Special issue of ADAM devoted to the
International Workshop on Symmetries
of Graph and Networks 2018
We are delighted to present this special issue of the Art of Discrete and Applied Mathemat-
ics (ADAM), on topics presented or related to topics covered at the TSIMF workshop on
‘Symmetries of Graphs and Networks’, held at Sanya, on the beautiful semi-tropical island
province of Hainan (China), in January 2018.
This workshop added to the series of conferences and workshops on symmetries of
graphs and networks initiated at BIRS (Canada) in 2008 and progressed in Slovenia every
two years from 2010 to 2016.
It was attended by 50 mathematicians from China and other parts of the world (includ-
ing Australia, Canada, New Zealand, Slovakia, Slovenia, South Korea and the USA), many
of whom gave lectures on a range of topics involving the symmetries of graphs and maps,
including Cayley graphs, arc-transitive graphs and digraphs, covering graphs, regular maps
on surfaces, and regular Cayley maps, plus related topics such as graph embeddings and
skew morphisms of groups.
Participants very much enjoyed the venue, which is similar in style to the BIRS facilities
in Banff and Oaxaca and the institute at Oberwolfach, giving plenty of opportunity for
interactions between participants, and stimulating further research on the topics covered.
This issue contains a number of interesting papers resulting from or associated with the
workshop. We would like to thank the authors for their valuable contributions. Also we are
very grateful to the TSIMF for its generous financial support and administrative assistance
for the workshop, and to the chief editors of ADAM for making this special issue possible.
Marston Conder (University of Auckland, New Zealand) and
Yan-Quan Feng (Beijing Jiaotong University, China)
Principal organisers of the workshop and Editors of this issue
i
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.01
https://doi.org/10.26493/2590-9770.1254.266
(Also available at http://adam-journal.eu)
Digraphs with small automorphism groups
that are Cayley on two nonisomorphic groups∗
Luke Morgan †
Centre for the Mathematics of Symmetry and Computation,
Department of Mathematics and Statistics (M019), The University of Western Australia,
35 Stirling Highway, Crawley, 6009, Australia
Current address: University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia,
and University of Primorska, IAM, Muzejski trg 2, 6000 Koper, Slovenia
Joy Morris ‡
Department of Mathematics and Computer Science, University of Lethbridge,
Lethbridge, AB T1K 3M4, Canada
Gabriel Verret
Department of Mathematics, The University of Auckland,
Private Bag 92019, Auckland 1142, New Zealand
Received 5 November 2018, accepted 30 April 2019, published online 9 January 2020
Abstract
Let Γ = Cay(G,S) be a Cayley digraph on a group G and let A = Aut(Γ). The
Cayley index of Γ is |A : G|. It has previously been shown that, if p is a prime, G is a
cyclic p-group and A contains a noncyclic regular subgroup, then the Cayley index of Γ is
superexponential in p.
We present evidence suggesting that cyclic groups are exceptional in this respect. Specif-
ically, we establish the contrasting result that, if p is an odd prime and G is abelian but not
cyclic, and has order a power of p at least p3, then there is a Cayley digraph Γ on G whose
Cayley index is just p, and whose automorphism group contains a nonabelian regular sub-
group.
∗We thank the referee for their comments on the paper. The first and third authors also thank the second author
and the University of Lethbridge for hospitality.
†The first author was supported by the Australian Research Council grant DE160100081.
‡The second author was supported by the Natural Science and Engineering Research Council of Canada grant
RGPIN-2017-04905.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 3 (2020) #P1.01
Keywords: Cayley digraphs, Cayley index.
Math. Subj. Class.: 05C25, 20B25
1 Introduction
Every digraph and group in this paper is finite. A digraph Γ consists of a set of vertices
V(Γ) and a set of arcs A(Γ), each arc being an ordered pair of distinct vertices. (Our
digraphs do not have loops.) We say that Γ is a graph if, for every arc (u, v) of Γ, (v, u) is
also an arc. Otherwise, Γ is a proper digraph.
The automorphisms of Γ are the permutations of V(Γ) that preserve A(Γ). They form
a group under composition, denoted Aut(Γ).
Let G be a group and let S be a subset of G that does not contain the identity. The
Cayley digraph on G with connection set S is Γ = Cay(G,S), the digraph with vertex-set
G and where (u, v) ∈ A(Γ) whenever vu−1 ∈ S. The index of G in Aut(Γ) is called the
Cayley index of Γ.
It is well-known that a digraph is a Cayley digraph onG if and only if its automorphism
group contains the right regular representation of G. A digraph may have more than one
regular subgroup in its automorphism group and hence more than one representation as a
Cayley digraph. This is an interesting situation that has been studied in [2, 9, 12, 13, 15],
for example.
Let p be a prime. Joseph [7] proved that if Γ has order p2 and Aut(Γ) has two regular
subgroups, one of which is cyclic and the other not, then Γ has Cayley index at least pp−1.
The second author generalised this in [11], showing that if p > 3, Γ has order pn and
Aut(Γ) has two regular subgroups, one of which is cyclic and the other not, then Γ has
Cayley index at least pnp−p−n+1. A simpler proof of this was later published in [1]. Kovács
and Servatius [8] proved the analogous result when p = 2.
The theme of the results above is that if Aut(Γ) has two regular subgroups, one of
which is cyclic and the other not, then Γ must have “large” Cayley index. The goal of this
paper is to show that cyclic p-groups are exceptional with respect to this property, at least
among abelian p-groups. More precisely, we prove the following.
Theorem 1.1. Let p be an odd prime and let G be an abelian p-group. If G has order at
least p3 and is not cyclic, then there exists a proper Cayley digraph on G with Cayley index
p and whose automorphism group contains a nonabelian regular subgroup.
It would be interesting to generalise Theorem 1.1 to nonabelian p-groups and to 2-
groups. More generally, we expect that “most” groups admit a Cayley digraph of “small”
Cayley index such that the automorphism group of the digraph contains another (or even a
nonisomorphic) regular subgroup. At the moment, we do not know how to approach this
problem in general, or even what a sensible definition of “small” might be. (Lemma 3.2
shows that the smallest index of a proper subgroup of either of the regular subgroups is a
lower bound – and hence that the Cayley index of p in Theorem 1.1 is best possible.) As
an example, we prove the following.
E-mail addresses: luke.morgan@famnit.upr.si (Luke Morgan), joy.morris@uleth.ca (Joy Morris),
g.verret@auckland.ac.nz (Gabriel Verret)
L. Morgan et al.: Digraphs with small automorphism groups that are Cayley on two . . . 3
Proposition 1.2. Let G be a group generated by an involution x and an element y of order
3, and such that Z6  G  Z3 oZ2. IfG has a subgroupH of index 2, then there is a Cayley
digraph Γ with Cayley index 2 such that Aut(Γ) contains a regular subgroup distinct from
G and isomorphic to H × Z2.
This paper is laid out as follows. Section 2 includes structural results on cartesian prod-
ucts of digraphs that will be required in the proofs of our main results, while in Section 3
we collect results about automorphism groups of digraphs. Section 4 consists of the proof
of Theorem 1.1. Finally, in Section 5 we prove Proposition 1.2 and consider the case of
symmetric groups.
