voltages project nontrivially to Cq .
Also, there is nothing in either voltage that could cancel the single occurrence of γ′s, so
both voltages also project nontrivially to Cs.
Therefore, in order to apply the Factor Group Lemma 2.4, it suffices to show that at
least one of these voltages projects nontrivially to Cr. If not, then both projections are
trivial, so
akγraγra · γ
′
ra · a
−k−3 = akγraγraγ
′
ra
−k−2
= V(C1)r
= e
= V(C2)r
= akγraγra
2γ′ra
−k−3
= akγraγra · aγ
′
r · a
−k−3,
which implies γ′ra = aγ
′
r. This contradicts the fact that a does not centralize any nontrivial
element of G′ (see Lemma 2.6(3)(a)).
Subcase 3.3. Assume ℓ ̸= (p − 3)/2. We may assume that the preceding case does not
apply. In particular, then b ̸= c, so k ̸= ℓ, so we have ℓ ≤ (p − 5)/2, which implies
k + ℓ ≤ p− 3. Therefore, we have the following two hamiltonian cycles in Cay(G;A):
C1 = (b, a
−(k−1), b, c, a−(ℓ−1), c, ap−k−ℓ−2) (A)
and
C2 = (b, a
−(k−1), b, a, c, a−(ℓ−1), c, ap−k−ℓ−3).
Their voltages are
V(C1) = ba
−(k−1)bca−(ℓ−1)cap−k−ℓ−2
= akγ · a−(k−1) · akγ · aℓγ′ · a−(ℓ−1) · aℓγ′ · ap−k−ℓ−2
= ak · γqγraγqγr · a
ℓ · γ′rγ
′
saγ
′
rγ
′
s · a
p−k−ℓ−2 (B)
and
V(C2) = ba
−(k−1)baca−(ℓ−1)cap−k−ℓ−3
= akγ · a−(k−1) · akγ · a · aℓγ′ · a−(ℓ−1) · aℓγ′ · ap−k−ℓ−3
= ak · γqγraγqγr · a
ℓ+1 · γ′rγ
′
saγ
′
rγ
′
s · a
p−k−ℓ−3.
D. W. Morris: On hamiltonian cycles in Cayley graphs of order pqrs 7
In each product, the two occurrences of γq cannot cancel each other, and the two occur-
rences of γ′s cannot cancel each other (because a does not invert any nontrivial element
of G′), so both voltages project nontrivially to Cq and Cs.
Therefore, in order to apply the Factor Group Lemma 2.4, it suffices to show that at
least one of these voltages projects nontrivially to Cr. If not, then both projections are
trivial, so, much as in Subcase 3.2, we have
akγraγra
ℓ · γ′raγ
′
ra · a
p−k−ℓ−3 = ak · γraγr · a
ℓ · γ′raγ
′
r · a
p−k−ℓ−2
= V(C1)r
= e
= V(C2)r
= ak · γraγr · a
ℓ+1 · γ′raγ
′
r · a
p−k−ℓ−3
= akγraγra
ℓ · aγ′raγ
′
r · a
p−k−ℓ−3,
so γ′raγ
′
ra = aγ
′
raγ
′
r.
Now, note that ⟨γ′r⟩ ◁ G (since every subgroup of the cyclic normal subgroup G
′ is
normal), so there is some γ̂ ∈ ⟨γ′r⟩, such that aγ
′
r = γ̂a. With this notation, the conclusion
of the preceding paragraph tells us that γ′rγ̂a
2 = γ̂a2γ′r. Since γ
′
r and γ̂ are in the abelian
group ⟨γ′r⟩, we know that they commute with each other, so this implies γ
′
ra
2 = a2γ′r.
However, since a generates G, and |G| = p is odd, we know that a2 also generates G.
Hence, we see from Lemma 2.6(3)(a) that a2 does not centralize any nontrivial element
of G′. This contradicts the conclusion of the preceding paragraph.
Subcase 3.4. Assume p = 7. We may assume Subcase 3.2 does not apply, so 1 < ℓ < k ≤
(p− 1)/2. Since (p− 1)/2 = (7− 1)/2 = 3, we conclude that ℓ = 2 and k = 3. Thus, we
have
p = 7, c = a2, and b = a3.
Now, the Cayley graph Cay(G; a, b, c) is the complete graph K7, so it has many hamil-
tonian cycles. In particular, we have the hamiltonian cycle
C = (c, a2, c, a−1, b, a−1).
Since G′ is a cyclic, normal subgroup of G (and |a| = p = 7), there is some α ∈ Z+, such
that α7 ≡ 1 (mod qrs) and
a g = gα a for all g ∈ G′.
Also, we may write γ′r = γ
x
r for some x ∈ Z
+. With this notation, the voltage of C is
V(C) = ca2ca−1ba−1
= a2γ′ · a2 · a2γ′ · a−1 · a3γ · a−1
= a2γ′a4γ′a2γa−1
= (γ′)α
2+α6γα
= (γ′rγ
′
s)
α2+α6(γq γr)
α
= (γxr γ
′
s)
α2+α6(γq γr)
α
= γαq γ
(α2+α6)x+α
r (γ
′
s)
α2+α6 .
8 Art Discrete Appl. Math. 5 (2022) #P3.12
Since α7 ≡ 1 (mod qrs), we know that α7 ≡ 1 (mod q) and α7 ≡ 1 (mod s), so it
is easy to see that α ̸≡ 0 (mod q) and α2 + α6 ̸≡ 0 (mod s). Hence, it is clear that
V(C) projects nontrivially to Cq and Cs. Therefore, if Theorem 2.4 does not apply, then
this voltage projects trivially to Cr, which means
(α2 + α6)x+ α ≡ 0 (mod r). (C)
We also have the hamiltonian cycle
C ′ = (b, a−2, b, c, a−1, c).
It is easy to see that the voltage of this hamiltonian cycle projects nontrivially to Cq and Cs.
(As in Subcase 3.3, this is because the only occurrences of γq come from the two occur-
rences of b, and the only occurrences of γs come from the two occurrences of c. None of
these can cancel each other, because no element of G inverts γq or γs.) Therefore, if V(C
′)
does not generate G′, then this voltage must project trivially to Cr.
Note that the hamiltonian cycle C ′ can be obtained from the hamiltonian cycle C1 of
Equation (A), with p = 7, k = 3, and ℓ = 2, by deleting the final term ap−k−ℓ−2 =
a7−3−2−2 = a0 of C1. Therefore, the formula (B) for the voltage of C1 also yields the
voltage of C ′ (if we delete the irrelevant final term ap−k−ℓ−2 = a0 = e), so we have
e = V(C1)r
= ak · γraγr · a
ℓ · γ′raγ
′
r
= a3 · γraγr · a
2 · γxr aγ
x
r
= γα
3+α4+α6x+α7x
r ,
so
α3 + α4 + α6x+ α7x ≡ 0 (mod r).
Multiplying by α4, and recalling that α7 ≡ 1 (mod r), we conclude that
(1 + α) + α3(1 + α)x ≡ 0 (mod r).
Dividing this equation by 1 + α yields 1 + α3x = 0 (in Zr), so x = −1/α
3. Plugging this
into Equation (C) yields −(α2 + α6)/α3 + α = 0 in Zr, so (multiplying by −α) we have
1 + α4 − α2 ≡ 0 (mod r). Recall that we also know α7 − 1 ≡ 0 (mod r). Since the
polynomial 1 + x4 − x2 is relatively prime to x7 − 1, this is impossible. More concretely,
we have
0 = (α6 − α5 + α4 − α3 − 1) · 0 − (α3 − α2) · 0
≡ (α6 − α5 + α4 − α3 − 1)(1 + α4 − α2)− (α3 − α2)(α7 − 1) (mod r)
= −1,
which is an obvious contradiction.
Subcase 3.5. Assume that the preceding cases do not apply. Since Subcase 3.3 does not
apply, we must have ℓ = (p− 3)/2. Since ℓ < k ≤ (p− 1)/2, this implies
k =
p− 1
2
= 1 +
p− 3
2
= 1 + ℓ.
D. W. Morris: On hamiltonian cycles in Cayley graphs of order pqrs 9
Therefore
a c = a aℓ = a1+ℓ = ak = b.
Putting c into the role of a will usually give us new values for k and ℓ. Indeed, if
Subcase 3.3 does not apply after this change (that is, after replacing the triple (a, b, c) with
the appropriate triple (c, aϵ1 , bϵ2) or (c, bϵ1 , aϵ2), with ϵ1, ϵ2 ∈ {±1}), then either
a±1 = c(p−3)/2 and b
±1
= c(p−1)/2 = a±1 c
or
b
±1
= c(p−3)/2 and a±1 = c(p−1)/2 = b
±1
c.
However, we have
a−1 c = aℓ−1 = a(p−5)/2 /∈ {a±(p−1)/2} = {b
±1
}
and
b c = ak+1 = a(p+1)/2 /∈ {a±1}.
Therefore, c(p−3)/2 must be either a or b
−1
. Noting that (since a c = b) we have b = a c
and a−1 = b
−1
c, this implies that the only possibilities are that either
a = c(p−3)/2 and b = a c = c(p−1)/2 (D)
or
b
−1
= c(p−3)/2 and a−1 = b
−1
c = c(p−1)/2. (E)
If (D) holds, then
a = c(p−3)/2 =
(
aℓ
)(p−3)/2
=
(
a(p−3)/2
)(p−3)/2
= a(p−3)
2/4,
so (p − 3)2/4 ≡ 1 (mod p), so 9 ≡ 4 (mod p), which implies p = 5. However, since
Subcase 3.2 does not apply, we know that c ̸= a, so ℓ > 1. This means (p − 3)/2 ≥ 2, so
p ≥ 7. This contradicts our recent previous conclusion that p = 5.
So (E) must hold. Then b
−1
= c(p−3)/2. Since b = ak = a(p−1)/2 and c = aℓ =
a(p−3)/2, this means
−(p− 1)/2 ≡ (p− 3)2/4 (mod p).
so 2 ≡ 9 (mod p), which implies p = 7. So Subcase 3.4 applies.
References
[1] C. C. Chen and N. F. Quimpo, On strongly Hamiltonian abelian group graphs, in: Combina-
torial Mathematics, VIII (Geelong, 1980), Springer, Berlin-New York, volume 884 of Lecture
Notes in Math., pp. 23–34, 1981, doi:10.1007/bfb0091805.
[2] E. Durnberger, Every connected Cayley graph of a group with prime order commutator group
has a Hamilton cycle, in: Cycles in Graphs (Burnaby, B.C., 1982), North-Holland, Amster-
dam, volume 115 of North-Holland Math. Stud., pp. 75–80, 1985, doi:10.1016/s0304-0208(08)
72997-9.
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[3] E. Ghaderpour and D. W. Morris, Cayley graphs on nilpotent groups with cyclic commutator
subgroup are Hamiltonian, Ars Math. Contemp. 7 (2014), 55–72, doi:10.26493/1855-3974.280.
8d3.
[4] C. Godsil and G. Royle, Algebraic Graph Theory, volume 207 of Graduate Texts in Mathemat-
ics, Springer-Verlag, New York, 2001, doi:10.1007/978-1-4613-0163-9.
[5] M. Hall, Jr., The Theory of Groups, The Macmillan Company, New York, N.Y., 1959, https:
//store.doverpublications.com/0486816907.html.
[6] K. Kutnar, D. Marušič, D. W. Morris, J. Morris and P. Šparl, Hamiltonian cycles in Cayley
graphs whose order has few prime factors, Ars Math. Contemp. 5 (2012), 27–71, doi:10.26493/
1855-3974.177.341.
[7] F. Maghsoudi, Cayley graphs of order 6pq and 7pq are Hamiltonian, Art Discrete Appl. Math.
5 (2022), Paper No. 1.10, 55, doi:10.26493/2590-9770.1389.fa2.
[8] D. W. Morris, Cayley graphs on groups with commutator subgroup of order 2p are hamiltonian,
Art Discrete Appl. Math. 1 (2018), Paper No. 1.04, 31, doi:10.26493/2590-9770.1240.60e.
[9] D. W. Morris and W. K. Cayley graphs of order kp are hamiltonian for k < 48, Art Discrete
Appl. Math. 3 (2020), Paper No. 2.02, doi:10.26493/2590-9770.1250.763.
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(1984), 293–304, doi:10.1016/0012-365x(84)90010-4.
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nian, Ars Math. Contemp. 8 (2015), 1–28, doi:10.26493/1855-3974.330.0e6.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.13
https://doi.org/10.26493/2590-9770.1535.8ad
(Also available at http://adam-journal.eu)
Regular Leonardo polyhedra:
Mathematics and art
Jürgen Bokowski
Department Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7,
D-64289 Darmstadt, Germany
Jörg M. Wills
Department Mathematik, University of Siegen, Emmy-Noether-Campus,
D-57068 Siegen, Germany
Received 22 March 2022, accepted 24 March 2022, published online 12 August 2022
Abstract
Five known combinatorial regular polyhedra with genus g ⩾ 2 and with the rotation
group of a Platonic solid are described, along with their history. They are close (hyperbolic)
analogues of the Platonic solids. The protagonists of their discovery are Leonardo da Vinci
and two mathematicians: Alicia Boole Stott and Harold Scott MacDonald Coxeter. For a
sixth candidate we discovered that there is so far only a Kepler-Poinsot type available. This
is a correction of a result from 1986.
Keywords: Polyhedral manifold, Leonardo polyhedra, regular map, Klein’s quartic
Math. Subj. Class.: 52B70
Leonardo da Vinci
In 1498 Leonardo was invited to illustrate the mathematician’s book De Divina Pro-
portione of Luca Pacioli, [16]. Leonardo gladly accepted this invitation and then spent four
years, of course with interruptions, busy with it. One reason for this was for sure that he
wanted to perfect his knowledge of the central perspective. Another reason may have been
to impress his colleagues and competitors, and certainly also potential customers.
Leonardo’s work proceeded in three phases. First he drew the five Platonic solids and
the 13 Archimedian bodies in the well-known way. In the second and decisive phase, he
E-mail addresses: juergen@bokowski.de (Jürgen Bokowski), wills@mathematik.uni-siegen.de (Jörg M.
Wills)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.13
Figure 1: Drawing by Leonardo da Vinci for the work of Luca Pacioli.
only drew the edge skeleton and replaced the edges with slender struts formed by con-
vex polygons, predominantly by quadrilaterals (vacuus). With this he had created the first
polyhedra (with genus g ⩾ 2).
In the third phase he made the polyhedron even more complicated and on each hole he
placed the edge skeleton of a pyramid (elevatus vacuus). It is known that these construc-
tions have made a great impression on his contemporaries and later generations. The best
known is probably Kepler’s Harmonici Mundi, a nice but wrong explanation of the plane-
tary system via the Platonic solids. For mathematicians, Leonardo’s polyhedrons were of
little interest because they are not regular. The first two combinatorially regular polyhedra
in the style of Leonardo only were found four hundred years later by the mathematician
Alicia Boole Stott. At least now it’s time for some basic definitions.
Definitions
For this work we understand a polyhedron to be a compact topological 2-manifold
without boundary, embedded in the three-dimensional Euclidean space E3 and made up of
convex polygons.
The embedding means that no self-intersections are allowed. The convex polygons of
the cells of the manifold exclude general topological embeddings, maintain a close rela-
tionship with the Platonic solids and form the faces, edges, and vertices of the polyhedron.
A polyhedron with the rotational symmetry of a Platonic solid and with genus g ⩾ 2
is called Leonardo polyhedron. This definition, of course, includes polyhedra with the full
symmetry group of a Platonic solid.
The numbers of vertices, edges, and faces f0, f1, and f2 of a polyhedron are given by
Euler’s polyhedron formula f2 − f1 + f0 = 2 − 2g = χ. Here g is the genus and χ is the
Euler characteristic.
J. Bokowski and J. M. Wills: Regular Leonardo polyhedra: Mathematics and art 3
If all faces of a polyhedron are p-gons and all vertices are q-valent, we call the poly-
hedron locally regular or equivelar. We use the Schläfli symbol {p, q} to describe this. A
triple consisting of a face, an edge of this face, and a vertex of this edge is called a flag.
Local regularity is a rather weak restriction. For example there are infinitely many locally
regular Leonardo polyhedra. So we do not try to survey them all here. We consider the
much stronger condition of regularity.
A polyhedron is called regular if its group of automorphisms acts transitively on its
flags. This elegant definition can be rather complicated to use in practice. In the next
sections we use some equivalent methods to prove regularity.
Figure 2: Regular Leonardo Polyeder of Alicia Boole Stott with symmetrical orthogonal
projections.
4 Art Discrete Appl. Math. 5 (2022) #P3.13
Alicia Boole Stott
Alicia Boole Stott, 1860 – 1940, was the daughter of George Boole, one of the pioneers
of set theory. She received her mathematics education from her parents and grandparents,
since for women in those years studying mathematics was very difficult or completely
impossible.
In 1910 she wrote a paper presenting the first two regular Leonardo polyhedra, [2].
Construction and proof were as simple as they were ingenious: just like Leonardo, she
started from regular convex polytopes, not in dimension 3 but instead in dimension 4.
More precisely, she used projections into the 3-dimensional space, the so-called Schlegel
diagrams. Like Leonardo, she only took the edge skeleton and replaced it with slender
struts bounded by convex quadrilaterals. Furthermore, she examined only the two self-dual
polytopes among the six regular ones in E4, i.e. the 4-simplex and the 24-cell. Obviously
the two resulting polyhedra are locally regular. All of their faces are quadrilaterals, and
q = 6 for the smaller of these, depicted in Figure 3, whereas q = 8 for the larger polyhedron
of Figure 2.