2 Cartesian products
The main result of this section is a version of a result about cartesian products of graphs
due to Imrich [6, Theorem 1] that is adapted to the case of proper digraphs. Imrich’s proof
can be generalised directly to all digraphs, but his proof involves a detailed case-by-case
analysis for small graphs, which can be avoided by restricting attention to proper digraphs.
Generally, there are two notions of connectedness for digraphs: a digraph is weakly
connected if its underlying graph is connected, and strongly connected if for every ordered
pair of vertices there is a directed path from the first to the second. In a finite Cayley
digraph, these notions coincide (see [4, Lemma 2.6.1] for example). For this reason, we
will refer to Cayley digraphs as simply being connected or disconnected.
The complement of a digraph Γ, denoted by Γ, is the digraph with vertex-set V(Γ),
with (u, v) ∈ A(Γ) if and only if (u, v) 6∈ A(Γ), for every two distinct vertices u and v of
Γ. It is easy to see that a digraph and its complement have the same automorphism group.
Given digraphs Γ and ∆, the cartesian product Γ∆ is the digraph with vertex-set
V(Γ) × V(∆) and with ((u, v), (u′, v′)) being an arc if and only if either u = u′ and
(v, v′) ∈ A(∆), or v = v′ and (u, u′) ∈ A(Γ). For each u ∈ V(Γ), we obtain a copy ∆u of
∆ in Γ∆, the induced digraph on {(u, v) | v ∈ V(∆)}. Similarly, for each v ∈ V(∆),
we obtain a copy Γv of Γ in Γ∆ (defined analogously).
A digraph Γ is prime with respect to the cartesian product if the existence of an isomor-
phism from Γ to Γ1Γ2 implies that either Γ1 or Γ2 has order 1, so that Γ is isomorphic
to either Γ1 or Γ2.
It is well known that, with respect to the cartesian product, graphs can be factorised
uniquely as a product of prime factors. Digraphs also have this property.
Theorem 2.1 (Walker, [14]). Let Γ1, . . . ,Γk,Γ′1, . . . ,Γ′` be weakly connected prime di-
graphs. If α is an isomorphism from Γ1 · · ·Γk to Γ′1 · · ·Γ′`, then k = ` and there
exist a permutation π of {1, . . . , k} and isomorphisms αi from Γi to Γ′π(i) such that α is
the product of the αis (1 6 i 6 k).
Theorem 2.1 is a corollary of [14, Theorem 10], as noted in the “Applications” section
of [14], but is more commonly proved by replacing a digraph Γ by its shadow S(Γ) (by
replacing arcs and digons with edges), noting that S(Γ1Γ2) = S(Γ1)S(Γ2), and
using the unique prime factorisation for graphs with respect to the cartesian product.
We now present the version of Imrich’s result that applies to proper digraphs. During
the refereeing process of this article, we were made aware of [5, Theorem 1], which is
similar to the theorem below.
4 Art Discrete Appl. Math. 3 (2020) #P1.01
Theorem 2.2. If Γ is a proper digraph, then at least one of Γ or Γ is prime with respect
to cartesian product and is weakly connected.
Proof. We first make a key observation. Let u = (x, y) and v = (x′, y) be distinct vertices
of a cartesian product X Y lying in the same copy Xy of X . If w = (z, y′) is a vertex
adjacent to both u and v (with no specification on the direction of the arcs), we claim that
w ∈ Xy . Indeed, since w ∼ u, if y 6= y′, then z = x and y ∼ y′ in Y . It follows that
w = (x, y′) cannot be adjacent to v = (x′, y) in X Y since x 6= x′. Hence y = y′ and
w ∈ Xy .
Suppose that either Γ or Γ is not weakly connected, say Γ. Then Γ is weakly connected.
We may assume that Γ admits a non-trivial factorisation as Γ = X Y (so that X and Y
have at least two vertices).
Suppose there exist a weakly connected component C of Γ and a copy of X , Xy say,
such that Xy has two vertices that are not in C, u and v say. By the key observation applied
to Γ, every vertex of C is also in Xy . Let c ∈ C and let y′ ∈ Y with y′ 6= y. Then, since
C is contained in Xy , there are at least two vertices of Xy′ with no arc in Γ in common
with c. Applying the key observation yields that c is in Xy
′
, which is a contradiction. We
may thus assume that every weakly connected component of Γ misses at most one vertex
from each copy of X . In particular, Γ has exactly two weakly connected components, and
therefore X has exactly two vertices. A symmetric argument yields that Y also has two
vertices and thus Γ has four vertices, and the result can be checked by brute force.
Now we may assume that both Γ and Γ are weakly connected. Towards a contradiction,
assume that Γ = AB and that ϕ is an isomorphism from Γ to C D, where A, B, C,
and D all have at least 2 vertices.
Since Γ is a proper digraph, without loss of generality so is A, and A has an arc (a, a′)
such that (a′, a) is not an arc of A. Let b be a vertex of B.
Pick b′ to be a vertex ofB distinct from b. We claim thatϕ((a, b)), ϕ((a′, b)), ϕ((a, b′)),
ϕ((a′, b′)) all lie in some copy of either C or D. The digraph in Figure 1 is the subdigraph
of Γ under consideration.
(a; b) (a0; b)
(a; b0) (a0; b0)
Figure 1: A subdigraph of Γ.
Since every arc in C D lies in either a copy of C or D, we may assume that the
arc from ϕ((a, b)) to ϕ((a′, b′)) lies in some copy Cd of C, say ϕ((a, b)) = (c, d) and
ϕ((a′, b′)) = (c′, d), with c, c′ ∈ V(C) and d ∈ V(D). Towards a contradiction, suppose
that ϕ((a′, b)) /∈ Cd. Then the arc from ϕ((a′, b)) to (c, d) must lie in Dc, so ϕ((a′, b)) =
(c, d′) for some vertex d′ of D. Since there is a path of length 2 via ϕ((a, b′)) from (c′, d)
L. Morgan et al.: Digraphs with small automorphism groups that are Cayley on two . . . 5
to (c, d′) and since ϕ((a, b′)) 6= (c, d) = ϕ((a, b)), we must have ϕ((a, b′)) = (c′, d′). But
now we have an arc from (c, d′) to (c, d) and an arc from (c′, d) to (c′, d′), so arcs in both
directions between d and d′ inD. This implies that there are arcs in both directions between
(c, d) = ϕ((a, b)) and (c, d′) = ϕ((a′, b)), a contradiction. Hence ϕ((a′, b)) ∈ Cd, and by
the observation in the first paragraph, we have ϕ((a, b′)) ∈ Cd also. This proves the claim.
By repeatedly applying the claim, all elements of ϕ({a, a′} × V(B)) lie in some copy
of C or D, say, Cd. Let a′′ ∈ V(A) − {a, a′} and let b and b′ be distinct vertices of B.
By the definitions of cartesian product and complement, there are arcs in both directions
between (a′′, b) and (a, b′) and between (a′′, b) and (a′, b′). Thus, by the observation in the
first paragraph, ϕ((a′′, b)) also lies in Cd. This shows that every vertex of Γ lies in Cd, so
D is trivial. This is the desired contradiction.
Remark 2.3. Imrich’s Theorem [6, Theorem 1] states that, for every graph Γ, either Γ or
Γ is prime with respect to the cartesian product, with the following exceptions: K2K2,
K2K2, K2K2K2, K4K2, K2K−4 , and K3K3, where Kn denotes the com-
plete graph on n vertices and K−4 denotes K4 with an edge deleted. These would therefore
be the complete list of exceptions to Theorem 2.2 if we removed the word ‘proper’ from
the hypothesis.