Figure 3: Regular Leonardo polyhedron of Alicia Boole Stott derived from the 4-simplex.
A short YouTube video of the regular Leonardo polyhedron of genus 73 shown in Figure 2
can be seen at https://youtu.be/d3Im8BLxVoM
The proof of regularity is also so simple that we can sketch it here. Consider first the
smaller polyhedron. It has the same symmetry group (of order 24) as the tetrahedron. The
five possible Schlegel diagrams are isomorphic. And because of self-duality, the automor-
phism group of the polyhedron doubles in size. Thus one sees that the automorphism group
of the polyhedron has order 24× 5× 2 = 240, which is exactly the number of flags of the
polyhedron. (Note that only the identity can fix a particular flag.) Likewise, but with larger
numbers, for the other polyhedron, we get a group of order 48× 24× 2 = 2304 and that is
the number of flags. In this way, the first two regular Leonardo polyhedra were found. In
J. Bokowski and J. M. Wills: Regular Leonardo polyhedra: Mathematics and art 5
1914, Alicia Boole Stott received an honorary doctorate from the University of Groningen
for her life’s work. And in 1930 she met H.S.M. Coxeter. She was then 70 years old and
Coxeter a 23 year old college student.
Harold Scott MacDonald Coxeter
H.S.M. Coxeter, 1907 – 2003, was one of the most important geometers of the 20th
century. He published countless works on geometry, algebra, and related fields, he wrote
quite a few books and had many students. For the two Leonardo polyhedra by Alicia
Boole-Stott he found (using standard methods) the two dual polyhedra [5].
For both polyhedra, Coxeter initially specified only 4-dimensional embeddings. How-
ever, the illustration in Figure 4 shows an embedding in 3-dimensional space, which was
given by Schulte and Wills [18] based on Coxeter’s result. In contrast, for the second poly-
hedron it turned out during our recent investigation that a 3-dimensional version through
an obvious construction with a Schlegel diagram leads in the end to self-intersections. A
3-dimensional model without self-intersections has not yet been found.
Figure 4: Coxeter’s regular Leonardo Polyhedron of type {6, 4|3}. On the right two
hexagons have been removed to show the interior.
We describe an idea for a construction of Coxeter’s {8, 4|3} starting with Coxeter’s 4-
dimensional version. We use the edge skeleton of the 24-cell represented within a Schlegel
diagram. We mark the midpoints of the 96 edges. Eight edges are incident to each of the
24 vertices of the 24-cell. We connect their centers appropriately and get an (affine) cube
in each case that contains a vertex in the interior. The 24 cubes touch each other in pairs
at the corners. When we enlarge these cubes simultaneously, and when we leave out the
intersections, we have that polyhedra {8.4|3}. Or do we?
Initially, everything works fine. However, in the end the six outer cubes cause self
intersections. We see from the pictures that a simultaneous magnification of six squares
(those that would cause self-intersections) would not lead to octagons.
6 Art Discrete Appl. Math. 5 (2022) #P3.13
Thus we have so far three regular Leonardo polyhedra. These polyhedra also have the
nice and easily controlled property that all geodesics, i.e., the shortest paths around a hole
or around a strut, are of equal (combinatorial) length, that is, they are all triangles. Coxeter
has already shown that this is sufficient for showing the regularity of the four examples.
This theorem is part of a much more general method, the beginnings of which go back to
Felix Klein (the Eightfold way) and which is used today to characterize regular maps and
polyhedra. This exhausted Leonardo’s approach. And it was almost 50 years before the
next regular polyhedra (with g ⩾ 2) were found.
Figure 5: Futile steps for constructing Coxeter’s regular Leonardo Polyhedron {8, 4|3} in
E3.
New developments
It would be beyond the scope of this article to describe the more recent developments
even halfway completely. We therefore limit ourselves to the essential facts. Starting in
1982, an infinite number of regular polyhedra were discovered, including two mutually
dual infinite series. But only two regular Leonardo polyhedra were among them. All these
regular polyhedra were derived from regular maps (Riemann surfaces, algebraic curves).
Regular maps are the main generalization of the Platonic solids.
The first (and most famous) regular maps were made by the Munich mathematicians
Felix Klein (1879), Walter von Dyck (1880), [7, 8], Adolf Hurwitz (1893), [11], and Robert
Fricke (1890). Throughout the 20th century countless new regular maps were added,
J. Bokowski and J. M. Wills: Regular Leonardo polyhedra: Mathematics and art 7
especially by Coxeter, [6]. In recent decades, Marston Conder, using a globally networked
computer program Magma, has enumerated all regular orientable maps of genus up to
g = 100. For these group theory contributions, see [3, 4].
This provides a huge arsenal of regular maps to generate corresponding regular poly-
hedra. But it can be very difficult to generate a polyhedron embedded in E3 given only a
combinatorial description of a particular regular map.
For a difficult example see [1]. And there are theorems excluding entire classes of
polyhedra, such as the Ringel-Youngs theorem.
Figure 6: Regular Leonardo Polyeder due to Schulte and Wills [17], based on Klein’s
quartric.
Figure 7: Regular Leonardo Polyeder based on Fricke-Klein, by Grünbaum et al., see
[9, 10].
8 Art Discrete Appl. Math. 5 (2022) #P3.13
The only two new regular Leonardo polyhedra are the Klein polyhedron, [12, 13],
{3, 7}8 with g = 3 and the rotation group of the tetrahedron as well as the Fricke-Klein
polyhedron, [14], {3, 8}12 with g = 5 and the rotation group of the octahedron. Both
polyhedra have geodesics of length 4.
Also, both are triangulations, and therefore their duals do not exist by a theorem of
Cauchy. Because the Klein map is so famous, one nevertheless can see the dual as a curved
surface constructed, not as a polyhedron but from Carrara marble, on the Berkeley campus,
[15].
And how does it continue? There are certainly many more regular polyhedra, maybe
infinite series, too. Whether there is a sixth Leonardo polyhedron, or even more, is an open
problem.
There does exist a sixth regular Leonardo polyhedron of type {8, 4|3} in 3-space with-
out self-intersections. This will be explained in detail in a forthcoming paper in the ADAM
journal of these two authors.
ORCID iDs
Jürgen Bokowski https://orcid.org/0000-0003-0616-3372
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14 (1879), 428–471, doi:10.1007/bf01677143.
J. Bokowski and J. M. Wills: Regular Leonardo polyhedra: Mathematics and art 9
[13] F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften
Grade, 1884, http://resolver.sub.uni-goettingen.de/purl?PPN5167
62672.
[14] F. Klein and R. Fricke, Vorlesungen über die Theorie der elliptischen Modulfunktionen, 1890.
[15] S. Levy (ed.), The Eightfold Way. The Beauty of Klein’s Quartic Curve, volume 35 of Math.
Sci. Res. Inst. Publ., Cambridge University Press, Cambridge, 1999, www.msri.org/com
munications/books/Book35/.
[16] L. Pacioli, De Divina Proportione (Disegni di Leonardo da Vinci 1500-1503), 1967.
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.14
https://doi.org/10.26493/2590-9770.1468.a2e
(Also available at http://adam-journal.eu)
On polynomials of small degree over finite fields
representing quadratic residues*
Shaofei Du
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
Kai Yuan†
School of Mathematics and Information Science, Yantai University, Yantai 264005, China
Dedicated to Marston Conder for his 65th birthday
Received 20 September 2021, accepted 13 June 2022, published online 12 August 2022
Abstract
Let F be a finite field of odd characteristic. It was proved by Madden and Vélez in
[Pacific J. Math. 98 (1982), 123–137] that except for finitely many exceptions of F , for
every polynomial f(x) ∈ F [x] which is not of the form αg(x)2 or αxg(x)2, where α ∈
F , there exists a primitive root β ∈ F such that f(β) is a nonzero square in F . When
this theoretical result is concretely used, it is necessary to determine such finitely many
exceptions. In this paper, the possible exceptions are determined, provided the degree of
f(x) is less than 9. Remark that our arguments do not hold when f(x) is of degree 9 and
also the capacity at this stage of a computer is not enough to do it. Moreover, such kinds of
results have been used to solve some combinatorial problems.
Keywords: Finite field, polynomial, quadratic residues.
Math. Subj. Class.: 12E99
1 Introduction
In early eighties, motivated by a question posed by Alspach, Heinrich and Rosenfeld [1]
in the context of decompositions of complete symmetric digraphs, Madden and Vélez [5]
investigated polynomials that represent quadratic residues at primitive roots. They proved
that, with finitely many exceptions, for any finite field F of odd characteristic, for every
*This work is supported by the National Natural Science Foundation of China (12071312, 12101535).
†Corresponding author.
E-mail addresses: dushf@mail.cnu.edu.cn (Shaofei Du), pktide@163.com (Kai Yuan)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.14
polynomial f(x) ∈ F [x] of degree r ≥ 1 which has a nonzero constant term and is not of
the form αg(x)2 or αxg(x)2, there exists a primitive root β such that f(β) is a nonzero
square in F . It is the purpose of this paper to refine their result for polynomials of degree
r where 2 ≤ r ≤ 8 and find such possible exceptions. Moreover, we refine the result in [3]
when F is of prime order and f(x) is of degree 4.
Our main results is the following theorem.
Theorem 1.1. Let F be a finite field of prime power order pn, where pn is odd and not
given in the sets Sr listed in Table 1. Then for every polynomial f(x) ∈ F [x] of degree
r that is not of the form αg(x)2 or αxg(x)2, there exists a primitive root β ∈ F such that
f(β) is a nonzero square in F .
2 Preliminary results
Let J = [q1 = 2, q2, . . . , qm] be an increasing sequence of prime numbers with qi < qj
for 1 ≤ i < j ≤ m. As in [5], we define the following functions with J :
d(u,m) = 2(1− 1
qu
)(1− 1
qu+1
) · · · (1− 1
qm
), (2.1)
cr(u,m) = 2r
√
(
q1q2 · · · qu−1
ququ+1 · · · qm
). (2.2)
Let k(m) be the unique integer such that d(k(m) − 1,m) ≤ 1 < d(k(m),m). Then
k(m) ≥ 2.
The following propositions is our starting point.
Proposition 2.1 ([5, Corollary 1]). Let F be a finite field with pn elements. If s and t are
integers such that
(i) s and t are coprime,
(ii) a prime q divides pn − 1 if and only if q divides st, and
(iii) 2φ(t)/t > 1 + (rs− 2)pn/2/(pn − 1) + (rs+ 2)/(pn − 1),
then, given any polynomial f(x) ∈ F [x] of degree r, square-free and with nonzero constant
term, there exists a primitive root γ ∈ F such that f(γ) is a nonzero square in F .
Proposition 2.2 ([5, Lemma 5]). Let [2 = q1, q2, . . . , qm] be a finite increasing sequence
of primes satisfying m ≤ 2k(m) + 1. Then m ≤ 9 and qk(m)−1 ≤ 5. In fact, the sequence
must satisfy one of the followings:
(i) k(m) = 4, qk(m)−1 = 5 and m = 9,
(ii) k(m) = 3, qk(m)−1 = 5 and m ≤ 7,
(iii) k(m) = 3, qk(m)−1 = 3 and m ≤ 7, or
(iv) k(m) = 2, qk(m)−1 = 2 and m ≤ 5.
S. Du et al.: On polynomials of small degree over finite fields representing quadratic residues 3
Set T = { 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79,
97, 101, 103, 109, 127, 131, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271,
277, 281, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 457, 463, 541, 547,
571, 601, 613, 631, 661, 673, 691, 727, 751, 757, 811, 859, 883, 967, 991, 1009, 1021,
1051, 1093, 1171, 1201, 1231, 1291, 1321, 1381, 1471, 1531, 1951, 2311, 2521, 2731,
3361, 3571, 4621, 4831 }.
Proposition 2.3 ([3]). Let F be a finite field of prime order p, where p is an odd prime not
given in T . Then for every polynomial f(x) ∈ F [x] of degree 4 that has a nonzero constant
term and is not of the form αg(x)2, there exists a primitive root β ∈ F such that f(β) is a
square in F .
3 The proof of Theorem 1.1
In the following lemma, we give a generalization of [5, Lemma 3] when r ≤ 8. Moreover,
Lemma 3.1 plays an indispensable role in programming and computing.
Lemma 3.1. Let Jq = [q1 = 2, q2, q3, . . . , qk(m) = q, qk(m)+1, . . . , qm] be a finite in-
creasing sequence of primes satisfying m ≥ 2k(m) + 2, and let r ≤ 8. Then
d(k(m) + 1,m)− cr(k(m) + 1,m) > 1. (3.1)
Proof. It suffices to show that this is true for r = 8. Since 2 ≤ k(m) ≤ m2 − 1, we have
m ≥ 6. Since
d(k(m) + 1,m) = (1 +
1
qk(m) − 1
)d(k(m),m) > 1 +
1
qk(m) − 1
,
(3.1) holds if
1 +
1
qk(m) − 1
− 2r( q1q2 · · · qk(m)
qk(m)+1qk(m)+2 · · · qm
)
1
2
> 1,
which may be rewritten in the following form
q2q3 · · · qk(m)(qk(m) − 1)2 <
1
512
qk(m)+1 · · · qm−1qm, (3.2)
in view of the fact that r = 8 and q1 = 2.
We divide the proof into two cases, depending on whether (k(m),m) ∈ T = {(2, 6),
(3, 8), (4, 10), (5, 12)} or not.
CASE 1. (k(m),m) ̸∈ T .
We shall in fact prove a more general result:
q2q3 · · · ql(ql − 1)2 <
1
512
ql+1 · · · qm−1qm,
where m ≥ 7, (l,m) ̸∈ T and l ≤ m2 − 1 is any integer. To show this, let Ω be the
increasing sequence of all prime numbers. For the above prime ql, let Iql = [w1 =
2, w2, w3, . . . , wl = ql, wl+1, . . . , wm] be a subsequence of Ω not missing any prime in
Ω from w2 to wm. Then we may get the desired result if the following (3.3) holds:
4 Art Discrete Appl. Math. 5 (2022) #P3.14
w2w3 · · ·wl(wl − 1)2 <
1
512
wl+1 · · ·wm−1wm, (3.3)
where m ≥ 7, (l,m) ̸∈ T and l ≤ m2 − 1 is any integer. If wm ≥ 512, then (3.3) is clearly
true. So we only need to consider primes that are smaller than 512. Note that m ≥ 2l + 2,
it suffices to show that (3.3) is true for (l,m) ∈ {(2, 7), (6, 14)}.
A computer search shows that (3.3) holds for all primes wm ≤ 512 with (l,m) ∈
{(2, 7), (6, 14)}. Suppose now that (3.3) is true for (l,m). Then we have
w2w3w4 · · ·wlwl+1(wl+1 − 1)2 = w2(w3 · · ·wlwl+1(wl+1 − 1)2)
< w2(wl+2wl+3 · · ·wmwm+1)
< (wl+2wl+3 · · ·wmwm+1)wm+2.
Therefore, (3.3) is true for (l,m) ̸∈ T with m ≥ 2l + 2. Hence (3.2) holds, and so
does (3.1).
CASE 2. (k(m),m) ∈ T .
Let Ω be the increasing sequence of all prime numbers. Let Iq = [w1 = 2, w2, w3, . . . ,
wk(m) = q, wk(m)+1, . . . , wm] be a subsequence of Ω not missing any prime in Ω from
w2 to wm. Also, let d(u,m)
′ and cr(u,m)
′ be the corresponding values for Iq as defined
by functions d and cr in (2.1) and (2.2). Then one can easily see that d(k(m) + 1,m)
′ ≤
d(k(m) + 1,m) and that cr(k(m) + 1,m)
′ ≥ cr(k(m) + 1,m), and so (3.1) holds for Jq
if it holds for Iq .
From (3.3), we assume wm ≤ 512. MAGMA[2] searching for all the primes less than
512 shows that (3.1) holds for Iq with (k(m),m) ∈ T . This completes the proof of
Lemma 3.1. □
Remark 3.2. For r = 9, (3.1) may not hold. For example, let p = 180181. Then p is
prime and the increasing sequence of prime divisors of p − 1 = 4 · 9 · 5 · 7 · 11 · 13 is
J = [2, 3, 5, 7, 11, 13]. For k(m) = 2, (3.1) does not hold for J .
Let sp(r) denote the smallest prime number that is greater than 8r2. Then sp(2) =
37, sp(3) = 73, sp(4) = 131, sp(5) = 211, sp(6) = 293, sp(7) = 397 and sp(8) = 521.
With a similar proof to [5, Lemma 7], we have the following lemma.
Lemma 3.3. Suppose r ∈ {3, 4, 5, 6, 7, 8} and p is an odd prime. Let [2 = q1, q2, . . . , qm]
be a finite increasing sequence of primes satisfying m ≤ 2k(m) + 1, and let pn − 1 =
qi11 q
i2
2 · · · qimm with qm ≥ sp(r). Then there exist s and t such that
(i) s and t are coprime,
(ii) a prime q divides pn − 1 if and only if q divides st, and
(iii) 2φ(t)/t > 1 + (rs− 2)√pn/(pn − 1) + (rs+ 2)/(pn − 1).
In order to proceed with the proof of Theorem 1.1 we now need to identify all those
increasing sequences [2 = q1, q2, . . . , qm] with qm < 8r
2 for which one cannot choose
s = q1q2 · · · qu and t = qu+1qu+2 · · · qm so as to satisfy Lemma 3.3(iii). Since Lemma 3.1
holds for each qm, we can assume that for each of these sequences Proposition 2.2 applies.
With the help of MAGMA[2], we have the following proposition.
S. Du et al.: On polynomials of small degree over finite fields representing quadratic residues 5
Proposition 3.4. Let [2 = q1, q2, . . . , qm] be a finite increasing sequence of primes sat-
isfying m ≤ 2k(m) + 1, and let pn − 1 = qi11 qi22 · · · qimm with qm < 8r2 where r ∈
{2, 3, 4, 5, 6, 7, 8}. If pn is not listed in the set
S′2 = {3, 5, 7, 9, 11, 13, 19, 25, 31, 37, 43, 49, 61, 67, 79, 103, 121, 127, 151, 169,
181, 211, 241, 271, 331, 421, 631}
or Sr where 3 ≤ r ≤ 8 in Table 1, then there exist s and t such that
(i) s and t are coprime,
(ii) a prime q divides pn − 1 if and only if q divides st, and
(iii) 2φ(t)/t > 1 + (rs− 2)√pn/(pn − 1) + (rs+ 2)/(pn − 1).
Remark 3.5. Let S′4 = {9, 25, 27, 49, 81, 121, 169, 289, 343, 361, 529, 625, 841, 961,
1681}. For r = 4, the numbers in S′4 were missed to discuss in [3], but their result is still
true.
We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1. We may assume without loss of generality that f(x) is square free,
since leaving out a square factor does not affect the validity of the conclusion. It follows by
Proposition 2.1 that a polynomial f(x) represents a nonzero square at some primitive root
in F if there exist s and t satisfying the following three conditions:
(i) s and t are coprime,
(ii) a prime q divides pn − 1 if and only if q divides st, and
(iii) 2φ(t)/t > 1 + (rs− 2)√pn/(pn − 1) + (rs+ 2)/(pn − 1).
Now, we show that such s and t exist for all odd primes p that are not listed in the set Sr.
Let [q1 = 2, q2, . . . , qm] be an increasing sequence of prime divisors of p
n − 1. If
m ≤ 2k(m) + 1 then Lemma 3.3 applies for qm ≥ 8r2, and Proposition 3.4 applies for
qm < 8r
2.
Suppose now that m ≥ 2k(m) + 2. Then, by Lemma 3.1, we have
d(k(m) + 1,m) > 1 + cr(k(m) + 1,m).
Let s = q1q2 · · · qk(m) and t = qk(m)+1 · · · qm. Then we have 2φ(t)/t = d(k(m) + 1,m),
and
cr(k(m) + 1,m) = 2r ·
√
q1q2 · · · qk(m)
qk(m)+1qk(m)+2 · · · qm
=
2rs√
q1q2 · · · qm
≥ 2rs√
pn − 1 .
Since s is even and r(p− 1) ≥ rs ≥ 3, we may apply [5, Lemma 6] to see that
(rs− 2)√p
p− 1 ≤
rs√
p− 1 .
6 Art Discrete Appl. Math. 5 (2022) #P3.14
It follows that
2φ(t)
t
= d(k(m) + 1,m) ≥ 1 + cr(k(m) + 1,m) ≥ 1 +
2rs√
pn − 1
≥ 1 + (rs− 2)
√
pn
pn − 1 +
rs+ 2
pn − 1 .
(Note that the last inequality holds since pn ≥ 7.) □
Remark 3.6. For the case r = 2, the number 103 was missed to discuss in [5, Theorem 3],
but the result is still true. With the help of a computer, S′2 may be restricted to S2, see
below.
Remark 3.7. We are sure that Theorem 1.1 has lots of applications. For instance, in a
paper of Kutnar, Marusic and the first author [4], the case for r = 4 in Theorem 1.1 (with
an independent proof) is used to prove the existence of Hamilton cycles in vertex-transitive
graphs of order a product of two primes.
Table 1: The Sets Sr .
Set Degree orders of the finite fields
S2 2 3, 5, 7, 9, 11, 13, 19, 25, 31
S3 3 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 49, 61, 67, 71, 73, 79,
97, 103, 109, 121, 127, 139, 151, 157, 163, 169, 181, 199, 211, 223, 229, 241,
271, 307, 313, 331, 337, 361, 379, 397, 421, 463, 541, 547, 571, 601, 631, 661,
691, 751, 841, 991, 1021, 1051, 1171, 1321, 1471, 1681, 2311, 2731
S4 4 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61,
67, 71, 73, 79, 81, 97, 101, 103, 109, 121, 127, 131, 139, 151, 157, 163, 169,
181, 193, 199, 211, 223, 229, 241, 271, 277, 281, 283, 289, 307, 313, 331,
337, 343, 349, 361, 367, 373, 379, 397, 409, 421, 457, 463, 529, 541, 547, 571,
601, 613, 625, 631, 661, 673, 691, 727, 751, 757, 811, 841, 859, 883, 961, 967,
991, 1009, 1021, 1051, 1093, 1171, 1201, 1231, 1291, 1321, 1381, 1471, 1531,
1681, 1951, 2311, 2521, 2731, 3361, 3571, 4621, 4831
S5 5 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71,
73, 79, 81, 83, 89, 97, 101, 103, 109, 113, 121, 127, 131, 139, 151, 157, 163,
169, 181, 191, 193, 199, 211, 223, 229, 241, 271, 277, 281, 283, 289, 307, 313,
331, 337, 343, 349, 361, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487,
491, 499, 523, 529, 541, 547, 571, 577, 601, 607, 613, 619, 625, 631, 643, 661,
673, 691, 709, 727, 733, 739, 751, 757, 811, 829, 841, 859, 883, 911, 919, 937,
961, 967, 991, 1009, 1021, 1051, 1093, 1123, 1171, 1201, 1231, 1291, 1303,
1321, 1327, 1381, 1429, 1471, 1483, 1531, 1597, 1621, 1681, 1723, 1741,
1801, 1831, 1849, 1861, 1933, 1951, 2011, 2131, 2143, 2161, 2221, 2251,
2281, 2311, 2341, 2371, 2521, 2551, 2731, 2851, 2971, 3061, 3121, 3301,
3361, 3511, 3571, 3631, 4201, 4621, 4831, 5041, 6091, 8191, 9241, 11551
S. Du et al.: On polynomials of small degree over finite fields representing quadratic residues 7
S6 6 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67,
71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137,
139, 151, 157, 163, 169, 181, 191, 193, 197, 199, 211, 223, 229, 239, 241, 251,
271, 277, 281, 283, 289, 307, 311, 313, 331, 337, 343, 349, 361, 367, 373, 379,
397, 409, 421, 433, 439, 457, 463, 487, 491, 499, 523, 529, 541, 547, 571, 577,
601, 607, 613, 619, 625, 631, 643, 661, 673, 691, 701, 709, 727, 733, 739, 751,
757, 769, 787, 811, 823, 829, 841, 853, 859, 877, 883, 907, 911, 919, 937, 961,
967, 991, 997, 1009, 1021, 1033, 1051, 1063, 1069, 1093, 1117, 1123, 1129,
1171, 1201, 1231, 1249, 1291, 1303, 1321, 1327, 1369, 1381, 1429, 1453,
1471, 1483, 1531, 1597, 1621, 1681, 1723, 1741, 1783, 1801, 1831, 1849,
1861, 1933, 1951, 2011, 2017, 2131, 2143, 2161, 2221, 2251, 2269, 2281,
2311, 2341, 2371, 2401, 2437, 2521, 2551, 2671, 2731, 2791, 2851, 2857,
2971, 3001, 3061, 3121, 3181, 3271, 3301, 3331, 3361, 3391, 3481, 3511,
3541, 3571, 3631, 3691, 3697, 3721, 4201, 4591, 4621, 4831, 4951, 5041,
5281, 5881, 6007, 6091, 6271, 6301, 7351, 7411, 7561, 7591, 8191, 8581,
9241, 9661, 9871, 10711, 11551, 11971, 16381, 18481
S7 7 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61,
67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131,
137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199,
211, 223, 229, 239, 241, 243, 251, 271, 277, 281, 283, 289, 307, 311, 313,
331, 337, 343, 349, 361, 367, 373, 379, 397, 401, 409, 421, 431, 433, 439,
457, 461, 463, 487, 491, 499, 521, 523, 529, 541, 547, 571, 577, 601, 607,
613, 619, 625, 631, 643, 661, 673, 691, 701, 709, 727, 733, 739, 751, 757, 769,
787, 811, 823, 829, 841, 853, 859, 877, 883, 907, 911, 919, 937, 961, 967, 991,
997, 1009, 1021, 1033, 1039, 1051, 1063, 1069, 1087, 1093, 1117, 1123, 1129,
1153, 1171, 1201, 1213, 1231, 1237, 1249, 1279, 1291, 1297, 1303, 1321,
1327, 1331, 1369, 1381, 1399, 1423, 1429, 1453, 1471, 1483, 1489, 1531,
1549, 1567, 1597, 1621, 1657, 1681, 1723, 1741, 1783, 1801, 1831, 1849,
1861, 1873, 1933, 1951, 2011, 2017, 2029, 2053, 2113, 2131, 2143, 2161,
2179, 2221, 2251, 2269, 2281, 2311, 2341, 2347, 2371, 2377, 2401, 2437,
2521, 2551, 2647, 2671, 2689, 2707, 2731, 2791, 2851, 2857, 2971, 3001,
3037, 3061, 3067, 3109, 3121, 3181, 3271, 3301, 3319, 3331, 3361, 3391,
3433, 3481, 3511, 3541, 3571, 3613, 3631, 3691, 3697, 3721, 3823, 3931,
4021, 4051, 4111, 4159, 4201, 4231, 4261, 4441, 4561, 4591, 4621, 4651,
4831, 4951, 5041, 5101, 5281, 5521, 5701, 5851, 5881, 6007, 6091, 6121,
6241, 6271, 6301, 7351, 7411, 7561, 7591, 7921, 8191, 8581, 8821, 8971,
9241, 9661, 9871, 10501, 10711, 11131, 11551, 11971, 12391, 14281, 16381,
17851, 18481, 21841, 25411, 43891
8 Art Discrete Appl. Math. 5 (2022) #P3.14
S8 8 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61,
67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131,
137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211,
223, 227, 229, 233, 239, 241, 243, 251, 271, 277, 281, 283, 289, 307, 311, 313,
331, 337, 343, 349, 361, 367, 373, 379, 397, 401, 409, 419, 421, 431, 433, 439,
457, 461, 463, 487, 491, 499, 521, 523, 529, 541, 547, 571, 577, 601, 607, 613,
617, 619, 625, 631, 643, 661, 673, 691, 701, 709, 727, 733, 739, 751, 757, 761,
769, 787, 811, 823, 829, 841, 853, 859, 877, 881, 883, 907, 911, 919, 937, 961,
967, 991, 997, 1009, 1021, 1033, 1039, 1051, 1063, 1069, 1087, 1093, 1117,
1123, 1129, 1153, 1171, 1201, 1213, 1231, 1237, 1249, 1279, 1291, 1297,
1303, 1321, 1327, 1331, 1369, 1381, 1399, 1423, 1429, 1447, 1453, 1459,
1471, 1483, 1489, 1531, 1543, 1549, 1567, 1579, 1597, 1609, 1621, 1627,
1657, 1663, 1669, 1681, 1693, 1699, 1723, 1741, 1747, 1753, 1759, 1777,
1783, 1789, 1801, 1831, 1849, 1861, 1867, 1873, 1879, 1933, 1951, 1993,
1999, 2011, 2017, 2029, 2053, 2089, 2113, 2131, 2143, 2161, 2179, 2209,
2221, 2251, 2269, 2281, 2311, 2341, 2347, 2371, 2377, 2381, 2401, 2437,
2521, 2551, 2647, 2671, 2689, 2707, 2731, 2791, 2809, 2851, 2857, 2887,
2971, 3001, 3037, 3061, 3067, 3109, 3121, 3163, 3169, 3181, 3271, 3301,
3319, 3331, 3361, 3391, 3433, 3481, 3499, 3511, 3529, 3541, 3571, 3613,
3631, 3691, 3697, 3721, 3739, 3823, 3907, 3931, 4021, 4051, 4111, 4159,
4201, 4231, 4243, 4261, 4327, 4441, 4561, 4591, 4621, 4651, 4663, 4789,
4801, 4831, 4861, 4951, 4999, 5011, 5041, 5101, 5281, 5431, 5521, 5581,
5641, 5701, 5791, 5821, 5851, 5881, 6007, 6091, 6121, 6151, 6211, 6241,
6271, 6301, 6361, 6451, 6469, 6661, 6841, 6961, 7351, 7411, 7561, 7591,
7921, 8161, 8191, 8581, 8779, 8821, 8971, 9241, 9661, 9871, 9901, 10501,
10711, 11131, 11311, 11551, 11971, 12211, 12391, 12541, 12601, 13441,
14071, 14281, 15121, 15331, 15541, 16381, 17161, 17851, 18061, 18481,
19321, 19531, 21841, 24571, 25411, 32341, 34651, 43891, 46411, 51871
ORCID iDs
Shaofei Du https://orcid.org/0000-0001-6725-9293
Kai Yuan https://orcid.org/0000-0003-1858-3083
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.15
https://doi.org/10.26493/2590-9770.1395.a37
(Also available at http://adam-journal.eu)
Finite and infinite vertex-transitive cubic graphs
and their distinguishing cost and density*
Wilfried Imrich†
Montanuniversität Leoben, 8700 Leoben, Austria
Thomas Lachmann
Institut für Finanzmathematik und Angewandte Zahlentheorie,
Johannes Kepler Universität Linz, 4040 Linz, Austria
Thomas W. Tucker
Colgate University, Hamilton NY 13346, USA
Gundelinde M. Wiegel
Institut für Mathematische Strukturtheorie, Technische Universität Graz,
8010 Graz, Austria
Dedicated to Marston Conder on the occasion of his 65th birthday.
Received 3 December 2020, accepted 20 May 2022, published online 30 August 2022
Abstract
A set S of vertices in a graphGwith nontrivial automorphism group is distinguishing if
the identity mapping is the only automorphism that preserves S as a set. If such sets exist,
then their minimum cardinality is the distinguishing cost ρ(G) of G. A closely related
concept is the distinguishing density δ(G). For finite G it is the quotient of ρ(G) by the
order of G.
We consider connected, vertex-transitive, cubic graphs G and show that either ρ(G) ≤
5 or ρ(G) = ∞ and δ(G) = 0 if G has one or three arc-orbits, or two arc-orbits and
vertex-stabilisers of order at most 2.
For the case of two arc-orbits and vertex stabilizers of order > 2 we show the existence
of finite graphs with ρ(G) > 5 and infinite graphs with δ(G) > 0.
*We are deeply indebted to the reviewers for their insightful comments that led to numerous expository and
mathematical changes. To one reviewer with a particularly detailed report we also owe a major mathematical
correction. Likewise we thank Gabriel Verret for having drawn our attention to Split Praeger–Xu graphs, and the
Austrian Science Fund for support of the second and fourth author (FWF: F5505-N26 and Y-901 for the second
and FWF: W1230 for the fourth).
†Corresponding author.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.15
We also prove that two well known results about finite, vertex-transitive, cubic graphs
hold without the finiteness condition and construct infinitely many cubic GRRs.
Keywords: Distinguishing number, distinguishing cost, vertex-transitive cubic graphs, automorphisms,
infinite graphs, asymmetrizing set.
Math. Subj. Class.: 05C15, 05C10, 05C25, 05C63, 03E10.
1 Introduction
This paper is concerned with automorphism breaking of graphs by vertex colorings. Fol-
lowing Albertson and Collins [1], we call a vertex coloring of a graph G distinguishing if
the identity is the only automorphism of G that preserves it. The smallest number of colors
needed is the distinguishing number D(G) of G. One says such a coloring breaks the au-
tomorphisms of G. When D(G) = 2 each of the two colors induces a set of vertices which
is preserved only by the identity automorphism. We call such sets distinguishing, but also
use the term asymmetrizing, which was introduced in 1977 by Babai [2], when referring to
results predating Albertson and Collins’ paper [1].
The cardinality of a smallest distinguishing set of a graphG is the 2-distinguishing cost.
It was introduced by Boutin [4] in 2008 and denoted ρ(G). Clearly 0 < ρ(G) ≤ |V (G)|/2.
Although we cannot talk of the cost unless we already know that D(G) = 2, when it is
clear from the context, we refer to ρ(G) as the distinguishing cost, or simply as the cost,
without adding that D(G) = 2.
For 2-distinguishable graphs G the distinguishing density δ(G), or simply the density,
is defined in Section 3. For finite G it is the quotient ρ(G)/|V (G)|.
Note that D(G) is 1 for asymmetric graphs and 2 for almost all other finite graphs,
because almost all finite graphs that are not asymmetric have just one automorphism, a
transposition of two vertices. Such graphs can be distinguished by coloring one of the two
vertices black, and all other vertices white. This means that ρ(G) = 1 for almost all finite
graphs that are not asymmetric.
Here we investigate the distinguishing cost and density of graphs of maximum valence
3, which we call subcubic. This is a rich class of graphs that is fully classified with respect
to the distinguishing number, see [16], but little is known about its cost and density.
In Section 2 we construct finite, connected, subcubic graphs, whose costs take on nu-
merous values between 1 and |V (G)|/2, and in Section 3 infinite, connected graphs of
maximum valence 3 with various densities between 0 and 1/2. With the exception of the
obvious bounds 1 ≤ ρ ≤ |V (G)|/2 for the cost, and 0 ≤ δ ≤ 1/2 for the density, there
seem to be few restrictions on the values.