Remark 2.4. While most of our results apply only to finite digraphs, Theorem 2.2 also
applies to infinite ones (as does Imrich’s Theorem). The proof is the same.
Corollary 2.5. Let Γ1 be a proper Cayley digraph on G with Cayley index i1 and let
Γ2 be a connected Cayley digraph on H with Cayley index i2. If i1 > i2, then at least
one of Γ1Γ2 or Γ1Γ2 has automorphism group equal to Aut(Γ1)× Aut(Γ2) and, in
particular, is a proper Cayley digraph on G×H with Cayley index i1i2.
Proof. By Theorem 2.2, one of Γ1 and Γ1 is connected and prime with respect to the carte-
sian product, say Γ1 without loss of generality. Clearly, we have Aut(Γ1) × Aut(Γ2) 6
Aut(Γ1Γ2). Since i1 > i2, Γ1 cannot be a cartesian factor of Γ2. It follows by Theo-
rem 2.1 that every automorphism of Γ1Γ2 is a product of an automorphism of Γ1 and an
automorphism of Γ2, so that Aut(Γ1)×Aut(Γ2) = Aut(Γ1Γ2).
3 Additional background
The following lemma is well known and easy to prove.
Lemma 3.1. Let G be a group, let S ⊆ G and let α ∈ Aut(G). If Sα = S, then α induces
an automorphism of Cay(G,S) which fixes the vertex corresponding to the identity.
The next lemma is not used in any of our proofs, but it shows that the Cayley indices in
Theorem 1.1 and Theorem 1.2 are as small as possible.
Lemma 3.2. If Cay(G,S) has Cayley index i and Aut(Cay(G,S)) has at least two reg-
ular subgroups, then G has a proper subgroup of index at most i.
Proof. Let A = Aut(Cay(G,S)) and let H be a regular subgroup of A different from
G. Clearly, G ∩ H is a proper subgroup of G and we have |A| > |GH| = |G||H||G∩H| hence
i = |A : G| = |A : H| > |G : G ∩H|.
6 Art Discrete Appl. Math. 3 (2020) #P1.01
If v is vertex of a digraph Γ, then Γ+(v) denotes the out-neighbourhood of v, that is,
the set of vertices w of Γ such that (v, w) is an arc of Γ.
Let A be a group of automorphisms of a digraph Γ. For v ∈ V(Γ) and i > 1, we use
A
+[i]
v to denote the subgroup of Av that fixes every vertex u for which there is a directed
path of length at most i from v to u.
Lemma 3.3. Let Γ be a connected digraph, let v be a vertex of Γ and let A be a transitive
group of automorphisms of Γ. If A+[1]v = A
+[2]
v , then A
+[1]
v = 1.
Proof. By the transitivity ofA, we haveA+[1]u = A
+[2]
u for every vertex u. Using induction
on i, it easily follows that, for every i > 1, we have A+[i]v = A
+[i+1]
v . By connectedness,
this implies that A+[1]v = 1.
Lemma 3.4. Let p be a prime and let A be a permutation group whose order is a power
of p. If A has a regular abelian subgroup G of index p and G has a subgroup M of index
p that is normalised but not centralised by a point-stabiliser in A, then A has a regular
nonabelian subgroup.
Proof. Let Av be a point-stabiliser in A. Note that A = Go Av and that |Av| = p. Since
M is normal in G and normalised by Av , it is normal in A and has index p2. Clearly,
M o Av 6= G hence A/M contains at least two subgroups of order p and must therefore
be elementary abelian.
Let α be a generator of Av and let g ∈ G −M . By the previous paragraph, we have
(gα)p ∈ M . Let H = 〈M, gα〉. Since M is centralised by g but not by α, it is not
centralised by gα hence H is nonabelian. Further, we have |H| = p|M | = |G|, so that H
is normal in A. If H was non-regular, it would contain all point-stabilisers of A, and thus
would contain α and hence also g. This would give G = 〈M, g〉 6 H , a contradiction.
Thus H is a regular nonabelian subgroup of A.
4 Proof of Theorem 1.1
Throughout this section, p denotes an odd prime. In Section 4.1, we show that Theorem 1.1
holds when G ∼= Z3p. In Sections 4.2 and 4.3, we subdivide abelian groups of rank 2 and
order at least p3 into two families, and show that the theorem holds for all such groups.
Finally, in Section 4.4, we explain how these results can be applied to show that the theorem
holds for all abelian groups of order at least p3.
4.1 G ∼= Z3p
Write G = 〈x, y, z〉, let α be the automorphism of G that maps (x, y, z) to (xy, yz, z), let
S = {xαi , yαi : i ∈ Z}, let Γ = Cay(G,S), and let A = Aut(Γ). Note that Γ is a proper
digraph (this will be needed in Section 4.4).
It is easy to see that, for i ∈ N, we have xαi = xyiz(
i
2), yα
i
= yzi and zα
i
= z. In
particular, α has order p and |S| = 2p. By Lemma 3.1, G o 〈α〉 6 A. We will show that
equality holds.
Using the formulas above, it is not hard to see that the induced digraph on S has exactly
2p arcs: (xα
i
, xα
i+1
) and (yα
i
, xα
i+1
), where i ∈ Zp. Thus, for every s ∈ S,A1,s = A+[1]1 .
By vertex-transitivity, Au,v = A
+[1]
u for every arc (u, v).
L. Morgan et al.: Digraphs with small automorphism groups that are Cayley on two . . . 7
Let s ∈ S. We have already seen that A1,s = A+[1]1 . Let t ∈ S. From the structure of
the induced digraph on S = Γ+(1), we see that t has an out-neighbour in S, so that both
t and this out-neighbour are fixed by A1,s. It follows that A1,s fixes all out-neighbours of
t. We have shown that A1,s = A
+[2]
1 . By Lemma 3.3, it follows that A1,s = 1. Since the
induced digraph on S is not vertex-transitive and α ∈ A1, the A1-orbits on S have length
p. Hence |A1| = p|A1,s| = p. Thus, Γ has Cayley index p and A = G o 〈α〉. Finally,
we apply Lemma 3.4 with M = 〈y, z〉 to deduce that A contains a nonabelian regular
subgroup.
4.2 G ∼= Zpn × Zp with n > 2
Choose x, y ∈ G of order pn and p respectively so that G = 〈x, y〉. Let x0 = xp
n−1
, let α
be the automorphism of G that maps (x, y) to (xy, x0y), let S = {xα
i
, yα
i
: i ∈ Z} and
let Γ = Cay(G,S). Again, note that Γ is a proper digraph.
Since n > 2, x0 is fixed by α. It follows that, for i ∈ N, we have xα
i
= xyix
(i2)
0 and
yα
i
= yxi0. In particular, α has order p and |S| = 2p.
Using these formulas, it is not hard to see that the induced digraph on S has exactly
2p arcs: (xα
i
, xα
i+1
) and (yα
i
, xα
i+1
), where i ∈ Zp. The proof is now exactly as in the
previous section, except that we use M = 〈xp, y〉 when applying Lemma 3.4.