Beginning with Section 4 we focus on connected, vertex-transitive, cubic graphs. For
them the situation is completely different and the main topic of interest. Hence, after Sec-
tion 3 all graphs will be connected, vertex-transitive and cubic, unless it is otherwise stated
or clear from the context.
For their investigation we classify them by the number of arc-orbits and treat the classes
separately. Arcs are ordered pairs (u, v) of adjacent vertices u, v in G, and the orbit of an
E-mail addresses: imrich@unileoben.ac.at (Wilfried Imrich), thomas.lachmann@jku.at (Thomas
Lachmann), ttucker@colgate.edu (Thomas W. Tucker), wiegel@math.tugraz.at (Gundelinde M. Wiegel)
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 3
arc (u, v) under the action of Aut(G) is the set
O((u, v)) = {(α(u), α(v)) |α ∈ Aut(G)}.
Recall that to any two vertices u,w of a vertex-transitive graphs there is an α ∈ Aut(G)
such that w = α(u). Therefore any arc of the form (w, z) is in the orbit of (u, α−1(z)).
If G is cubic, then there are only three possibilities for α−1(z), and thus the number of
arc-orbits in a vertex-transitive, cubic graph is 1, 2 or 3.
A closely related, but weaker concept, is that of edge-orbits. One says two edges uv and
xy are in the same edge-orbit if there exists an automorphism α that maps the unordered
pair {u, v} into the unordered pair {x, y}. The relationship between arc- and edge-orbits is
described in Section 4.
Graphs with only one arc-orbit are called arc-transitive. Four of them are not 2-
distinguishable. All other arc-transitive graphsG are 2-distinguishable with cost ≤ 5 unless
G is the infinite 3-valent tree, which has infinite cost and zero density; see Theorem 6.1. If
G has three arc-orbits, then each automorphism of G that fixes a vertex is the identity, so
ρ(G) = 1.
For graphs with two arc-orbits we will show in Corollary 4.2 that one of the orbits
consists of pairs (u, v) and (v, u), where the edges uv are independent and meet each vertex
of G. Because complete matchings of a graph G are defined as sets of independent edges
that meet each vertex of G, we call this orbit the matching orbit and its edges matching
edges.
If one removes the matching edges, but not their endpoints from G, then the vertices in
the remaining graph have valence 2. This implies that its connected components are cycles
or 2-sided infinite paths. Because G is vertex-transitive, all components must be either
2-sided infinite paths or cycles of the same lengths. We call this orbit the cycle-orbit.
Let uv be a matching edge in a graph with two arc-orbits. Then u, v have no common
neighbors unless G is the K4, which is not 2-distinguishable. If x, y are the neighbors of
u different from v, and w, z the neighbors of v different from u, then Figure 1 depicts the
subgraph of G consisting of the edges ux, yx,uv,vw and vz. It need not be induced unless
G has neither 3- nor 4-cycles. If there are 3-cycles, then they must be in the cycle-orbit
and then both uxy and vwz are 3-cycles. If there are 4-cycles, then there may be a pair of
independent edges between the sets {x, y} and {w, z}. It is also possible that u and v are
in the same cycle of the cycle-orbit.
x
y
u v
w
z
Figure 1: Edges incident with a matching edge uv in a graph with two arc-orbits.
The pairs of arcs (u, x), (u, y) and (v, w), (v, z) are in the cycle-orbit. Hence there
exists an α ∈ Aut(G) that fixes u and interchanges x, y, and a β ∈ Aut(G) that fixes
v and interchanges w, z. If each automorphism that fixes u and interchanges x, y also
interchanges w, z, then we callG rigid, and otherwise flexible. Note thatG is rigid exactly
when the order of the group induced by Aut(G) on the subgraph formed by a matching
edge and its four incident edges is 2.
4 Art Discrete Appl. Math. 5 (2022) #P3.15
Other important concepts are the girth and the motion of a graph G. The girth g(G) is
the size of a smallest cycle of G, and the motion m(G) is the smallest number of vertices
moved by any non-trivial automorphism of G.
Our results pertaining to cost and density and some of the ensuing questions can be
summarized as follows:
Let G be a connected, vertex-transitive, cubic graph different from K4, K3,3, the cube
and the Petersen graph.
1. If G has three arc-orbits, then G has trivial vertex stabilizers and ρ(G) = 1.
2. If G is arc-transitive, then ρ(G) ≤ 5, except for the infinite cubic tree, which has
infinite distinguishing cost and density 0. See Theorem 6.1.
For the existence of certain infinite, arc-transitive cubic graphs of girth 6 see Ques-
tion 6.9.
3. If G is rigid with two arc-orbits, then ρ(G) ≤ 3, see Theorem 7.1.
4. If G is finite, flexible with two arc-orbits and of girth 3, then ρ(G) ≤ 3. See Theo-
rem 8.1.
For the existence of such graphs compare Proposition 8.2.
5. For two arc-orbit, flexible graphs G of girth 4 we construct families of connected
finite graphs with cost > 5 and infinite graphs of density δ(G) = 1/m(G) > 0.
For both families we have δ(G) ≤ 1/4. See Theorems 9.10 and 9.11.
We do not know whether other connected, two arc-orbit, flexible graphs G of girth
≥ 4 with positive density or finite cost > 5 exist. See Question 9.12 and Lemma 9.3.
If they do not exist, then all finite or infinite, vertex-transitive, cubic graphs on at
least 20 vertices have density ≤ 1/4. See Questions 2.1 and 3.5.
6. We also construct infinite, two arc-orbit, flexible, cubic graphs G of all girths with
ρ(G) = ∞ and δ(G) = 0. See Theorem 7.6 and Lemma 9.3.
We do not know whether two arc-orbit, flexible, cubic graphs G, whose cycle-orbits
consist of 2-sided infinite paths, exist. See Question 7.5.
Furthermore, we prove a number of other results. In Section 2 we provide examples
of infinite graphs with finite and infinite distinguishing cost. In Section 3 we define and
give examples of the distinguishing density of infinite graphs. In these sections the graphs
are not necessarily cubic or vertex-transitive. In Section 4 we show that edge-transitivity
implies arc-transitivity not only for finite, but also for infinite, vertex-transitive, cubic
graphs. Section 5 treats vertex-transitive, cubic graphs with three arc-orbits, that is, vertex-
transitive graphs with trivial vertex stabilizers. Such graphs are called Graphical Regular
Representations, GRRs for short. In Section 6 Theorem 6.2 extends a result about the ex-
istence of connected, vertex-transitive, cubic graphs of girth at most 5 to infinite graphs.
Section 9 is concerned with flexible cubic graphs of girth 4. As a byproduct we construct
a class of infinitely many cubic GRRs with two edge-orbits. The paper ends with a new
characterization of Split Praeger–Xu graphs.
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 5
We conclude the introduction with the remark that Babai’s paper [2] was not the only
one on automorphism breaking that predates Albertson and Collins [1]. After Babai, it was
notably Polat and Sabidussi [24, 25] and Polat [22, 23], who published deep and interesting
results on the asymmetrization of graphs of any cardinality.
Moreover, the concept of asymmetrization, or distinguishing, extends to groups of per-
mutations on a set. As most results on asymmetrizing sets of groups are also relevant for
graphs, let us mention that Gluck [13] showed in 1983 that permutation groups of odd or-
der can be asymmetrized by two colors, and that Cameron, Neumann, and Saxl [6] proved
in 1984 that all but finitely many primitive permutation groups other than An, Sn can be
asymmetrized by two colors. In 1997 Seress [28] classified the remaining ones, taking
recourse to the classification of finite simple groups.
2 Cost
We continue with examples on bounds on the distinguishing cost for finite graphs and show
that infinite graphs can have finite or infinite cost.
It is important to keep in mind that the distinguishing cost is only defined for graphs
that are 2-distinguishable. In other words, the results on cost and density only hold for such
graphs. For cubic graphs this is not a strong restriction, because the only vertex-transitive
cubic graphs that are not 2-distinguishable are the K4, the K3,3, the cube and the Petersen
graph, where the K4 has distinguishing number 4, and the others 3.
If the distinguishing number of a graph G is 2, then V (G) can be partitioned into
two sets V1, V2, where the stabilizer of either one is the trivial subgroup. This means, if
α ∈ Aut(G) and α(Vi) = Vi for i = 1 or i = 2, then α = id. Recall from the introduction
that either of the sets V1 or V2 is called distinguishing, and that the smallest possible size
of such a set is the distinguishing cost ρ(G) of G.
To fix ideas we shall always choose a minimum distinguishing set of G, color its ver-
tices black and all others white. Thus ρ(G) is the minimal number of black vertices needed
to break all automorphisms.
Clearly 0 < ρ(G) ≤ |V (G)|/2. The upper bound can easily be attained. The simplest
example is a single edge. To construct larger graphs with ρ(G) =
⌊
|V (G)|
2
⌋
we consider
binary trees:
Let Bk be the binary tree of height k. Bk is defined as a rooted tree, whose root has
valence 2, whose vertices of distance k from the root are leaves, and where all other vertices
have valence 3. Bk has 2
k+1 − 1 vertices, ρ(Bk) = 2
k − 1 and δ(Bk) = 1/2− 1/(2
k+1 −
1) = ⌊|V (G)|/2⌋. By attaching a path to the root one obtains a graph with smaller density.
It can be made arbitrarily small.
For cost |V (G)|/2 we consider the graph Gk consisting of two binary trees Bk, whose
roots are connected by an edge. Clearly ρ(Gk) = 2
k+1 − 1 = |V (Gk)|/2.
If the graphs are vertex-transitive, then we have stricter bounds. In Theorem 6.1 we
show that ρ(G) ≤ 5 for finite, connected, arc-transitive, cubic graphs that are different
from K4, K3,3, the cube and the Petersen graph. Hence, for finite arc-transitive graphs G
on at least 20 vertices δ(G) ≤ 1/4.
There are 24 connected, vertex-transitive graphs of order < 20, see [27]. As a brief
application of results of the paper we show that exactly five of these 24 graphs have den-
sity > 1/4: two are Möbius ladders, two circular ladders, and the fifth is the Heawood
graph. A k-Möbius ladder consists of a cycle of length 2k, together with k diagonals
that connect opposite vertices of the cycle. The k diagonals are the rungs of the ladder.
6 Art Discrete Appl. Math. 5 (2022) #P3.15
A circular ladder or k-ladder is a prism over a k-cycle, that is, the Cartesian product of Ck
by K2. For k ≤ 9 they have < 20 vertices. If they are 2-distinguishable, their cost is 3 by
Theorem 7.1, and thus the density is 3/(2k). Since we are interested in density > 1/4 we
can restrict attention to k ≤ 5.
The 2-Möbius ladder is the K4, the 3-Möbius ladder the K3,3, the 4-Möbius ladder
has density 3/8 and the 5-Möbius ladder density 3/10. The 2-ladder does not exist, the
3-ladder is the prism over a triangle and has density 1/2, the 4-ladder is the cube and the
5-ladder has density 3/10.
For the Heawood graph, which has girth 6 and 14 vertices we show in Section 6 that its
cost is at most five. However, it is relatively easy to see that its cost is at least 4, and hence
its density > 1/4.
The Petersen graph, the Heawood graph, the eight Möbius ladders, and the seven prisms
of order < 20 account for 17 vertex-transitive cubic graphs of order < 20.
Two of the remaining seven graphs have girth 3. If one contracts their triangles to single
vertices one obtains the K4 and the K3,3. By Theorem 8.1 the costs of the uncontracted
graphs are 3 and the densities 1/4 and 1/6. Two other graphs are the crossed 3- and 4-
ladders of densities 1/4, see Theorem 9.11. The remaining three graphs have girth 6. One
is the Pappus graph. It has 18 vertices, and its cost is 3 by the remark after Lemma 6.8.
Another graph on 18 vertices has cost 1. It is the smallest GRR, i.e. graph with three
arc-orbits, see Section 5. The last graph has 16 vertices, and thus cannot have three arc-
orbits. By Corollary 4.2 it also cannot have two arc-orbits. Therefore it is arc-transitive.
By Lemma 6.8 and the following remarks it has cost ≤ 3, because it is not the Heawood
graph.
Question 2.1. Are there finite, connected, vertex-transitive, cubic graphs on at least 20
vertices with distinguishing densities > 1/4?
If such graphs exist, then they must be flexible by Lemma 5.1, Theorem 6.1 and Theo-
rem 7.1.
The cost can be finite for infinite graphs. This was studied in [3], where it was shown
that the cost ρ(G) of connected, locally finite, infinite graphs G is finite if and only if
Aut(G) is countable. (In this statement it is not needed to explicitly assume 2-distinguishability,
because all locally finite graphs with countable automorphism group are 2-distinguishable,
see [18].)
An example for an infinite graph with countable automorphism group and finite cost is
the infinite ladder of Figure 2. It has distinguishing cost 3, as is easily verified.
Figure 2: The infinite ladder with a distinguishing coloring.
Contrariwise, the closely related chain of quadranglesQ of Figure 3, also called infinite
crossed ladder, is an example for an infinite, connected graph with uncountable automor-
phism group, and hence infinite distinguishing cost. Q can be formally defined as a graph
consisting of the 2-sided infinite paths . . . u−2u−1u0u1u2 . . . and . . . v−2v−1v0v1v2 . . .,
with the edges u2nv2n+1, v2nu2n+1, where n ∈ Z.
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 7
To see that Aut(Q) is uncountable, observe that we obtain an automorphism of Q for
any integer n by simultaneously interchanging u2n−1 with v2n−1, and u2n−1 with u2n,
while fixing all other vertices. This automorphism interchanges the pair of matching edges
u2n−1u2n and v2n−1v2n. Because Q has infinitely many such pairs of matching edges,
and because the edges in each pair can be interchanged independently of the other pairs,
Aut(Q) has at least 2ℵ0 elements, and is thus uncountable.
Figure 3: The chain of quadrangles Q with a distinguishing coloring.
A distinguishing 2-coloring is displayed in the figure. The black vertices are u0 and
u2n+1, where n is an integer different from −1. Because u0, u1 are the only adjacent black
vertices each automorphism must map the set {u0, u1} into itself. As there is a black vertex
of distance 2 from u1, namely u3, but no black vertex of distance 2 from u0, the vertices
u0, u1, u3 are fixed individually. It is easy to see that this implies that all four-cycles are
fixed setwise, and thus also all pairs of matching edges u2n−1u2n and v2n−1v2n. Each
such pair has exactly one black vertex, hence all of its vertices must be fixed. Because
V (Q) is partitioned by the sets of endpoints of these pairs of matching edges the coloring
is distinguishing.
3 Density
To define the distinguishing density we follow [17] and begin with the density of sets of
vertices. Let S be a set of vertices of a graph G, v ∈ G, and BG(v, n) = {w ∈ G :
d(v, w) ≤ n} be the ball of radius n with center v. If no ambiguity arises, we also write
B(v, n) for BG(v, n). Then
δv(S) = lim sup
n→∞
| B(v, n) ∩ S |
| B(v, n) |
is the density of S at v. If δv(S) exists for all vertices, which is the case for locally finite
graphs, then
δ(S) = sup{δv(S) : v ∈ V (G)}
is the density of S in G. Finally,
δ(G) = inf{δ(S) |S is a distinguishing set of G}
is the distinguishing density of G.
Note that δ(G) = ρ(G)/|V (G)| for finite graphs and zero for infinite graphs with finite
distinguishing cost.
Under certain conditions all values of δv(G), v ∈ V (G), are equal. Then δ(G) is equal
to δv(G) for arbitrarily chosen v ∈ V (G). We have two such conditions, one for density
zero, and a stricter one for positive density. First the condition for density zero see [17,
Lemma 1].
8 Art Discrete Appl. Math. 5 (2022) #P3.15
Lemma 3.1. Let G be a connected graph and v, w ∈ V (G). Suppose there is a constant
c such that for all n ∈ N we have |B(w, n + 1)| < c|B(w, n)|. If G has distinguishing
density zero at v, then G has distinguishing density zero.
For an example of a 2-distinguishable graph that does not have distinguishing density
zero, but nevertheless distinguishing density zero at some vertex v, see [17]. That graph
does not satisfy the conditions of the lemma.
The infinite tree Tk, where each vertex has degree k, is an example of a graph with
distinguishing density zero. For the proof and for many other examples of graphs with
distinguishing density zero we refer to [17].
For positive distinguishing density we have the following lemma.
Lemma 3.2. Let G be a connected graph, S ⊆ V (G), v ∈ V (G), and
lim sup
n→∞
|B(v, n)|
|B(v, n+ 1)|
= 1. (3.1)
If there is a vertex u in G with δu(S) = a ≥ 0, then δw(S) = a for all w ∈ V (G).
Proof. We first observe that (3.1) implies that
lim sup
n→∞
|B(v, n)|
|B(v, n+ d)|
= 1, (3.2)
for all natural numbers d.
Let δv(S) = a, w ∈ G, and d = d(v, w). Clearly
|B(v, n)| ≤ |B(w, n+ d)| ≤ |B(v, n+ 2d)|,
and hence
lim sup
n→∞
|B(v, n)|
|B(v, n+ 2d)|
≤ lim sup
n→∞
|B(w, n+ d)|
|B(v, n+ 2d)|
≤ lim sup
n→∞
|B(v, n+ 2d)|
|B(v, n+ 2d)|
= 1.