4.3 G ∼= Zpn × Zpm with n > m > 2
Choose x, y ∈ G of order pn and pm respectively so that G = 〈x, y〉. Let x0 = xp
n−1
,
let y0 = yp
m−1
, let α be the automorphism of G that maps (x, y) to (xy0, yx0), let S =
{xαi , yαi , (xy−1)αi : i ∈ Z}, let Γ = Cay(G,S), and let A = Aut(Γ). Again, note that
Γ is a proper digraph.
Since n > m > 2, x0 and y0 are both fixed by α. It follows that, for i ∈ N, we have
xα
i
= xyi0, and y
αi = yxi0. In particular, α has order p and |S| = 3p. By Lemma 3.1,
Go 〈α〉 6 A.
Using the formulas above, it is not hard to see that the induced digraph on S has exactly
2p arcs: ((xy−1)α
i
, xα
i
) and (yα
i
, xα
i
), where i ∈ Zp. It follows that |A1 : A1,x| = p.
We will show that A1,x = 1, which will imply that A = Go 〈α〉.
Let X = {xαi : i ∈ Z} = x〈y0〉, Y = {xα
i
: i ∈ Z} = y〈x0〉 and Z = {(xy−1)α
i
:
i ∈ Z} = xy−1〈x−10 y0〉. It follows from the previous paragraph that X is an orbit of A1
on S.
Note that the p elements of Y 2 = y2〈x0〉 are out-neighbours of every element of Y .
Similarly, the p elements of Z2 are out-neighbours of every element of Z. On the other
hand, one can check that an element of Y and an element of Z have a unique out-neighbour
in common, namely their product. This shows that Y and Z are blocks for A1. We claim
that Y and Z are orbits of A1.
Let Y1 = Y and, for i > 2, inductively define Yi =
⋂
x∈Yi−1 Γ
+(x). Define Zi
analogously. Let g ∈ A1. By induction, Y gi ∈ {Yi, Zi}, and Y
g
i = Zi if and only if
Y g = Z. Note that Yi = Y i = yi〈x0〉 and Zi = Zi = xiy−i〈x−10 y0〉. If n = m, then
1 ∈ x0y−10 〈x
−1
0 y0〉 = Zpm−1 , but 1 /∈ y0〈x0〉 = Ypm−1 , so Y and Z are orbits for A1. We
may thus assume that n > m. Note that y0 ∈ y0〈x0〉 = Ypm−1 , and y0 is an in-neighbour
of x ∈ X . However, Zpm−1 = xp
m−1
y−10 〈x
−1
0 y0〉. Since n > m, we see that no vertex of
8 Art Discrete Appl. Math. 3 (2020) #P1.01
Zpm−1 is an in-neighbour of a vertex ofX . Again it follows that Y and Z are orbits for A1.
Considering the structure of the induced digraph on S, it follows that, for every s ∈ S,
A1,s = A
+[1]
1 . By vertex-transitivity, Au,v = A
+[1]
u for every arc (u, v). Since elements of
Y and Z have an out-neighbour in S, A1,x fixes the out-neighbours of elements of Y and
Z. Furthermore, for every i ∈ Z, xyi0y is a common out-neighbour of xyi0 and y, hence it
is fixed by A1,x. Thus, every element of X has an out-neighbour fixed by A1,x. It follows
that A1,x fixes all out-neighbours of elements of X and thus A1,x = A
+[1]
1 = A
+[2]
1 . By
Lemma 3.3, it follows that A1,x = 1. As in Section 4.1, we can also conclude |A1| = p,
Γ has Cayley index p and A = G o 〈α〉. Finally, applying Lemma 3.4 with M = 〈xp, y〉
implies that A contains a nonabelian regular subgroup.
4.4 General case
Recall that G is an abelian p-group that has order at least p3 and is not cyclic. By the
Fundamental Theorem of Finite Abelian Groups, we can write G = G1 × G2, where G1
falls into one of the three cases that have already been dealt with in this section.
(More explicitly, if G is not elementary abelian, then we can take G1 isomorphic to
Zpn × Zpm with n > 2 and m > 1. If G is elementary abelian, then, since |G| > p3, we
can take G1 isomorphic to Z3p.)
We showed in the previous three sections that there exists a proper Cayley digraph Γ1
on G1 with Cayley index equal to p and whose automorphism group contains a nonabelian
regular subgroup.
Note that every cyclic group admits a prime, connected Cayley digraph whose Cayley
index is 1. (For example, the directed cycle of the corresponding order.) Since G2 is a
direct product of cyclic groups, applying Corollary 2.5 iteratively yields a proper connected
Cayley digraph Γ on G1 × G2 with automorphism group Aut(Γ1) × G2. In particular, Γ
has Cayley index p and its automorphism group contains a nonabelian regular subgroup.
This concludes the proof of Theorem 1.1.
In fact, the proof above yields the following stronger result.
Theorem 4.1. Let G be an abelian group. If there is an odd prime p such that the Sylow p-
subgroup of G is neither cyclic nor elementary abelian of rank 2, then G admits a proper
Cayley digraph with Cayley index p whose automorphism group contains a nonabelian
regular subgroup.
5 Proof of Proposition 1.2
We begin with a lemma that helps to establish the existence of regular subgroups.
Lemma 5.1. Let G be a group with nontrivial subgroups H and B such that G = HB
and H ∩ B = 1, and let Γ = Cay(G,S) be a Cayley digraph on G. If S is closed under
conjugation by B, then Aut(Γ) has a regular subgroup distinct from the right regular
representation of G and isomorphic to H ×B.
Proof. Let A = Aut(Γ). For g ∈ G, let `g and rg denote the permutations of G induced
by left and right multiplication by g, respectively. Similarly, for g ∈ G, let cg denote the
permutation of G induced by conjugation by g. For X 6 G, let RX = 〈rx : x ∈ X〉.
Let LB = 〈`b : b ∈ B〉 and CB = 〈cb : b ∈ B〉. Note that RH 6 A. For every g ∈ G,
L. Morgan et al.: Digraphs with small automorphism groups that are Cayley on two . . . 9
rgcg−1 = `g . For all b ∈ B, we have rb ∈ A and, since S is closed under conjugation by
B, cb−1 ∈ A hence LB 6 A.
Let K = 〈LB , RH〉. If K = RG, then LB 6 RG which implies that CB 6 RG,
contradicting the fact that RG is regular. Thus K 6= RG. Note that LB and RH commute.
Suppose that k ∈ RH ∩ LB , so k = rh = `b for some h ∈ H and some b ∈ B. Thus
h = 1rh = 1k = 1`b = b.
Since H ∩ B = 1, this implies k = 1. It follows that RH ∩ LB = 1 and hence
K = RH × LB ∼= H × B. Finally, suppose that some k = rh`b ∈ K fixes 1. It follows
that 1rh`b = 1 = bh so that b ∈ H , a contradiction. This implies that K is regular, which
concludes the proof.
We now prove a general result, which together with Lemma 5.1 will imply Proposi-
tion 1.2.
Proposition 5.2. Let G be a group generated by an involution x and an element y of
order 3, let S = {x, y, yx} and let Γ = Cay(G,S). If G is isomorphic to neither Z6 nor
Z3 o Z2 ∼= Z23 o Z2, then Γ has Cayley index 2.