By (3.2)
lim sup
n→∞
|B(w, n+ d)|
|B(v, n+ 2d)|
= 1. (3.3)
Clearly
|B(v, n) ∩ S| ≤ |B(w, d+ n) ∩ S| ≤ |B(v, n+ 2d) ∩ S|,
and hence
|B(v, n) ∩ S|
|B(v, n)|
|B(v, n)|
|B(v, n+ 2d)|
≤
|B(w, n+ d) ∩ S|
|B(w, n+ d)|
|B(w, n+ d)|
|B(v, n+ 2d)|
≤
|B(v, n+ 2d) ∩ S|
|B(v, n+ 2d)|
.
By (3.2) the lim sup of the term on the left is equal to the lim sup of the term on the right,
and by (3.3)
δw(S) = lim sup
n→∞
|B(w, n+ d) ∩ S|
|B(w, n+ d)|
= lim sup
n→∞
|B(v, n) ∩ S|
|B(v, n)|
= δv(S)
for all w ∈ V (G). As we assume the existence of a vertex u ∈ V (G) with δu(S) = a, we
infer that δw(S) = a for all w ∈ V (G), i.e. δ(G) = a.
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 9
Lemma 3.2 will be useful in Subsection 3.1. Interestingly we do not need it for the
chain of quadrangles.
Lemma 3.3. The distinguishing density of the chain of quadrangles Q is 1/4.
Proof. In Section 2 we already observed that any distinguishing set S of Q must contain
at least one vertex in each pair of matching edges connecting two quadrangles. It is easy
to see that this implies that δv(S) ≥ 1/4 for arbitrary S and v. Hence δ(G) ≥ 1/4. If we
choose for S the set of black vertices in Figure 3, then δv(S) = 1/4 for all v, and therefore
the distinguishing density of Q is indeed 1/4.
Let us note nonetheless that Q satisfies the conditions of Lemma 3.2, because
|BQ(v, n+ 1)| = |BQ)(v, n)|+ 4
for n ≥ 4.
We wish to add that it is relatively easy to construct graphs with non-zero distinguishing
density that are not vertex-transitive, see [17].
3.1 Upper bounds for the density
For infinite graphs the situation is similar. If we take a one-sided infinite path P and connect
each of its vertices by an edge to the root of a copy of the binary tree Bk, then we obtain
a graph, say Hk, of distinguishing cost ρ(Hk) = 1/2− 1/2
k+1. In this way we can reach
densities that are arbitrarily close to 1/2, but not 1/2.
To reach density 1/2 we have to be more careful. Let P1/2 be the graph obtained from
the one-sided infinite path P = v1v1v2 . . . as follows: Connect v1 to the root of B1 by
an edge, then each of v2, v3 by an edge to the root of distinct copies of B2. Continue by
connecting v4, v5, v6 to the roots of distinct copies of B3, and so on.
Lemma 3.4. δ(P1/2) = 1/2.
Proof. Each minimal 2-distinguishing coloring of P1/2 leaves P and its neighbors white.
The remaining vertices come in pairs {u, v} of vertices of equal distance from v1, where
one vertex has to be colored black, and the other white. Let S be the set of black vertices
of such a distinguishing coloring. Then |B(v1, n) ∩ S| = (|B(v1, n)| − 2n − 1)/2, and
|B(v, n) ∩ S|/|B(v1, n)| = 1/2 − (2n − 1)/(2 · |B(v, n)|). The supremum of the latter
expression is 1/2 if the growth of B(v1, n) is more than linear. This is achieved by our
construction. It is not hard to see that P1/2 satisfies the conditions of Lemma 3.2. We
conclude that ρ(P1/2) = 1/2.
It seems that one can construct connected, subcubic infinite graphs with arbitrary den-
sity between 0 and 1/2 in similar ways.
But we know of no infinite, connected, vertex-transitive, cubic graph with density larger
than 1/4.
Question 3.5. Are there infinite, connected, vertex-transitive, cubic graphs with distin-
guishing densities larger than 1/4?
As we already observed after Question 2.1, any examples of such graphs must be flexi-
ble.
10 Art Discrete Appl. Math. 5 (2022) #P3.15
4 Arc- versus edge-orbits
This section describes the relationship between arc- and edge-orbits, respectively between
arc- and edge-transitivity.
Edge-transitive graphs need not be vertex-transitive, as the K2,3 shows. But even a
vertex-transitive graph G that is also edge-transitive need not be arc-transitive. The exis-
tence of such graphs was shown in 1966 by Tutte [33]. He also proved that any finite graph
of this type must be regular of even degree. The first examples were given by Bouwer [5],
and the smallest graph of this type is the Doyle graph. It is quartic and has 27 vertices, see
[15] and [26] .
By Tutte’s result each finite, vertex-transitive, cubic graph that is edge-transitive is arc-
transitive. In 1989 Thomassen and Watkins [30] extended Tutte’s result to infinite graphs
of subexponential growth. They proved that each vertex- and edge-transitive graph of odd
valence with subexponential growth is 1-transitive. For infinite cubic graphs we do not
need the growth condition:
Theorem 4.1. Let G be a finite or infinite, connected, vertex-transitive, cubic graph. If it
is edge-transitive, then it is also arc-transitive.
Proof. Let G satisfy the assumptions of the Theorem. If it is not arc-transitive, then it has
two or three arc-orbits.
Let u be an arbitrary vertex of G with the incident edges uv, uw, ux. Suppose first
that G has three arc-orbits. By edge-transitivity there is an α ∈ Aut(G) that maps uv into
uw and a β ∈ Aut(G) that maps uw into ux. Because G has three arc-orbits α(u) = w,
α(v) = u, and β(w) = u, β(u) = x. Hence βα maps the arc (u, v) into the arc (u, x), a
contradiction.
Suppose now that G has two arc-orbits, where (u,w) and (u, x) are in the same arc-
orbit. Let O1 be the arc-orbit of (u, v), O2 the orbit of (u,w), and D be the digraph on
V (G) whose arcs are the arcs of O2. Because G is edge-transitive at least one of the arcs
(u, v) or (v, u) must be in O2. By assumption (u, v) ̸∈ O2, hence (v, u) ∈ O2. Therefore,
in D, the vertex u has one incoming arc and two outgoing arcs. By vertex-transitivity
this holds for all vertices of G. But then G cannot be finite, because the total number of
incoming arcs in D has to be the same as the total number of outgoing arcs.
Hence G is infinite. It cannot be a tree, otherwise it would be the infinite cubic tree
T3, which has only one arc-orbit. Therefore G has a cycle, say C. By vertex-transitivity
we can assume that u is in C. Because, u has two outgoing arcs in D, one of them, say
(u,w), must be in C. Continuing this argument one sees that C is a directed cycle in D.
By edge-transitivity each arc is in directed cycle in D.
Let C ′ be a cycle containing (u, x). Clearly both C and C ′ contain the arc (v, u). Let
P be the longest path in C ∩ C ′ that contains (v, u). One of its endpoints is u, let the
other one be z. Clearly z has one outgoing arcs in D, and two incoming arcs, which is not
possible.
Corollary 4.2. Let G be a finite or infinite, connected, vertex-transitive, cubic graph. If
it has two arc-orbits, then one of the arc-orbits consists of pairs (u, v), (v, u), where the
edges uv are independent and meet each vertex of G.
Proof. Let G be a finite or infinite, connected, vertex-transitive, cubic graph with two arc-
orbits and u be an arbitrary vertex of G with the incident edges uv, uw, ux. As in the proof
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 11
of Theorem 4.1 we assume that (u, v) is inO1 and that (u,w), (u, x) are inO2. In the proof
of the theorem the assumption that (v, u) was in O2 led to a contradiction to the number of
arc-orbits of G. Hence, (v, u) ∈ O1. By vertex-transitivity each vertex y ∈ V (G) meets
exactly two arcs of the form (y, z), (z, y) ∈ O1. Clearly the corresponding edges yz are
independent and meet each vertex of G.
Corollary 4.3. Let G be a finite or infinite, connected, vertex-transitive, cubic graph. If it
has two arc-orbits, then it has two edge-orbits, but if it has three arc-orbits, it may have
two or three edge-orbits.
Proof. By definition each edge-orbit consists of the undirected arcs of the union of one or
more arc-orbits. We have seen that G has only one arc-orbit if G has only one edge-orbit.
Hence, if G has two arc-orbits, they must be in different edge-orbits, and thus G has two
edge-orbits.
If G has three arc-orbits, then it has two or three edge-orbits, because the case of one
edge-orbit is not possible by Theorem 4.1.
Recall from the introduction that we called O1 the matching orbit. The other orbit
was called cycle-orbit. By definition O1 and O2 are arc-orbits, but by the corollary they
correspond to edge-orbits. By abuse of language we do not distinguish between O1 or
O2 as and arc- or edge-orbits. In this sense O2 is the disjoint union of cycles of the same
length or of 2-sided infinite paths.
5 Three arc-orbits
Let G be a connected, vertex-transitive, cubic graph with three arc-orbits. If we fix a vertex
v, then all neighbors of v are also fixed. As G is connected, this implies that all vertices of
G are fixed if one is fixed. To break its automorphisms it suffices to color one vertex black
and leave all others white. The distinguishing cost is 1.
Lemma 5.1. The distinguishing cost of connected, vertex-transitive, cubic graphs with
three arc-orbits is 1.
Recall that Graphical Regular Representations, or GRRs, are vertex-transitive graphs
with trivial vertex stabilizers. They have been widely investigated and although GRRs are
abundant, it is interesting to explicitly describe special classes.
From Corollary 4.3 we know that the case of three arc-obits allows two or three edge-
orbits. The smallest example of a GRR has 18 vertices. It has two edge-orbits and girth 6.
The smallest example for the case of three edge-orbits is the truncated cuboctahedron. It
is the skeleton of an Archimedean solid that was already described by Kepler, where each
vertex is in one cube, one hexagon and one octagon. For both graphs we refer to a list of
cubic GRRs with at most 120 vertices from 1981 by Coxeter, Frucht and Powers, see [8].
A series of infinitely many such graphs was constructed by Godsil in 1983 [14], the
smallest of order 19!/2. In Section 9 we also construct infinitely many such graphs, see
Corollary 9.5. The smallest has 48 vertices and is also listed in [8].
Let us note in passing that the same publication lists two GRRs of girth 5, both of them
on 110 vertices, whereas there is only one finite 2-distinguishable arc-transitive cubic graph
of girth 5, the dodecahedron, and no infinite one. See Theorem 6.2.
We continue with graphs with one and two arc-orbits. Clearly the cost in these cases is
at least 2.
12 Art Discrete Appl. Math. 5 (2022) #P3.15
6 One arc-orbit
By definition such graphs are arc-transitive. Tutte [31] also calls them symmetric, but the
notation is not uniform and symmetric is also used for edge-transitive graphs that are not
arc-transitive. To avoid confusion we only speak of arc-transitive graphs. But we need a
refinement of the concept.
Following Tutte [31] we call a sequence of distinct vertices v0, v1, . . . , vs ∈ V (G) an
s-arc if vivi−1 ∈ E(G) for 1 ≤ i ≤ s, but vi−1 ̸= vi+1 for 1 ≤ i < s. Then G is s-arc-
transitive if Aut(G) is transitive on the set of all s-arcs on G. Moreover, we call G s-arc-
regular if for any two s-arcs v0v1 . . . vs and w0w1 . . . ws there is a unique automorphism
φ that maps v0v1 . . . vs into w0w1 . . . ws and respects the order of the vertices.
In this section we shall prove the following theorem.
Theorem 6.1. Let G be an arc-transitive, cubic graph different from K4, K3,3, the cube
and the Petersen graph. If it has finite girth, then ρ(G) ≤ 5, otherwise it is the infinite
cubic tree T3, which has infinite distinguishing cost and distinguishing density 0.
We will break up the proof of Theorem 6.1 into several parts, but begin with the remark
that there is only one connected acyclic cubic graph. It is the infinite cubic tree T3, of which
we already mentioned that it has distinguishing density zero.
It thus remains to prove the theorem for graphs with finite girth. We begin with a
structure theorem about graphs of girth at most 5. For girth 3 and 4 the theorem is folklore,
for finite graphs of graphs of girth 5 it was shown by Glover and Marušić [12]. Here is a
concise proof for all cases.
Theorem 6.2. The only connected, arc-transitive, cubic graphs of girth at most 5 are K4,
K3,3 the cube, the Petersen graph and the dodecahedron.
Proof. Let G be a connected, arc-transitive, cubic graphs of girth at most 5. Because we
forbid multiple edges the smallest girth is 3 and the arcs incident to an arbitrary vertex
induce a K1,3. Let a, b, c be the arcs incident with a vertex v of G. We show first that there
is an automorphism α of G that rotates a, b, c. In other words, there is an automorphism α
whose cycle representation of its action on {a, b, c} is (abc) or (acb). If this were not the
case, there would exist automorphisms whose actions on {a, b, c} are of the type (ab), (ac),
because of arc-transitivity. Then their product (ab)(ac) = (abc) is the desired rotation.
Let x, y, z be the endpoints of a, b, c different from v, and let the notation be chosen
such that there exists an automorphism α of G that rotates the arcs a, b, c and their end-
points.
x y z
x2 y1 y2 z1 z2x1
v
Figure 4: Basic structure for girth 5.
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 13
Girth 3. If G has a triangle we can assume without loss of generality that it is vxy. By
applying α twice we see that G also contains the edges yz and zx. Hence G = K4.
Girth 5. We first observe that the neighbors of x, y, z that are different from v are
distinct. Let them be x1, x2, y1, y2, z1, z2 and vxx1y1y a pentagon, see Figure 4.
Rotating clockwise it is seen that there is a pentagon vyα(x1)z1z, where α(x1) ∈
{y1, y2}. Suppose α(x1) = y1. Notice that we have two pentagons that share two edges.
The rotation moves z1 into a neighbor of x. It must be x2, because we have no triangles.
By further rotations we obtain the 6-cycle x1y1z1x2y2z2, see Figure 5. Clearly G is the
Petersen graph.
x
v z
z1
x2
y z2
y1
y2
x1
Figure 5: The Petersen graph.
Suppose now that no two pentagons share two edges. Then α(x1) = y2, which implies
α(x2) = y1. Furthermore, we can choose the notation such that α(y1) = z2, and α(y2) =
z1. Thus z2 = α(y1) = α
2(x2), and hence α(z2) = x2 and α(z1) = x1. By rotation we
obtain the pentagons vyy2z2z and vzz1x2x from vxx1y1y. Let H be the union of these
pentagons, see Figure 6. Note that v is in three pentagons, hence, by vertex-transitivity,
this holds for all vertices. More important, any two incident arcs determine exactly one
pentagon.
v
x = α(z)
z = α(y)
z1 = α(y2)
x2 = α(z2)
x
′
2
z
′
1
z2 = α(y1)
z
′
2
y
′
2
y
′
1
y1 = α(x2)
y = α(x)
x
′
1
x1 = α(z1)
p
Figure 6: The subgraph H (solid lines).
14 Art Discrete Appl. Math. 5 (2022) #P3.15
v
x
z
z1
x2
x
′
2
z
′
1
z2
z
′
2
y
′
2
y
′
1
y1
y
x
′
1
x1
r
p
Figure 7: The graph H ′.
The vertices of valence 2 in H form the set S = {x1, y1, y2, z2, z1, x2}. If u ∈ S, let
u′ be the third neighbor of u. Let S′ = {x′1, y
′
1, y
′
2, z
′
2, z
′
1, x
′
2}. By the girth condition and
the condition that two pentagons share only one edge we infer that the sets S′ and H are
disjoint and that any two vertices of S′ distinct.
Now consider x. It is in two pentagons in H , but has to be in a third in G. Neither x1x
′
1
nor x2x
′
2 can be in the first two pentagons, hence x1x
′
2 ∈ E(G). Similarly we find that
z′1z
′
2 and y
′
1y
′
2 are in E(G). The new graph, say H
′, is depicted in Figure 7.
Consider the path x′2x2z1z
′
1. Its edges must be in a pentagon that is not in H
′, which is
only possible if there exists a vertex, say p, that is adjacent to x′2 and z
′
1. Because G has
girth 5 the vertex p cannot be a neighbor of x′1, nor of z
′
2, and because G is cubic it cannot
be any vertex of H ′.
Now consider the neighbors of x′1, p and z
′
2 that are not in V (H
′) ∪ {p}, and denote
them by q, r, s. Because of the girth condition and because G is cubic they are pairwise
distinct and not in V (H ′) ∪ {p}.
Finally, because all vertices have to be in three pentagons we find four additional edges
that complete the graph to a dodecahedron.
Corollary 6.3. The only 2-distinguishable, finite or infinite, connected, arc-transitive, cu-
bic graph of girth at most 5 is the dodecahedron. Its distinguishing cost is 3.
Proof. We already know thatK4,K3,3, the cube and the Petersen graph are not 2-distinguish-
able. Hence, by the theorem, the only 2-distinguishable graph of girth at most five is the
dodecahedron. It is easily seen that its distinguishing cost is 3.
For girth > 5 we will invoke a result of Tutte for finite graphs and its extension to
infinite graphs by Djokovic and Miller. First the result from [32] for finite graphs.
Theorem 6.4. Let G be a connected, finite, arc-transitive, cubic graph. Then G is s-arc-
regular for some s ≤ 5.
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 15
And now the extension to infinite graphs from [9].
Theorem 6.5. Every connected, infinite, arc-transitive, cubic graph is s-arc-regular for
some s ≤ 5 with the exception of the infinite cubic tree.
We begin with a result for 1-arc-regular graphs.