Proof. Clearly, Γ is connected. Since G is not isomorphic to Z6, we have yx 6= y. In
particular, we have |S| = 3. If yx = y−1, then G ∼= Sym(3) and the result can be checked
directly. We therefore assume that yx 6= y−1. Since G  Z3 o Z2, we have yxy 6= yyx.
We have that Γ+(x) = {1, yx, xy}, Γ+(y) = {xy, y2, yxy} and Γ+(yx) = {yx, yyx,
(y2)x}. One can check that the only equalities between elements of these sets are the ones
between elements having the same representation. In other words,
|{1, yx, xy, y2, yxy, yyx, (y2)x}| = 7.
(For example, if yx = yxy, then xy
−1
= yx, contradicting the fact that x and y have
different orders.)
Let A = Aut(Γ) and let cx denote conjugation by x. Note that cx ∈ A1. We first show
that A+[1]1 = 1. It can be checked that y
2 is the unique out-neighbour of y that is also an
in-neighbour of 1, hence it is fixed by A+[1]1 , and so is (y
2)x by analogous reasoning. We
have seen earlier that xy is the unique common out-neighbour of x and y, hence it too is
fixed by A+[1]1 , and similarly for yx. Being the only remaining out-neighbours of y, y
xy
must be also fixed, and similarly for yyx. Thus A+[1]1 = A
+[2]
1 . Since Γ is connected,
Lemma 3.3 implies that A+[1]1 = 1.
Note that x is the only out-neighbour of 1 that is also an in-neighbour, hence it is fixed
by A1, whereas cx interchanges y and yx. It follows that |A1| = |A1 : A+[1]1 | = 2 and Γ
has Cayley index 2, as desired.
Proof of Proposition 1.2. Let Γ = Cay(G, {x, y, yx}). By Proposition 5.2, Γ has Cayley
index 2. Since |G : H| = 2 and y has order 3, we have y ∈ H . As 〈x, y〉 = G, we
have x /∈ H and G = H o 〈x〉. Clearly, {x, y, yx} is closed under conjugation by x. It
follows by Lemma 5.1 that Aut(Γ) has a regular subgroup distinct from G and isomorphic
to H × 〈x〉 ∼= H × Z2.
10 Art Discrete Appl. Math. 3 (2020) #P1.01
It was shown by Miller [10] that, when n > 9, Sym(n) admits a generating set con-
sisting of an element of order 2 and one of order 3; this is also true when n ∈ {3, 4}. In
these cases, we can apply Proposition 1.2 with H = Alt(n) to obtain a Cayley digraph
on Sym(n) that has Cayley index 2 and whose automorphism group contains a regular
subgroup isomorphic to Alt(n)× Z2.
A short alternate proof of this fact can be derived from a result of Feng [3]. This yields
a Cayley graph and is valid for n > 5.
Proposition 5.3. If n > 5, then there is a Cayley graph on Sym(n) with Cayley index 2,
whose automorphism group contains a regular subgroup isomorphic to Alt(n)× Z2.
Proof. Let T = {(1 2), (2 3), (2 4)} ∪ {(i i + 1) : 4 6 i 6 n − 1} and let Γ =
Cay(Sym(n), T ). Note that all elements of T are transpositions. Let Tra(T ) be the trans-
position graph of T , that is, the graph with vertex-set {1, . . . , n} and with an edge {i, j}
if and only if (i j) ∈ T . Note that Tra(T ) is a tree and thus T is a minimal generating
set for Sym(n) (see for example [4, Section 3.10]). Let B = 〈(1 3)〉. Since n > 5,
Aut(Tra(T )) = B. It follows by [3, Theorem 2.1] that Aut(Γ) ∼= Sym(n) o B. In
particular, Γ has Cayley index 2.
Note that Sym(n) = Alt(n)oB and that T is closed under conjugation byB. Applying
Lemma 5.1 with H = Alt(n) shows that Aut(Γ) has a regular subgroup isomorphic to
Alt(n)× Z2.
ORCID iDs
Luke Morgan https://orcid.org/0000-0003-2396-5430
Gabriel Verret https://orcid.org/0000-0003-1766-4834
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.02
https://doi.org/10.26493/2590-9770.1291.c54
(Also available at http://adam-journal.eu)
On strongly sequenceable abelian groups
Brian Alspach , Georgina Liversidge
School of Mathematical and Physical Sciences, University of Newcastle,
Callaghan, NSW 2308, Australia
Received 4 February 2019, accepted 7 August 2019, published online 22 July 2020
Abstract
A group is strongly sequenceable if every connected Cayley digraph on the group ad-
mits an orthogonal directed cycle or an orthogonal directed path. This paper deals with the
problem of whether finite abelian groups are strongly sequenceable. A method based on
posets is used to show that if the connection set for a Cayley digraph on an abelian group
has cardinality at most nine, then the digraph admits either an orthogonal directed path or
an orthogonal directed cycle.
Keywords: Strongly sequenceable, abelian group, diffuse poset, sequenceable poset.
Math. Subj. Class. (2020): 05C25
1 Introduction
The Cayley digraph
−−→
Cay(G;S) on the groupG has the elements ofG for the vertex set and
an arc (g, h) from g to h whenever h = gs for some s ∈ S, where S ⊂ G and 1 6∈ S. The
set S is called the connection set. It is easy to see that left-multiplication by any element
of G is an automorphism of
−−→
Cay(G;S) which implies that the automorphism group of
−−→
Cay(G;S) contains the left-regular representation of G.
A given s ∈ S generates a spanning digraph of
−−→
Cay(G;S) composed of vertex-disjoint
directed cycles of length |s|, where |s| denotes the order of s. We call this subdigraph a
(1, 1)-directed factor because the in-valency and out-valency at each vertex is 1. Hence,
there is a natural factorization of
−−→
Cay(G;S) into |S| arc-disjoint (1, 1)-directed factors.
This is the Cayley factorization of
−−→
Cay(G;S) and is denoted F(G;S).
Let
−−→
Cay(G;S) be a Cayley digraph on a group G. A subdigraph
−→
Y of
−−→
Cay(G;S) of
size |S| (the size is the number of arcs in
−→
Y ), is orthogonal to F(G;S) if
−→
Y has one arc
E-mail addresses: brian.alspach@newcastle.edu.au (Brian Alspach), gliv560@aucklanduni.ac.nz
(Georgina Liversidge)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 3 (2020) #P1.02
from each (1, 1)-directed factor of F(G;S). In order to simplify the language, we simply
say that
−−→
Cay(G;S) admits an orthogonal
−→
Y .
The complete digraph
−→
Kn may be viewed as a Cayley digraph
−→
K(G) on any group G
of order n by choosing the connection set to be G \ {1}. B. Gordon [13] defined a group G
to be sequenceable if
−→
K(G) admits an orthogonal Hamilton directed path (he used different
language). Gordon was motivated by looking for methods to produce row-complete Latin
squares and a sequenceable group gives rise to a row-complete Latin square.
From his work on the Heawood map coloring problem, G. Ringel [18] asked when
does
−→
K(G) admit an orthogonal directed cycle of length |G| − 1 (he also used different
language). A group G for which this holds was called R-sequenceable in [12].
So the two notions of a sequenceable group and an R-sequenceable group were mo-
tivated by quite disparate mathematical problems, but as we have seen they are closely
related. The topic of sequenceable and R-sequenceable groups has generated, and contin-
ues to generate, a considerable amount of research. There have been surveys [11, 17] and
many papers including [1, 2, 3, 4, 5, 6, 7, 12, 13, 14, 16].