Lemma 6.6. Each connected, 1-arc-regular cubic graph has distinguishing cost 2.
Proof. Let G be a connected, 1-arc-regular cubic graph. Then the action of Aut(G) on the
arcs incident with an arbitrary vertex v cannot have involutions. To see this, let a, b, c be
the arcs incident with v, and α ∈ Aut(G) induce the involution (cb) on {a, b, c}. Then
α(a) = a, which contradicts 1-regularity.
Furthermore, by Theorem 6.2 the girth of G must be at least 5, because the graphs
K4, K3,3, the cube, the Petersen graph and the dodecahedron are not 1-arc-regular. We
now choose two vertices u, v of distance 2 in G and color them black. Because g(G) > 4
there is a unique vertex w that is adjacent to both u and v. Therefore any color preserving
automorphism α fixes w. If it interchanges u with v, then it interchanges two arcs incident
with w and fixes the third, and thus induces an involution on the arcs incident with w. If α
fixes u, then it fixes the arc uw and is the identity by 1-arc-regularity.
The existence of such graphs is guaranteed by a result of Frucht [11], who provided an
example of a 1-arc-regular cubic graph of girth 12 with 432 vertices. It can be embedded
in a surface of genus 55 so as to form a map of 108 dodecagons.
For the girth of s-arc-regular cubic graphs we will use the bound
2s ≤ g(G) + 2 (6.1)
from [31].
We first consider girth 7 and then girth 6.
Lemma 6.7. Let G be a connected, s-arc-regular, cubic graph of girth at least 7. If s = 1,
then ρ(G) = 2, otherwise ρ(G) ≤ 3, unless s = 4 and g(G) = 7, then ρ(G) ≤ 4.
Proof. Because of arc-transitivity we can invoke Theorems 6.4 and 6.5. They imply that
our graphs are s-arc-regular for some s ≤ 5.
By Lemma 6.6 we can assume that s > 1. For s = 2 or 3 we choose a path uxvw in G.
This is possible because the girth is > 6. We color u, v, w black as visualized in Figure 8.
Each color preserving automorphism φ fixes u because it is the only black vertex without
black neighbors. As v and w have different distances from u, they are also fixed. Hence φ
fixes the s-arcs ux, uxv and uxvw, where s = 2, 3, respectively. By s-arc-regularity φ is
the identity.
Now, let s = 4. For girth g > 7 we choose a path uxyvw and color u, v and w black
as in Figure 8. Then we argue as before to prove that the 4-arc uxyvw is fixed by all color
preserving automorphisms. If the girth is 7 this coloring allows that both v and w have
distance 3 from u. In this case it suffices to color y black to fix the 4-arc uxyvw by all
color preserving automorphisms.
For s = 5 we first observe that the girth is at least 8 by Equation (6.1). We choose a
path uxyzvw of length 5 and color u, v and w black. If the girth is different from 9 this
coloring fixes the 5-arc uxyzvw. If the girth is 9, then v, w could be interchanged by color
preserving automorphisms.
16 Art Discrete Appl. Math. 5 (2022) #P3.15
u x v w
u x y v w
u x y z v w
s = 2; 3
s = 4
s = 5
u x y z v w
s = 5; g = 9
u x y v w
s = 4; g = 7
Figure 8: Colorings of s-arcs.
Lemma 6.8. Let G be a connected, arc-transitive, cubic graph of girth 6. Then G is at
most 4-arc-regular. If G is 1-arc-regular, then ρ(g) = 2, if G is 2 or 3-arc-regular, then
ρ(G) ≤ 3, otherwise ρ(G) ≤ 5.
Proof. ThatG is at most 4-arc-regular follows from Equation 6.1. WhenG is 1-arc-regular
we invoke Lemma 6.6 and when G is 2- or 3-arc-regular we can use the same coloring as
in the proof of Lemma 6.7.
v1
v2
v3
v4
v5
v0
v
Figure 9: Coloring for s = 4 and girth 6.
If G is 4-arc-regular we choose an arbitrary 6-gon v0 · · · v5 and color v0, . . . , v3 black
as well as the neighbor of v4 that is not in the cycle. Let this neighbor be v, see Figure 9.
Clearly v cannot be adjacent to any vertex of the cycle, because G has girth 6. Hence v is
fixed by the coloring. Furthermore, v cannot have distance 2 from v0, otherwise G would
have a cycle of length 5. As v has distance 2 from v3, this implies that the coloring fixes
v3. Invoking the girth condition we see that v4 is also fixed, because it is on a path of
length 2 between the fixed vertices v3 and v. This fixes the entire 4-arc v0v1v2v3v4. By
4-arc-regularity this coloring is distinguishing.
Together with Corollary 6.3, Lemma 6.7 and Theorem 6.5 this also completes the proof
of Theorem 6.1.
Let us mention that, except for the interchange of black and white, our coloring for the
4-arc-regular case is the same as that for a graph described in [16, Theorem 7.3].
We also observe that there exists only one finite, connected, 4-arc-regular, cubic graph
of girth 6. It is the Heawood graph, see [7, 10], also known as Tutte’s 6-cage [31]. It is not
know whether there exist infinite, connected, 4-arc-regular, cubic graphs of girth 6.
Similarly, by [7] there are only two finite, connected, 3-arc-regular cubic graphs of girth
6, namely the Pappus graph and the generalized Petersen graph GP (10, 3).
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 17
Question 6.9. Are there any infinite, 3-arc- or 4-arc-regular, cubic graphs of girth 6?
7 Two arc-orbits
Now we turn to connected, vertex-transitive, cubic graphs G with two arc-orbits, say O1
and O2. We choose the notation such that O1 is the matching orbit and O2 the cycle-orbit.
By the elements of O1 and O2 we mean their connected components, and recall that, by
Lemma 4.2, the case of two arc-orbits coincides with the case of two edge-orbits. Hence, in
this case we need not distinguish between arc- and edge-orbits, but we need the following
observation about the action Aut(G) on the elements of O1 and O2.
The group of automorphisms ofG acts on the elements ofO2 by reflections and vertex-
transitively, that is, Aut(G) induces the full group of automorphisms on each element of
O2. The full groups of automorphisms of finite cycles are called dihedral, and the full group
of 2-sides infinite paths is called infinite dihedral. It acts by reflections and translations.
On the elements of O1 the group Aut(G) acts by reflections.
First a Theorem about connected, vertex-transitive, cubic graphs with two arc-orbits
that are rigid. Recall that graphs with two arc-orbits are rigid if the order of the group
induced by Aut(G) on the subgraph formed by a matching edge and its four incident edges
is 2.
Theorem 7.1. The distinguishing cost of connected, vertex-transitive, rigid cubic graphs
G with two arc-orbits is 2, unless G is an infinite ladder, a k-ladder or a k-Möbius ladder
with k > 3. Then ρ(G) = 3.
Proof. Let G be a connected, vertex-transitive, rigid cubic graph with matching orbit O1
and cycle-orbitO2. Suppose there is a quadrangle uvwz, where uv and wz are inO1. If we
color u, v, w black and leave all other vertices white, then w is fixed by all color preserving
automorphism α, and hence also u, because uv is a matching edge. Hence α also fixes w,
and therefore the element of the cycle-orbit that contains uw, say A.
If an automorphism α of a rigid graph fixes an element A of O2, be it a cycle or a
2-sided infinite path, then α fixes all neighboring elements of A in O2, and because G is
connected all elements of O2, and thus the entire graph G.
It is easily seen that the only graphs that satisfy the assumptions of the lemma and that
contain such a quadrangle uvwz are an infinite ladder, or a k-ladder, resp. a k-Möbius
ladder. As the 3-ladder is the cube and the 3-Möbius ladder the K3,3, both of which are not
2-distinguishable, we have to require that k > 3.
In all other cases one considers a matching edge uv together with a neighbor w of
v, and colors u and w black. As there is no quadrangle uvwz in G, the vertex v is the
only common neighbor of u and w, and hence fixed. Now we conclude as before that the
coloring is distinguishing.
We have two corollaries, one about the order of the vertex-stabilizers of rigid and of
flexible graphs, and one about planar graphs.
Corollary 7.2. Let G be a connected, vertex-transitive, cubic graph with two arc-orbits.
When G is rigid, then the order of its vertex-stabilizers is 2, otherwise, that is when G is
flexible, it is at least 4.
Proof. The validity of the assertion for rigid G is clear by the proof of the theorem, and for
flexible G it follows directly from the definition.
18 Art Discrete Appl. Math. 5 (2022) #P3.15
Corollary 7.3. Let G be a 3-connected, planar, vertex-transitive, cubic graph with two
arc-orbits. Then G is rigid and ρ(G) = 2 unless G is a k-ladder or a k-Möbius ladder
with k > 3.
Proof. Let G satisfy the assumptions of the corollary. By Whitney’s theorem, which by
[29] also holds for infinite graphs, G is uniquely embeddable into the plane in the sense
that any automorphism that reverses the order of the edges incident with an arbitrary vertex
reverses this order for all vertices. Hence G is rigid, and ρ(G) = 2 by Theorem 7.1.
For flexible cubic graphs we have the following lemma.
Lemma 7.4. Let G be a connected, vertex-transitive, flexible cubic graph with two arc-
orbits, say O1 and O2, where O2 consists of finite cycles. Then:
(i) One arc-orbit of G, say O1, consists of arcs (u,v) and (v,u), where the edges uv form
a complete matching.
(ii) The number of cycles in O2 is a least 2.
(iii) O2 consists of finite cycles of equal lengths without diagonals.
(iv) The number of edges between pairs of adjacent cycles is either 1 for all pairs, or 2
for all pairs.
(v) If the number of edges between adjacent cycles is 2, then the edges between neigh-
boring cycles connect pairs of opposite vertices.
(vi) If the length ℓ of the cycles in the cycle-orbit is at least 6, then g(G) = ℓ.
Proof. Item (i) is Lemma 4.2. As usual we let O1 be the matching orbit.
For (ii) we begin with the case when O2 has only one connected component. Suppose
it is the cycle Cn = u0u1 . . . un−1 of length n. By (i) each vertex of Cn is incident with
exactly one edge of O1. Let u0uj be the edge of O1 that is incident with u0, and α be the
reflection ofO2 that fixes u0. Clearly α(uj) = un−j . BecauseG is cubic, u0uj = u0un−j ,
which is only possible if n is even, say n = 2k, and j = k. By vertex-transitivity the other
edges of O1 are of the form uiui+k. The resulting graph is the k-Möbius ladder. As we
have already seen, for k = 3 it is the K3,3, which is not 2-distinguishable, and for k > 3 it
is rigid. This proves (ii).
So we turn to the case when O2 has at least two cycles, and all cycles of O2 have the
same length, say ℓ. If a cycle has a diagonal, then this diagonal must be in O1. By vertex-
transitivity this implies all elements of O1 are diagonals, but then G is disconnected. This
proves (iii).
Now, let k be the number of edges between neighboring cycles. Clearly k is a divisor
of ℓ. When k = ℓ, then G is the prism over the k-cycle and rigid.
When 3 ≤ k < ℓ, consider a cycle u0u1 . . . uℓ inO2 and a neighboring cycle u
′
0u
′
1 . . . u
′
ℓ
in O2, and then the induced subgraph of G that contains the paths from uℓ−ℓ/k to uℓ and
from u′ℓ−ℓ/k to u
′
ℓ. It consists of two cycles of length 4ℓ/k + 2 with the common edge
u0u
′
0, which is in O1. When u0 is fixed and uℓ−ℓ/k is interchanged with uℓ, then u
′
ℓ−ℓ/k is
interchanged with u′ℓ. In this case G is rigid. This means that k is 1 or 2 if G is flexible,
which proves (iv).
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 19
If k = 2, then ℓ is even. Furthermore, the matching edges between two neighboring
cycles must connect pairs of opposite edges in the cycles. To see this, let C = v0v1 . . . vℓ
be an element of O2, C
′ a neighboring cycle, and v0, vj be the origins of the matching
edges form C to C ′. Because the edges v0vℓ−1 and v0v1 are in the same arc-orbit, there
is an automorphism α that fixes v0 and interchanges vℓ−1 with v1. Clearly α preserves C
′
and maps vj into vℓ−j , which is only possible if vℓ−j = vj , because there are only two
edges between C and C ′. Hence ℓ = 2j. This proves (v).
It is easy to check (vi).
Lemma 7.4 excludes graphs whose cycle-orbits O2 consist of 2-sided infinite paths.
If such a graph exists, then it is easily seen that O2 contains at least two connected com-
ponents, and that there can be at most one matching edge between any two components,
otherwise G would be rigid. Furthermore G cannot be G acyclic, because then it is the
infinite cubic tree T3, which has only one arc-orbit. This leads to the following question:
Question 7.5. Are there any two arc-orbit, flexible, cubic graphs G whose cycle-orbits
consist of 2-sided infinite paths?
It is not hard to show that the girth of any such graph, if it exists, must be at least 9.
We conclude this part with a theorem on tree-like graphs. We call an infinite, flexible
cubic graph G tree-like if its cycle-orbit consist of finite cycles, and if the graph G∗ that
is obtained from G by contraction of the cycles in the cycle-orbit to single vertices and by
replacement of double edges, if they occur, by single edges, is a tree of valence at least 3.
Theorem 7.6. Let G be a tree-like infinite, flexible cubic graph. Then ρ(G) = ∞ and
δ(G) = 0.
Proof. Suppose G satisfies the assumptions of the theorem, and let k be the length of the
cycles in the cycle-orbit. Then G∗ is an infinite tree Tr of valence r = k or r = k/2, where
r ≥ 3. By [17] Tr is 2-distinguishable, has uncountable automorphism group, infinite
distinguishing cost and distinguishing density zero. It is easy to see that the coloring c ofG
that is obtained from a distinguishing coloring c∗ of G∗ by coloring an arbitrary preimage
of each black vertex of G∗ black is distinguishing, has infinitely many black vertices, and
distinguishing density 0.
To complete the proof we have to show that ρ(G) is infinite. We know that G∗ has
uncountable group and any automorphism of G∗ is induced by one of G. If α and β are
two automorphisms ofG that induce different automorphisms ofG∗ then they cannot leave
all k-cycles invariant and must be distinct. Hence Aut(G) is uncountable.
Tree like graphs are the basis for the construction of connected, vertex-transitive, flex-
ible cubic graphs of girth four with two arc-orbits, infinite cost and zero density. See
Lemma 9.3.
8 Flexible graphs of girth 3
Let G be a connected, vertex-transitive, cubic graph with two arc-orbits and girth 3. Then
the cycle-orbits of G are triangles. Let G∗ be the graph obtained by contracting each such
triangle to a single vertex, making the matching edges of G the edges of G∗.
This implies that Aut(G) and Aut(G∗) are isomorphic. To see this, let α be an au-
tomorphism of a graph H . It induces an incidence preserving permutation on E(H) that
20 Art Discrete Appl. Math. 5 (2022) #P3.15
maps each edge uv of H into α(u)α(v). We denote it by αE and the group of incidence
preserving permutations of E(H) by AutE(H). If H is a nontrivial graph, then Aut(H)
and AutE(H) are isomorphic if and only if H has at most one isolated vertex and K2 is
not a component. It is not hard to prove directly, but actually it is a theorem about the
automorphism group of the line graph of a graph, see e.g. [19, Theorem 1.2].
This means that Aut(G∗) ∼= AutE(G
∗). Furthermore, each φ ∈ Aut(G) induces
an incidence preserving permutation φ∗E of E(G
∗). Clearly the mapping φ 7→ φ∗E is
bijective and thus Aut(G) and AutE(G
∗) are isomorphic. By the above this implies that
Aut(G) ∼= Aut(G∗). We also observe:
1. G∗ is arc-transitive. By Theorems 6.4 and 6.5 this implies that it is s-arc-regular for
some s ≤ 5, unless it is the infinite cubic tree.
2. If G∗ ̸= T3, then G
∗ is s-arc-regular for s > 2 if and only if G is flexible. In
particular, if G∗ is 3-connected and planar, then s = 2 and G is rigid. Observe
that the assertion about s for G∗ is a consequence of the unique embeddability of
3-connected planar graphs into the plane. Compare the proof of Corollary 7.3.
3. ρ(G) ≤ ρ(G∗), because if coloring the vertices of a set U ⊆ V (G∗) black distin-
guishesG∗, then coloring one vertex black in each triangle corresponding to a vertex
in U distinguishes G. A similar argument was used in the proof of Theorem 7.6.
4. In some cases one can show that ρ(G) < ρ(G∗) by using the extra freedom of three
choices for vertices in each triangle. Our approach is as follows. Let c be a coloring
of the vertices ofG and Aut(G)c be the group of the color preserving automorphisms
of G. Recall that Aut(G∗) ∼= Aut(G). So Aut(G)c is isomorphic to a subgroup,
say (Aut(G)c)
∗, of Aut(G∗). If G∗ is s-regular and if (Aut(G)c)
∗ fixes an s-arc,
then (Aut(G)c)
∗ must be the trivial group, and thus also Aut(G)c.
Therefore, if c is a 2-coloring with n black vertices, where the other vertices are
white, and n < ρ(G∗), then ρ(G) < ρ(G∗) if (Aut(G)c)
∗ is trivial.
Theorem 8.1. Let G be a finite, vertex-transitive, flexible cubic graph with two arc-orbits
and girth 3. Then ρ(G) = 2, unless g(G∗) = 4, 6, 8. In these cases G∗ is the K3,3, the
Heawood graph or the Tutte-Coxeter graph and ρ(G) = 3.