The following definition is a natural extension of sequenceable and R-sequenceable
groups.
Definition 1.1. A group G is strongly sequenceable if every connected Cayley digraph on
G admits either an orthogonal directed path or an orthogonal directed cycle.
An abelian group cannot be both sequenceable andR-sequenceable, but by allowing ei-
ther an orthogonal directed path or an orthogonal directed cycle in the definition of strongly
sequenceable, we guarantee that when an abelian group is strongly sequenceable, it is either
sequenceable or R-sequenceable.
The first author and T. Kalinowski have posed the following problem.
Research problem 1.2. Determine the strongly sequenceable groups.
2 Abelian groups
It is not difficult to verify that the non-abelian group of order 6 is not strongly sequenceable.
The only connection set for which it fails is the one giving
−→
K6.
There has been some work on the preceding problem for abelian groups. We use addi-
tive notation for abelian groups which is the case for the remainder of this paper. The first
author asked whether cyclic groups are strongly sequenceable in 2000. Bode and Harborth
[9] showed that the answer is yes for the cyclic group Zn whenever the the sum of the
elements in the connection set S is not 0 and either |S| = n− 1 or |S| = n− 2.
The same problem was discovered independently by Archdeacon, also restricted to
cyclic groups, and studied in [8]. The authors prove that all cyclic groups of order at most
25 are strongly sequenceable. They also show that there is an orthogonal directed path or
orthogonal directed cycle whenever the connection set S has cardinality at most 6 for all
cyclic groups..
Costa, Morini, Pasotti and Pellegrini [10] observed that almost all the methods em-
ployed for the previously cited work do not depend on the group being cyclic. Conse-
quently, their paper deals with abelian groups. They use computer verification to show that
all abelian groups of orders at most 23 are strongly sequenceable. They also look at the
problem in terms of the cardinality of S but with two restrictions, namely, they do not allow
B. Alspach and G. Liversidge: On strongly sequenceable abelian groups 3
S to contain any inverse pairs, that is, if g ∈ S, then −g 6∈ S, and they insist the elements
sum to 0. With these restrictions in place, they show that if |S| ≤ 9, the Cayley digraph
admits either an orthogonal directed cycle.
In some new work, Hicks, Ollis and Schmitt [15] restrict themselves to the case that the
group has prime order. They improve the Bode and Harborth result to include |S| = p− 3,
and they improve the cardinality of the connection set result to |S| ≤ 10. Thus, a circulant
digraph of prime order admits an orthogonal directed path or an orthogonal directed cycle
whenever its out-valency (and in-valency) is at most 10.
There is one obvious fact about a Cayley digraph on an abelian group we now observe.
For the connection set S, let ΣS denote the sum of the elements in S.
Proposition 2.1. Let
−→
X =
−−→
Cay(G;S) be a Cayley digraph on an abelian group G. When
−→
X admits an orthogonal directed cycle or directed path
−→
Y , then
−→
Y is a directed cycle if
ΣS = 0; otherwise, it is a directed path.
Proof. If we use one arc of each length s ∈ S and we start at vertex g, the directed trail
formed terminates at g + ΣS no matter in which order we choose the lengths because G is
abelian. From this it is easy to see that the proposition follows.
3 The associated poset
We use ⊆ for subset inclusion so that A ⊂ B means that A is a proper subset of B.
We define a poset P to be diffuse if the following properties hold:
• The elements ofP are subsets of a ground set Ω and the order relation is set inclusion;
• ∅ ∈ P;
• Every non-empty element of P has cardinality at least 2;
• If A,B ∈ P are disjoint, then A ∪B ∈ P;
• If A,B ∈ P and A ⊂ B, then B \A ∈ P; and
• If A,B ∈ P and A and B are not comparable, then |A4B| ≥ 3.
In order to simplify the discussion, if the ground set has cardinality at least 1 and the
empty set is the only element in the poset, we shall say this poset is diffuse.
Definition 3.1. Let
−→
X =
−−→
Cay(G;S) be a Cayley digraph on the abelian group G. The
associated poset P(
−→
X ) is defined as follows. The ground set is S and and the elements are
any non-empty subsets S′ of S such that ΣS′ = 0 plus the empty set.
Theorem 3.2. If
−→
X =
−−→
Cay(G;S) is a Cayley digraph on the abelian group G, then the
associated poset P(
−→
X ) is diffuse.
Proof. If S′ is a non-empty subset of G \ {0} whose elements sum to 0, then clearly S′
has at least two elements of S. If S′, S′′ ∈ P(
−→
X ), then the sum of the elements in each
of the subsets is 0. If the two subsets are disjoint, then the sum of the elements in their
union also is 0 implying that S′ ∪S′′ ∈ P(
−→
X ). If S′′ ⊂ S′ and both belong to P(
−→
X ), then
4 Art Discrete Appl. Math. 3 (2020) #P1.02
clearly the elements of S′ \ S′′ also sum to 0. This implies S′ \ S′′ ∈ P(
−→
X ). Finally, if
S′, S′′ ∈ P(
−→
X ) and they are not comparable, there must be at least one element of S′ not in
S′′ and vice versa. If the symmetric difference S′4S′′ has exactly two elements x, y ∈ S,
then x = y would hold because S is a subset of an abelian group. This is a contradiction
and the conclusion follows.
Given a sequence s1, s2, . . . , sn, a segment denotes a subsequence of consecutive en-
tries. The notation [si, sj ] is used for the segment si, si+1, . . . , sj , where i ≤ j.
Definition 3.3. Let P be a poset on a groundset Ω = {s1, s2, . . . , sk} with set inclusion
as the order relation. We say that P is sequenceable if there is a sequence a1, a2, . . . , ak of
all the elements of Ω such that no proper segment of the sequence is an element of P . The
sequence is called an admissible sequence.
We only require that proper segments are not elements of P in the preceding definition
because we wish to allow all of Ω to be an element of the poset and still have the poset
possibly be sequenceable.
Corollary 3.4. Let
−→
X =
−−→
Cay(G;S) be a Cayley digraph on the abelian group G. If the
associated poset P(
−→
X ) is sequenceable, then
−→
X admits either an orthogonal directed path
or an orthogonal directed cycle.
Proof. Let s1, s2, . . . , sk be an admissible sequence for P(
−→
X ). If we take a directed trail
of arcs of lengths s1, s2, . . . , sk in that order, it is easy to see that we obtain an orthogonal
directed path of length k if ΣS 6= 0, whereas, we obtain an orthogonal directed cycle of
length k when ΣS = 0.
Conjecture 3.5. Diffuse posets are sequenceable.
Because of Theorem 3.2 and Corollary 3.4, the truth of Conjecture 3.5 would imply
that abelian groups are strongly sequenceable. We do not prove the conjecture here, but it
does shift the work to looking at a restricted family of posets and getting away from the
structure of the groups.
4 The poset approach
Recall that an atom in a poset is an element that covers a minimal element of the poset.
When the empty set is an element, it is the unique minimal element so that the atoms are
the sets not containing any non-empty proper subset in the poset. As we are considering
posets whose elements are sets, we shall refer to an atom of cardinality t as a t-atom.