Proof. Suppose g(G∗) ≤ 5, so G∗ is K4, the cube, the dodecahedron, the Petersen graph,
or K3,3. The first three are 3-connected planar and thus G is rigid, contrary to assumption.
When G∗ is the Petersen graph or K3,3, then G is flexible, because both graphs are 3-
regular by [7].
By Theorems 6.4 and 6.5 G∗ is s-arc-regular with s ≤ 5. Given a G∗ with s ≤ 5, we
choose an s-arc whose arcs a∗1, . . . a
∗
s in G
∗ correspond to matching arcs a1, . . . as in G.
Set ai = uiu
′
i, and color u1, us black. Let u
∗
1, u
∗
s be the vertices in G
∗ corresponding to
u1, us in G.
If the girth of G∗ is at least 2s − 1 then u1u
′
1u2u
′
2 · · ·us is the unique shortest u1, us-
path in G. Because u1u
′
1 is a matching edge, but not u
′
s−1us each color preserving auto-
morphism ofGmust fix u1 and us. But if us is fixed, then u
′
s is also fixed. Letφ ∈ Aut(G)
fix u1 and u
′
s. Then it fixes a1, . . . as in G and φ
∗ fixes a∗1, . . . a
∗
s in G
∗. Because G∗ is
s-arc regular φ∗ is the identity, and thus also φ. Hence the coloring of G is distinguishing
and ρ(G) = 2, see Figure 10 for s = 5.
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 21
u1 u
0
1
u2 u
0
2 u3 u
0
3 u4
u
0
4
u5
u
0
5
Figure 10: Distinguishing coloring of G for s = 5, g(G) ≥ 9.
Therefore, cost 2 is assured for s = 3, 4, 5 when g(G∗) ≥ 5, 7, 9, respectively. This
leaves the cases g(G∗) = 4, 6, 8 for s = 3, 4 and 5.
If s = 3, then g(G∗) = K3,3, which is the only remaining graph g(G
∗) of girth at most
5. It is Tutte’s 4-cage. Furthermore, at the end of Section 6 we noted that the only finite,
4-arc-regular, cubic graph of girth 6 is the Heawood graph, also known as Tutte’s 6-cage.
Finally, in 1991 M. J. Morton [21] showed that the Tutte-Coxeter graph, also called Tutte’s
8-cage, is the only finite graph with s = 5 and girth 8.
Using the same notation for all three cages, we color u1, us and u
′
s black. As
dG(u1, us) = 2s − 2 and dG(u1, u
′
s) = 2s − 3, all black vertices have to be fixed by
any color preserving φ ∈ Aut(G). As before we conclude that φ∗ fixes a∗1, . . . a
∗
s in G
∗.
Hence the coloring of G is distinguishing, see Figure 11 for s = 4.
u1
u
0
1
u4
u
0
4
Figure 11: Distinguishing coloring of G for s = 4, g(G) = 6.
It remains to show that ρ(G) > 2 when G∗ is a Tutte 4, 6 or 8-cage, that is when
s = 3, 4 or 5. We first show that each path of length s− 1 is in two cycles of length 2s− 2.
Let C be a cycle of length 2s − 2 in G∗ and a∗1, . . . a
∗
s be the arcs of an s-arc in C. By
arc-transitivity there is α ∈ Aut(G∗) that fixes a∗1, . . . a
∗
s−1, but not a
∗
s . It maps C into
another cycle α(C). Clearly the C ∩ α(C) = a∗1, . . . a
∗
s−1. By arc-transitivity this implies
that any path of length s− 1 is in two cycles of length 2s− 2.
To show that ρ(G) > 2 it suffices to show that to any vertices u,v in G there is an
automorphism ofG that interchanges them or a nontrivial automorphism that fixes them. If
u,v are in the same triangle, then there clearly is an automorphism that interchanges them.
Because the diameter of the cages is s−1 we can thus assume that 1 < dG∗(u
∗, v∗) ≤ s−1.
Suppose dG∗(u
∗, v∗) = s − 1 and that u∗ and v∗ are the endpoints of an s − 1-arc
a∗1, . . . a
∗
s−1. Clearly there are at least three automorphisms of G
∗ that interchange u∗ and
v∗: one that reverses the s − 1-arc a∗1, . . . a
∗
s−1, another one that rotates C, and one that
rotates α(C). Therefore, if dG(u, v) = 2s− 2 or 2s, then an inflection interchanges u with
v, and if dG(u, v) = 2s− 1 a rotation interchanges them.
22 Art Discrete Appl. Math. 5 (2022) #P3.15
This leaves the case d = dG∗(u
∗, v∗) ≤ s− 2. If dG(u, v) = 2d− 1 or 2d+1 then we
can interchange u, v by an inflection. Hence we can assume that dG(u, v) = 2d. Then each
shortest u, v-path either begins or ends with a matching edge. We choose the notation such
that it begins with a matching edge. In G∗ there clearly is an s-arc a∗1, . . . a
∗
s where u
∗ is
the origin of a∗2 and v
∗ the origin of a∗d+2. Note that d+ 2 ≤ s. As before we let ai denote
the matching edge in G that corresponds to a∗i and set ai = uiu
′
i. By our assumptions
u = u2 and v = ud+2. There is an α
∗ ∈ Aut(G∗) that moves a∗1, but fixes a
∗
2, . . . , a
∗
s . The
automorphism α∗ uniquely extends to an α ∈ Aut(G) that fixes u2, u
′
2, . . . ud+2, u
′
d+2.
Figure 12 depicts the case when d is maximal, i.e. when v = ud+2 = us. Observe that α
interchanges the neighbors of u2 that are different from u
′
2, fixes all other depicted vertices,
but the non-matching edges incident with u′s can be interchanged.
α(u′
1
)
u
′
1
u2 u
′
2
us
asa1
Figure 12: Nontrivial α fixing u = u2 and v = us.
We conclude the section with a remark about the construction of connected, vertex-
transitive, flexible cubic graph with two arc-orbits and girth 3. The process of contracting
triangles in G to get G∗ can be reversed by truncation, namely replacing each vertex by a
triangle. In this case, we begin with G∗ and construct G by truncating G∗. This way every
cubic graph is a G∗ for some cubic graph G. For example, in the proof of Theorem 8.1, the
graph G with G∗ = K3,3 is flexible since K3,3 is. In general, we have
Proposition 8.2. LetG∗ be an s-arc regular cubic graph, where s ≥ 3. Then its truncation
G is flexible of girth 3.
Proof. Clearly, G has two arc-orbits, one orbit consisting of arcs in the triangles and a
matching orbit corresponding to arcs in G∗.
9 Flexible graphs of girth 4
In this section we mainly consider connected, flexible, vertex-transitive, cubic graphs G
with two arc-orbits and girth 4. We show that this class contains finite graphs of arbitrarily
large distinguishing cost and infinite graphs with positive distinguishing densities.
For our constructions we need two operations on G, one we call folding, and the other
unfolding. The first folds the 4-cycles of a graph G into edges of a new graph F (G), and
the second unfolds the matching edges of a graph G into 4-cycles of a new graph U(G).
Figure 13 shows how a matching edge uv of the graph on the left is unfolded into a
4-cycle in the graph on the right, respectively it shows how a 4-cycle in the graph on the
right is folded into an edge of the graph on the left.
To be more precise, let G be a cubic graph with a complete matching M and C be the
subgraph of G with the edge-set E(G) −M . The edges not in M form a subgraph C in
which each edge has valence 2. Hence, each component of C is a finite cycle or a 2-sided
infinite path. When G is vertex-transitive only one of these options is possible, and if there
are finite cycles, then all have the same length.
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 23
(c, ew)
(c, e)
(d, e)
(d, ez)
(b, e)
(b, ey)
(a, e)
(a, ex)w
c
d
z
v
b
a
y
xex
u
Figure 13: Unfolding and folding.
For a formal description of the new operations we denote incidence of edges a, b by
a|b, and non-incidence by a ∤ b. The unfolded graph U(G) is then defined by:
V (U(G)) = {(a, e) : a ∈ E(C), e ∈M,a|e},
E(U(G)) = {(a, e)(c, f) : a = c, a|e, a|f, e ̸= f or e = f, e|a, e|c, a ̸= c, a ∤ c}.
Each vertex u of G is incident with exactly one matching edge, say e, and two edges
from C, say a,b. It thus gives rise to two vertices of U(G), we say each vertex is split into
two vertices. Furthermore, if (a, e)(b, f) ∈ E(U(G)), then either a connects an endpoint
of e with one of f , or e connects an endpoint of a with one of b. Hence, identification of
the two vertices in each pair of split vertices is a homomorphism from U(G) onto G.
Clearly unfolding is well defined for all cubic graphs G with a matching M . The
folding operation, however, is only defined for cubic graphs G with a set of disjoint of
4-cycles. Given such a graph G, the folded graph F (G) is obtained by identification of
opposite vertices in the 4-cycles of G, and replacement of the ensuing multiple edges from
the 4-cycles by single edges.
Observe that we did not require vertex-transitivity for our operations.
Now we recall that any α ∈ Aut(G) induces an incidence preserving permutation on
E(G) that maps each edge uv of G into α(u)α(v). We denote it by αE and, as before,
we let AutE(G) be the group of incidence preserving permutations of E(G). We already
observed that Aut(G) ∼= AutE(G) if and only if G has at most one isolated vertex and K2
is not a component.
Lemma 9.1. LetG be a connected, vertex-transitive, cubic graph with a complete matching
M , and let C denote the subgraph of G consisting of the edges not in M . Then
(i) U(G) is a connected, cubic graph that consists of disjoint quadrangles, which form
a spanning subgraph, and a set of independent edges that form a complete matching.
The quadrangles arise from the edges in M , and the matching edges of U(G) are in
one-one correspondence with the edges of C.
(ii) If all quadrangles of G are components of C, then any two adjacent quadrangles in
U(G) are connected by exactly one edge.
(iii) There is an isomorphism of AutE(G) into Aut(U(G)). If G has three arc-orbits,
then U(G) is not vertex-transitive.
(iv) If G is vertex-transitive with two arc-orbits, then U(G) is also vertex-transitive, but
may have three arc-orbits, see Corollary 9.5.
24 Art Discrete Appl. Math. 5 (2022) #P3.15
(v) IfG is vertex-transitive and flexible with two arc-orbits, thenU(G) is vertex-transitive
with two arc-orbits, but may be rigid. See Corollary 9.8 and the remark after its
proof.
Proof. Let e = uv ∈ M , and a, b, c, d ∈ V (C), where a, b are incident with u, and c, d
incident with v. Then (a, e)(c, e)(b, e)(d, e) is a quadrangle in U(G). Note that (a, e) is
adjacent to (c, e) and (d, e) in U(G), but not to (b, e).
If a = ux ∈ M and ex is the matching edge incident with x, then (a, e)(a, ex) ∈
E(U(G)). Hence the the neighbors of (a, e) are (c, e), (d, e) and (a, ex). This proves (i).
For (ii) let A, B be two neighboring quadrangles in U(G) that are joined by (at least)
two edges. By (i) they are matching edges, say e, f . Let ae, af be the endpoints of e,
resp. f , in A and be,bf be the endpoints in B. If ae and af arise from the same vertex in
G, then G has a double edge or a triangle, depending on whether be and bf arise from the
same vertex in G or not. Hence ae, af , be, bf arise from different vertices in G. It also
means that aeaf and bebf arise from matching edges, while e, f arise from non-matching
edges. But then G has a quadrangle that is not in C, contrary to assumption. This proves
(ii).
To prove (iii) we observe that the mapping (a, e) 7→ (αE(a), αE(e)) is an automor-
phism of U(G), say α∗. It is then easy to see that the mapping αE 7→ α
∗ from AutE(G)
into Aut(U(G)) is an injective isomorphism. If G has three arc-orbits and if a, b are the
non-matching edges that are incident with a vertex v and if e is the matching edge incident
with v, then there is automorphism of G that fixes v and interchanges a with b, and thus no
automorphism of U(G) that maps (a, e) into (b, e). This proves (iii).
To prove (iv) consider two vertices (a, e), (b, f) of U(G). Let u be the common vertex
of a and e, and v the common vertex of b and f . IfG is vertex-transitive with two arc-orbits,
then there exists an automorphism α that maps u into v, where αE(e) = f and αE(a) is
incident with v. If αE(a) ̸= b, then there is a β ∈ Aut(G) that fixes v and maps αE(a)
into b. Then (βα)∗(a, e) = (b, f), and U(G) is vertex-transitive. This proves (iv).
Finally, consider a vertex (a, e) and its two non-matching neighbors, say (c, e), (d, e).
If e = uv, then c, d are non-matching edges of G that are incident with v. Because G is
flexible with two arc-orbits, there is an automorphism α that fixes a, u, e and v and where
αE interchanges c, d. Clearly α
∗ fixes (a, e) and interchanges (c, e) with (d, e).
Lemma 9.2. Let G be a cubic graph with a complete matching M and a set C of quadran-
gles, where any two neighboring quadrangles of C are connected by one edge of M . Then
there is a natural isomorphism between Aut(G) and Aut(U(G)).
Proof. Suppose G satisfies the conditions of the lemma. Then all quadrangles of G must
be in the set C. If not there must be a quadrangle uvwx that contains a matching edge, say
e = uv. Vertices u and v are in different quadrangles from C, say A and B. Then ux is in
A, vw in B, and wx is a second edge that connects A with B.
By Lemma 9.1 this implies that any two neighboring quadrangles in U(G) are con-
nected by exactly one edge, and thus the partition of E(U(G)) into a set of matching edges
and a set of quadrangles is unique. Hence the set of pairs of split vertices arising in the con-
struction of U(G) from G is exactly the set of pairs of opposite vertices of the quadrangles
in U(G). Thus there is only one way to fold U(G), and G = F (U(G)).
Each automorphism α of U(G) preserves the pairs of opposite vertices of the quadran-
gles in U(G) and thus acts as a permutation α′ on V (G). As adjacences are preserved α′
is an automorphism. Moreover, (ϕψ)′ = ϕ′ψ′, and thus ϕ 7→ ϕ′ is a homomorphism.
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 25
If there were two distinct automorphisms ϕ and ψ of U(G) that induce the same action
on G, they would have to coincide on these pairs as sets, but in at least one pair {u, v}
they would have to act differently. Then ψ−1ϕ(u) is not the identity on {u, v}, and thus
interchanges u and v. But then the matching edge eu incident with u has to be interchanged
with the matching edge ev incident with v. As eu and ev lead to different quadrangles,
not all pairs of opposite vertices in the quadrangles can be preserved. Hence ϕ 7→ ϕ′ is
injective, and hence also the mapping ϕ 7→ (ϕ′)E
The observation that Aut(G) ∼= AutE(G) and that there is an injective isomorphism
of AutE(G) into Aut(U(G)) completes the proof.
As an application of Lemma 9.1 we have the following lemma.
Lemma 9.3. There exist infinitely many connected, vertex-transitive, flexible cubic graphs
of girth four with cost ≤ 5 or with infinite cost and zero density.
Proof. LetG be the truncation of an s-transitive cubic graph, where s ≥ 3. By Lemmas 8.2
and 8.1 it is flexible with cost ≤ 5 or zero density. Then U(G) has girth 4 and it is easy to
see that it is flexible with cost ≤ 5 or zero density.
Similarly, if G is one of the tree-like graphs of Theorem 7.6, then U(G) is flexible of
girth 4 and its density is zero.
9.1 The graphs Q(n) and Qk(n)
In Section 2 we defined the chain of quadrangles Q, which we also called the infinite
crossed ladder, as a graph with vertex set V (Q) = {ui, vi : i ∈ Z} and edge set
E(Q) = {uiui+1, vivi+1, u2iv2i+1, v2iu2i+1 : i ∈ Z }.
If we replace Z in the definition of Q by Z2k and take indices modulo 2k, then we
obtain the crossed 2k-ladder, which we will denote by Qk. As the crossed 4-ladder is the
cube, which is not 2-distinguishable, we only consider crossed ladders Qk for k ≥ 3.
Q and Qk for k ≥ 3 consist of a set M of matching edges and a set C of disjoint
quadrangles, where adjacent quadrangles are connected by two matching edges. Note that
M consists of the edges of G that are not in the quadrangles and that no matching edge
is in a 4-cycle of G. Hence, if G is Q or a crossed 2k-ladder for k ≥ 3, then U(G) is
well-defined if we unfold with respect to M . By Lemma 9.1 U(G) also consists of a set
of matching edges and a set of quadrangles. This property is retained by all graphs that are
obtained by iterations of the unfolding process if we unfold with respect to the edges that
are not in the quadrangles.
We set Q(1) = Q, Qk(1) = Qk, and, for k ≥ 3, n > 1, iteratively define Qk(n) by
Qk(n) = U(Qk(n − 1)), and Q(n) = U(Q(n − 1)). We always unfold with respect to
the edges that are not in the quadrangles. They are uniquely defined, and hence so are also
Q(n) and Qk(n).
Q(1) and Qk(1), k ≥ 3, are vertex-transitive and flexible with two arc-orbits. As we
unfold with respect to the matching orbit we infer by Lemma 9.1 that Q(n) and Qk(n),
k ≥ 3, 2 ≤ n < k − 1, are connected, vertex-transitive, flexible cubic graphs with two
arc-orbits, where the cycle orbit consists of 4-cycles, and where any two adjacent 4-cycles
are connected by exactly one edge in the matching orbit.
Furthermore, by Corollary 9.8 we shall see that Qk(k − 1) is rigid with two arc-orbits.