Because of the properties possessed by diffuse posets, once we have a list of the atoms
we know all the elements of the poset. The elements are all possible unions of mutually
disjoint atoms. Note that the same element may arise in more than one way as a union of
atoms.
Given a poset P whose elements are subsets of a ground set Ω, then the poset induced
on a subset Ω′ ⊆ Ω is the collection of all members of P that lie entirely in Ω′. This poset
is denoted by P〈Ω′〉. Note that an induced subposet of a diffuse poset is itself diffuse.
Lemma 4.1. If every atom of a diffuse poset P is a 2-atom, then P is sequenceable.
B. Alspach and G. Liversidge: On strongly sequenceable abelian groups 5
Proof. The 2-atoms of P are mutually disjoint because P is diffuse. Let {a1, b1}, {a2, b2},
. . . , {at, bt} be the 2-atoms of P , and let x1, x2, . . . , xr be any elements not belonging to
atoms. Note that none of the elements x1, x2, . . . , xr belong to any element of P because
its non-empty elements are disjoint unions of atoms. The sequence
a1, a2, . . . , at, x1, x2, . . . , xr, b1, b2, . . . , bt
is admissible and the proof is complete.
We shall use the language of a segment belonging or not belonging to a diffuse poset P
and this refers to the set of elements in the segment belonging to P .
Lemma 4.2. Let P be a diffuse poset with ground set Ω. If Ω ∈ P and every diffuse poset
on a ground set of cardinality |Ω| − 1 is sequenceable, then for each s ∈ Ω, there is an
admissible sequence whose first term is s.
Proof. Let s be any element of the ground set Ω. The set Ω \ {s} does not belong to the
induced poset P ′ = P〈Ω \ {s}〉 because Ω ∈ P . The poset P ′ is diffuse and has an
admissible sequence a1, a2, . . . , at by hypothesis. We claim the sequence s, a1, a2, . . . , at
is admissible for P .
Any proper segment of the latter sequence not containing s is not in P because P ′ is
an induced poset. If there is a proper subsequence containing s belonging to P , then by
complementation and the fact that Ω ∈ P , the rest of the sequence belongs to P . But this
contradicts the fact that the sequence a1, a2, . . . , at is admissible for P ′. This concludes
the proof.
Lemma 4.3. Let P be a diffuse poset with ground set Ω. If there exists an element s ∈ Ω
such that Ω \ {s} ∈ P , s belongs to a single atom, and all diffuse posets on ground sets of
cardinality |Ω| − 2 are sequenceable, then P is sequenceable.
Proof. Let s and P satisfy the hypotheses. Let a1 be an element of the atom containing s
such that a1 6= s. By Lemma 4.2, there is an admissible sequence a1, a2, . . . , at for the
induced poset P〈Ω \ {s}〉. Consider the sequence a1, a2, . . . , at−1, s, at.
The set composed of the entire sequence is not in P because the latter is diffuse. Any
segment not containing s is not in P as the segment is part of an admissible sequence for
P〈Ω \ {s}〉. Thus, if there is a proper segment in P , it must contain s which implies it
must contain a1 because s belongs to only one atom. But the segment [a1, s] cannot be in
P because the cardinality of the symmetric difference of [a1, s] and Ω\{s} is 2. The result
follows.
Lemma 4.4. Let P be a diffuse poset with ground set Ω, where |Ω| ≥ 3. If there exists an
element s ∈ Ω such that Ω \ {s} is an atom, then P is sequenceable.
Proof. If s belongs to no atoms, then any sequence of the elements of Ω such that s is at
neither end is admissible. The preceding is the case when |Ω| = 3. If s belongs only to
a single atom A, then there must be x, y ∈ Ω such that x, y ∈ Ω \ A by the symmetric
difference condition. It is straightforward to verify that any sequence beginning x, s, y is
admissible by observing that neither x nor y can be in the atom A. Thus, we may assume
s belongs to at least two atoms.
6 Art Discrete Appl. Math. 3 (2020) #P1.02
Hence, we have that |Ω| > 4 and s belongs to an r-atom A with r ≥ 3 because an
element belongs to at most one 2-atom. Choose A so that r is maximum among all atoms
containing s. Note that |A| < |Ω| − 1 because P is diffuse. Let the elements of A be
s, s2, . . . , sr and let y 6= s be an element of Ω not belonging to A.
We claim the sequence π = s2, s, s3, . . . , sr−1, y, sr, . . . completed by any permutation
of the remaining elements is admissible for P . To verify this, first observe that no segment
beginning from the third entry or later belongs to P because the entries form a proper
subset of Ω \ {s} which is an atom. The elements of the entire sequence do not belong to
P because the poset is diffuse.
Any segment of the form [s2, x], where x ∈ {s, s3, . . . , sr−1}, is a proper subset of
A so that it does not belong to P . The segment [s2, y] does not belong to P because the
symmetric difference with A has cardinality 2. Finally, any segment of the form [s2, x],
where x is any element from sr or later in π, cannot be an atom because this contradicts
the choice of A. Thus, if it is in P , there would be an atom properly contained in Ω \ {s}.
The only segments remaining to check are those beginning with s. The argument for
these is essentially the same as for those beginning with s2. One difference is the segment
[s, sr] but it has cardinality 2 symmetric difference with A so cannot be in P . Another
difference is the segment [s, y]. If this segment is in P , then interchange sr−1 and sr in the
sequence and the new segment [s, y] cannot be in P because of the symmetric difference
condition. The switching argument just used requires that r ≥ 4, and when this holds the
rest of the argument is the same as the preceding paragraph completing the proof.
When r = 3, the sequence π begins s2, s, y, s3. The segment [s, y] ∈ P implies that
{s, y} is a 2-atom. However, there are at least two elements of Ω \ {s} not in A. So choose
one that does not form a 2-atom with s.
Lemma 4.5. Let P be a diffuse poset with ground set Ω, where |Ω| ≥ 4. If there exist
s1, s2 ∈ Ω such that Ω \ {s1, s2} is an atom, then P is sequenceable.
Proof. If |Ω| = 4, then either {s1, s2} also is a 2-atom in which case P is sequenceable
by Lemma 4.1, or neither s1 nor s2 are in atoms in which case P is easily seen to be
sequenceable. If |Ω| = 5, then either {s1, s2} is a 2-atom, just one of s1, s2 belongs to a
2-atom, both s1, s2 belong to 2-atoms, neither s1 nor s2 belong to a 2-atom, s1, s2 belong
to a 3-atom or s1, s2 belong to a 4-atom. It is easy to find an admissible sequence in all six
situations.
We assume |Ω| > 5 for the rest of the proof. Let A denote the atom Ω \ {s1, s2}. Let
M(s1),M(s2) and M(s1, s2) denote the collections of atoms containing s1 and not s2, s2
and not s1, and both s1, s2, respectively. We assume that at least one of M(s1) and M(s2)
contains a k-atom for k ≥ 3 as it is easy to verify that P is sequenceable when this is not
the case.
Given an atom A1 in M(s1) of maximum cardinality r + 1, r ≥ 2, stretching the
atom refers to a sequence of the form a1, s1, a2, . . . , ar−1, x, ar, where x is an element to
be named later and a1, a2, . . . , ar is any sequence of the distinct elements of A1 different
from s1. Let B = A \A1 = {b1, b2, . . . , bq} and note that q ≥ 2 because P is diffuse. We
first consider the case M(s2) = ∅.