26 Art Discrete Appl. Math. 5 (2022) #P3.15
As there is a unique matching orbit in Qk(k − 1) the graph Qk(k) = U(Qk(k − 1)) is
vertex-transitive by Lemma 9.1(v). That it has three arc-obits is asserted in Corollary 9.5.
Because we keep unfolding with respect to the edges that are not in quadrangles, the
graphs Qk(n) are still uniquely defined for n > k, but not vertex-transitive any more.
In the definition of F (G) we only merged multiple edges that arose from folded 4-
cycles, but not multiple edges that arise from matching edges. Such cases may occur, for
example when folding the chain of quadrangles Q, which we now call Q(1), see Figure 3.
The folding process maps Q(1) into a chain of alternating single and double edges and
Qk(1) into a cycle of alternating single and double edges. We set Q(0) = F (Q(1)) and
Qk(0) = F (Qk(1)). ForQ(0) see Figure 14, and for a part ofQ(0) and how it is unfolded,
see Figure 15.
u0 u1
Figure 14: Q(0).
e
f
g
h
e
h
gf
a
b
e; f; g; h a
b
Figure 15: A part G of Q(0) and how it is unfolded.
Clearly all unfolded graphs of Q(0) are isomorphic up to AutF (Q(0)), and the graphs
Qk(0) are isomorphic up to F (Qk(0)). Moreover, Aut(Q(1)) ∼= AutF (Q(0)), and
Aut(Qk(1)) ∼= AutF (Qk(0)). Recall that Q2(1) = U(Q2(0)) is the cube.
Theorem 9.4. The order of Aut(Qk(n)), k ≥ 3, 1 ≤ n ≤ k, is 2k · 2
k.
Proof. Clearly the order of AutE(Qk(0)) is 2k · 2
k. By Lemma 9.1(iii) this is also the
order of all groups Aut(Qk(n)), k ≥ 3, 1 ≤ n ≤ k.
Corollary 9.5. For k ≥ 3 the graphs Qk(k) are GRRs.
Proof. By the lemma |Aut(Qk(k))| = 2k · 2
k, which equals |V (Qk(k))|. Because Qk(k)
is vertex-transitive this implies that its vertex stabilizers are trivial. Hence Qk(k) has three
arc-orbits and is a GRR.
Note that the smallest such graph isQ3(3) and has 48 vertices. The graphQ4(4) already
has 128 vertices.
Finally, observe that the graphs Qk(n) for n > k are not vertex-transitive by
Lemma 9.1(iii).
One calls a partition of the vertex set of a vertex-transitive graph a system of imprim-
itivity if it is preserved by all automorphisms. The partition of V (G) into sets of size
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 27
one, {{v} : v ∈ V (G)} and into just one set of size |V (G)| are called trivial parti-
tions. We have seen that the sets of pairs of opposite vertices in the quadrangles of Q(1),
resp. Qk(1), k ≥ 3, form sets of imprimitivity. Each such pair is the preimage of a single
vertex in Q(0), resp. Qk(0), k ≥ 3.
Lemma 9.6. Let n ∈ N and G be one of the graphs Q(n) or Qk(n), k ≥ 3. Furthermore,
let φ denote the mapping from V (G) into V (F (G)). Then the preimages of the vertices
of Q(0), resp. Qk(0), k ≥ 3, with respect to φ
n, are a system of imprimitivity in Q(n),
resp. Qk(n), k ≥ 3.
Proof. The assertion of the lemma is true for n = 1. Suppose it is true for n ≥ 1. Let
G be Q(0) or Qk(0), k ≥ 3, and {φ
−n(v) : v ∈ V (G)}, where φ−n(v) is the preimage
of v with respect to φn. By the induction hypothesis {φ−(n−1)(v) : v ∈ V (G)} is a
system of imprimitivity in U (n−1)(G). The set φ−n(v) arises from φ−(n−1)(v) by splitting
each of its vertices into a pair of opposite vertices in the quadrangles of U(U (n−1)(G)).
The observation that Aut(U(U (n−1)(G))) preserves the set of these pairs concludes the
proof.
Let G be Q(0) or Qk(0), 3 ≤ k, and n ≥ 1. Then we call the sets φ
−n(v), v ∈ V (G),
the columns of Un(G), and denote them by cnv . Setting V (Q(0)) = Z and V (Qk(0)) =
Z2k, k ≥ 3, the columns of U
n(G) are thus cni , i ∈ Z or Z2k.
Lemma 9.7. Let n ∈ N and cn−n+2, . . . c
n
n+1 be 2n columns in Q(n) or Qk(n), where
k − 1 ≥ n. Then there exists an α ∈ Aut(Q(n)), respectively α ∈ Aut(Qk(n)), that
moves all vertices in columns cn−n+2, . . . , c
n
n+1 and fixes all other vertices.
Proof. We proceed by induction with respect to n and note that the assertion of the lemma
is true for n = 1. Let G be Aut(Q(n)) or Aut(Qk(n)) and suppose the assertion is true
for n ≥ 1. Then there exists an α ∈ Aut(G) that moves all vertices in cn−n+2, . . . , c
n
n+1
and fixes all other vertices.
Recall from Lemma 9.1, if G is a connected, vertex-transitive cubic graph with a com-
plete matching M , then Aut(G) ∼= AutE(G) ∼= Aut(U(G)). Furthermore, a vertex u
of G that is incident with a matching edge e and the non-matching edges f , g gives rise to
the vertices (f, e) and (g, e) in U(G). Each α ∈ Aut(G) then gives rise to an automor-
phism α∗ of U(G) defined by α∗(f, e) = (αE(f), αE(e)). This means that all elements in
cn+1−n+2, . . . , c
n+1
n+1 are moved by α
∗. As the edges between cn+1−n+1 and c
n+1
−n+2 are matching
edges, as well as the edges between cn+1n+1 and c
n+1
n+2, the vertices in c
n+1
−n+1 and c
n+1
n+2 are also
moved.
As α fixes all vertices in cn−n and c
n
n+3, as well as the vertices in the neighboring
columns, all edges incident with vertices in cn−n and c
n
n+3 are fixed, and thus α
∗ fixes all
vertices in cn+1−n and c
n+1
n+3.
It is easily seen that all other vertices of U(G) that are not in cn+1−n+1, . . . , c
n+1
n+2 are also
fixed. But, we wish to point out that columns cn+1−n and c
n+1
n+3 can be adjacent. In this case
n = k − 1, cn+1−n , c
n+1
n+3 are connected by matching edges and are the only columns whose
elements are fixed by α∗.
Corollary 9.8. The graphsQ(n) andQk(n), n ∈ N, k−2 ≤ n, are flexible, butQk(k−1)
is rigid.
28 Art Discrete Appl. Math. 5 (2022) #P3.15
Proof. Let α ∈ Aut(Q(n)) or Aut(Qk(n)) be the automorphism that moves all vertices in
the columns cn−n+2, . . . , c
n
n+1 and fixes all other vertices. Observe that the edges between
cn−n+2 and c
n
−n+3 are matching edges, and hence also the edges between c
n
−n and c
n
−n+1,
which are fixed. Let v be a vertex in cn−n+1. It is fixed and has two neighbors, say w,z in
cn−n+2 that are both moved. The other neighbor of v, say u is in c
n
−n, and uv is a matching
edge. Clearly the other two neighbors of u, say x,y are in cn−n−1 and fixed, unless k−1 = n,
because then c−n−1 = c−n+2k = cn+1, all of whose elements are moved, which means
that Qk(k − 1) is rigid.
The fact that Qk(n) is flexible for 1 ≤ n ≤ k − 2 and rigid for n = k − 1 is also a
consequence of Theorem 9.4 and Corollary 7.2. To see this, observe that |V (Q(k, n))| =
2k·2n. The graphs are vertex-transitive and the order of their automorphism groups is 2k·2k
by Theorem 9.4. Hence the order of their vertex-stabilizers is at least 4 when n ≤ k − 2
and 2 when n = k − 1. Now an application of Corollary 7.2 completes the argument.
Theorem 9.9. Let G be Q(n) or Qk(n), k − 1 ≥ n. Then the motion m(G) of G is
n · 2n+1, and to each automorphism α of G that stabilizes the columns of G there is a row
of 2n columns, all of whose vertices are moved by α.
Proof. Let G be Q(n) or Qk(n), n ≤ k − 1. By Lemma 9.7 there exists an automorphism
that moves all vertices in the 2n columns cn−n+2, . . . , c
n
n+1 of G. Because each column
contains 2n vertices, m(G) ≤ n · 2n+1.
If an automorphism of G moves a column into another one, then it has to move all
columns. This is easily seen, because after n foldings G maps onto Q(0) or Qk(0), whose
automorphisms move all vertices if they move at least one vertex. Hence, if not all columns
are stabilized, then at least 2k columns are moved, and thus at least 2k·2n > 2n·2n vertices.
We can thus restrict attention to automorphisms that stabilize the columns. We shall
show the existence of a row of 2n columns to each automorphism α of G, where α moves
all vertices in the columns, and where the edges between the outermost pairs of columns
are matching.
This is true for n = 1. We wish to show that it holds for n under the assumption that
it holds for n − 1 ≥ 1. Each automorphism of G is of the form α∗, where α ∈ F (G),
where F (G) is Q(n− 1) or Qk(n− 1), n ≤ k − 1. Hence there are 2n− 2 columns,all of
whose vertices are moved by α. By vertex-transitivity we can assume that the columns are
cn−1−n+3, . . . , c
n−1
n . (This also assures that the edges between the outermost pairs of columns
are matching.)
Then α∗ moves all vertices in cn−n+3, . . . , c
n
n. As the edges between columns c
n
−n+2,
cn−n+3, and c
n
n, c
n
n+1 are matching, all vertices in the columns c
n
−n+2, . . . , c
n
n+1 are moved.
Theorem 9.10. The graphs Q(n), n ∈ N, are connected, vertex-transitive, flexible cu-
bic graphs with two arc-orbits, girth 4, and motion m(Q(n)) = n · 2n+1. Furthermore,
ρ(Q(n)) = ∞ and δ(Q(n)) = 1/m(Q(n)).
Proof. By Lemma 9.7, Theorem 9.9 and Corollary 9.8 we only have to find a distinguishing
coloring with density 1/(n · 2n+1). We choose the columns cn2nj , j ∈ Z − {0} and color
one vertex in each column black. Then we choose a vertex in cn0 , say u. It is incident
with a matching edge e, say uv. Instead of coloring u, we color v black. Let c be this
coloring. Clearly the projection of c intoQ(0) distinguishes Aut(Q(0)), where Aut(Q(0))
W. Imrich et al.: Finite and infinite vertex-transitive cubic graphs and their distinguishing cost . . . 29
is the group of permutations of V (Q(0)) that preserve double and single edges. Hence any
color preserving automorphism of Q(n) stabilizes the columns, and thus fixes all black
vertices. But, because e is a matching edge, it also fixes u, and thus any color preserving
automorphism α fixes at least one vertex in any row of 2n columns. By Theorem 9.9 α
must be the identity.
Theorem 9.11. The graphs Qk(n), 1 ≤ n ≤ k − 2, are connected, vertex-transitive,
flexible cubic graphs with two arc-orbits, girth 4 and motion m(Q(n)) = n · 2n+1. Their
distinguishing cost is ρ(G) = ⌈ kn⌉, unless
k
n = 2; then ρ(G) = 3. In all cases δ(Qk(n)) ≤
1/4.
Proof. We choose the columns cn2nj , 1 ≤ j ≤ ⌈
k
n⌉, and color one vertex in each column
black. For cn0 we proceed differently. We choose a u ∈ c
n
0 . It is incident with a matching
edge e = uv, and instead of coloring u, we color v black. Let c be this coloring.
If k/n ̸= 2 the projection of c into Qk(0) distinguishes Aut(Qk(0)), hence any color
preserving automorphism ofQk(n) stabilizes the columns, and thus fixes all black vertices.
But, because e is a matching edge, it also fixes u, and thus any color preserving automor-
phism α fixes at least one vertex in any row of 2n columns. By Theorem 9.9 α must be the
identity.
If k/n = 2 we need three black vertices in Qk(0) to break Aut(Qk(0)). It is easily
seen, that it suffices to color one vertex in each of the columns cn0 , c
n
1 , and c
n
n black, in
order to obtain a distinguishing coloring. Hence ρ(G) ≤ 3.
But we still have to show that two black vertices do not suffice. Suppose two black
vertices suffice, say u,v. When k/n = 2 we have 4n columns, and because there are
automorphisms that move 2n contiguous columns and fix all vertices of the other columns,
we have to place the black vertices in columns of distance 2n, say u ∈ cn1 and v ∈ c
n
2n+1.
We do not need consider the case that the black vertices are in cn0 and c
n
2n, because there is
a reflection that interchanges the pair cn1 , c
n
2n+1 with c
n
0 , c
n
2n.
By vertex-transitivity there is an automorphism β with β(u) = v. Setw = β(v). If u =
w, then our coloring is not distinguishing, hence w ̸= u. If we can find an automorphism
that ψ that fixed v and maps w into u, then ψβ(u) = v and ψβ(v) = ψ(w) = u, and the
coloring is not distinguishing.
Let n be fixed and 1 ≤ m ≤ n. If m = 1, then cm1 consists of just two vertices, u and
w and one sees directly that one can interchange them while fixing the vertices in cmℓ , for
2m+ 1 ≤ ℓ ≤ 4n− 2m− 1. i.e. also v. We continue by induction with respect to m and
assume that for m− 1 ≥ 1 and any two vertices x, y ∈ cm−11 there is an automorphism ψ,
such that x = ψ(y) and where ψ fixes all vertices in cmℓ , 2m+1 ≤ ℓ ≤ 4n−2m−1. Given
u,w ∈ cm1 we consider their images under the folding homomorphism φ. Let x = φ(u)
and y = φ(w). There is an automorphism ψ of Q(m − 1, k) that maps y into x and fixes
cmℓ , 2m − 1 ≤ ℓ ≤ 4n − 2m + 1 pointwise. Then ψ
∗ moves the preimage of y into the
preimage of x and fixes all vertices in the columns cmℓ for 2m + 1 ≤ ℓ ≤ 4n − 2m − 1.
If ψ∗ does not move w into u, then it moves it into a vertex u′, where u, u′ are opposite
vertices in a quadrangle. But then there is an automorphism that moves u′ into u and fixes
all vertices in cmℓ for 2m+ 1 ≤ ℓ ≤ 4n− 2m− 1.
Question 9.12. We wonder whether there exist any infinite, connected, vertex-transitive
cubic graphs with positive density other than Q(n), where 1 ≤ n, or finite connected,
vertex-transitive cubic graphs with cost > 5 other than Qk(n), where 1 ≤ n ≤ k − 2 and
⌈ kn⌉ > 5. If they exist, they must be flexible with two arc-orbits.
30 Art Discrete Appl. Math. 5 (2022) #P3.15
9.1.1 Split Praeger–Xu graphs
The graphs Qk(n), 1 ≤ n ≤ k − 1, k ≥ 3, are also know as Split Praeger–Xu graphs
SPX(2, k, n), see [20]. Their vertex sets are Zn2 × Zk × {+,−} and the edge-sets consists
of the pairs
(i0, . . . , in−1, x,+)(i1, . . . , in, x+ 1,−) and (i0, . . . , in−1, x,+)(i0, . . . , in−1, x,−)
for ij ∈ Z2, x ∈ Zk.
(0; 0; 0;+)
(0; 1; 0;+)
(1; 0; 0;+)
(1; 1; 0;+)
(0; 0; 1;−)
(0; 1; 1;−)
(1; 0; 1;−)
(1; 1; 1;−)
(0; 0; 1;+)
(0; 1; 1;+)
(1; 0; 1;+)
(1; 1; 1;+)
(0; 0; 2;−)
(0; 1; 2;−)
(1; 0; 2;−)
(1; 1; 2;−)
(0; 0; 2;+)
(0; 1; 2;+)
(1; 0; 2;+)
(1; 1; 2;+)
(0; 0; 3;−)
(0; 1; 3;−)
(1; 0; 3;−)
(1; 1; 3;−)
(0; 0; 3;+)
(0; 1; 3;+)
(1; 0; 3;+)
(1; 1; 3;+)
Figure 16: Part of SPX(2, k, 2) for large k.
For SPX(2, k, 2), where k is large, compare Figure 16. We leave it to the reader to
verify that Qk(n) = SPX(2, k, n). Note that the graphs SPX(2, k, n), k ≥ 3, are flexible
for 1 ≤ n ≤ k − 2, rigid for n = k − 1 by Corollary 9.8, and not defined as SPX-graphs
for n = k.
By our preceding results onQk(n) the following theorem characterizes the Split Praeger–
Xu graphs.
Theorem 9.13. Let 1 ≤ n ≤ k − 1 and k ≥ 3. Then the Split Praeger–Xu graphs
SPX(2, k, n) are exactly those cubic graphs G that have a spanning subgraph consisting
of disjoint 4-cycles and can be folded onto Qk(0) by n foldings.
We could have based our presentation of Qk(n) on that of [20], but preferred the more
graph theoretic approach. It allowed us to directly treat the graphsQ(n), which are infinite,
and to illustrate the role of motion. Besides, it also led to a new series of GRRs.
ORCID iDs
Wilfried Imrich https://orcid.org/0000-0002-0475-9335
Thomas Lachmann https://orcid.org/0000-0003-1253-8003
Thomas W. Tucker https://orcid.org/0000-0002-7868-6925
Gundelinde M. Wiegel https://orcid.org/0000-0001-5394-1494
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