Start a sequence π by stretching the atom A1 and choose x = b1. Complete the se-
quence as ar, b2, b3, . . . , bq, s2. We now verify that π is admissible.
Because M(s2) is empty, the only possible proper segment ending with s2 that can be
in P is [s1, s2]. If it is in P , then it must be an atom and the theorem holds by Lemma 4.4.
B. Alspach and G. Liversidge: On strongly sequenceable abelian groups 7
So we consider only segments not ending with s2.
Any such segment beginning with an element different from a1 or s1 is a proper subset
of A which implies it is not in P . Almost all proper segments beginning with a1 or s1
are not in P because they either are proper subsets of A1 or violate the maximality of
|A1|. The exceptional segments are [a1, b1], [s1, ar] and [s1, b1]. The segments [a1, b1] and
[s1, ar] are not in P because the symmetric difference with A1 has cardinality 2 in both
cases. If [s1, b1] ∈ P , then interchange b1 and b2 in π. The resulting sequence is then
admissible.
Thus, we now consider the case that M(s2) 6= ∅. Because both M(s1) and M(s2) are
non-empty, we may assume that no atom in M(s2) has cardinality bigger than |A1|. Over
all atoms in M(s2), let ` be the largest cardinality of the intersections with B. Let A2 be
an atom of M(s2) of maxium cardinality intersecting B in ` elements.
Partition A into four subsets as follows:
• B1 = A1 \A2;
• B2 = A1 ∩A2;
• B3 = B \A2; and
• B4 = B ∩A2.
We present the argument for the caseB2 = B3 = ∅ in detail and use it to dispose of the
remaining cases fairly quickly. In this case we see that the atoms A1 and A2 are disjoint
and A1 ∪ A2 = Ω. This implies that Ω ∈ P which, in turn, implies that {s1, s2} is a
2-atom.
Using the same notation for the elements as above, define the sequence
π = a1, s1, a2, . . . , ar−1, b1, ar, b2, . . . , bq, s2.
We use π to find an admissible sequence.
Consider segments of the form [a1, x]. If x ∈ {s1, a2, . . . , ar−1}, then [a1, x] is not in
P because it is a proper subset of the atom A1. If x = b1, then the symmetric difference of
A1 and [a1, b1] has cardinality 2 which implies [a1, b1] 6∈ P .
For x ∈ {ar, b2, . . . , bq}, if [a1, x] ∈ P , then because the segment contains the ele-
ments of A1, the elements of the segment not belonging to A1 would have to be in P . But
this is impossible because they form a proper subset of A2. Hence, no proper segment
beginning with a1 belongs to P .
We move to segments beginning with s1, that is, of the form [s1, x]. If x ∈ {a2, a3, . . . ,
ar−1}, then it is a proper subset ofA1 so that it is not in P . If [s1, b1] ∈ P , then interchange
b1 and b2 (and their labels too). The new interval [s1, b1] does not belong to P . This
interchange is possible because q ≥ 2 as noted earlier.
The interval [s1, ar] cannot belong to P because the cardinality of the symmetric dif-
ference with A1 is 2. If x ∈ {b2, b3, . . . , bq}, the interval [s1, x] is not an atom because
it has cardinality bigger than |A1|. But the elements of the interval not belonging to an
atom containing s1 form a proper subset of A and cannot belong to P . Finally, the interval
[s1, s2] is not in P because Ω ∈ P .
Of the remaining intervals, the only ones which are not proper subsets of A are those
ending in s2 so that we now examine intervals of the form [y, s2]. Any such interval belong-
ing to P must be an atom otherwise the elements of the segment not in the atom containing
8 Art Discrete Appl. Math. 3 (2020) #P1.02
s2 is a proper subset of A. Thus, when y ∈ {b2, b3, . . . , bq}, [y, s2] is not in P because it is
a proper subset of A2.
The interval [ar, s2] is not inP because the cardinality of the symmetric difference with
A2 is 2. The intervals [y, s2], for y ∈ {a2, a3, . . . , ar−1, b1}, are not in P as they would
contradict the choice of A2 .
We see that most intervals are trivially eliminated as possible elements of P in the
preceding argument. There are several crucial intervals and they are all we discuss in the
remaining cases.
Now let B2 = ∅ and B3 6= ∅. Moreover, label the elements of B so that B3 =
{b1, b2, . . . , bq−`} and B4 = {bq−`+1, . . . , bq}. We modify the sequence π slightly de-
pending on the value of q − `. Here are the three scenarios. When q − ` = 1, let π be the
same through ar−1 and end the sequence as
ar−1, b2, ar, b1, b3, . . . , bq, s2.
When q − ` = 2, end the sequence as
ar−1, b1, ar, b3, b2, b4, . . . , bq, s2.
When q − ` > 2, end the sequence as
ar−1, b1, ar, b2, . . . , bq−`−1, bq−`+1, bq−`, bq−`+2, . . . , s2.
First consider segments of the form [y, s2] for y 6∈ {a1, s1} and for all three scenarios.
The interval [bq−`, s2] has symmetric difference of cardinality 2 with A2 so is not in P .
The interval [y, s2] for y ∈ {bq−`+2, . . . , bq} is a proper subset of A2 implying it is not in
P .
The interval [bq−`+1, s2] has bigger intersection with B than A2 so that it cannot be
an atom. This implies it is not in P as this would imply the existence of an atom properly
contained in A. We obtain essentially the same contradiction for all other values of y
distinct from a1 and s1. Notice that these intervals are eliminated independent of the choice
of ar ∈ A1 \ {s1}.
No segment of the form [a1, x] belongs to P in any of the three preceding scenarios
for the same reasons discussed earlier. This conclusion holds independent of the choice of
a1 ∈ A1 \ {s1}. The only problematic segments beginning with s1 are [s1, s2], [s1, b2] in
the first scenario, and [s1, b1] in the second and third scenarios.
If both [s1, s2] and [s1, b2], or both [s1, s2] and [s1, b1] are in P , then interchanging a1
and a2 results in an admissible sequence. If just [s1, s2] ∈ P , then interchange a1 and
ar to obtain an admiisible sequence. Finally, if just [s1, b2] or [s1, b1] belongs to P , then
interchange ar−1 and ar to obtain an admissible sequence.
The preceding interchanges require r ≥ 3 to hold so we consider the special subcase
r = 2 separately. This case means that A is a 3-atom {s1, a1, a2}. If s2 is in a 3-atom
{s2, bq−1, bq} and b1, . . . , bq−2 are the remaining elements ofB, then a1, s1, bq−1, a2, bq−2,
bq−3, . . . , b1, bq, s2 is an admissible sequence. When s2 is not in a 3-atom, then it is easy
to find an admissible sequence.
Now we examine the case that B2 6= ∅ and B3 = ∅. Label the elements of A1 so
that B1 = {a1, a2, . . . , at} and B2 = {at+1, at+2, . . . , ar}. Note that t ≥ 2 because
A2 contains all of B and |A2| ≤ |A1|. When t ≥ 3, modify the original sequence π by
B. Alspach and G. Liversidge: On strongly sequenceable abelian groups 9
interchanging the positions of at and at+1. The previous arguments for segments beginning
with a1 and s1 are valid and we look at segments of the form [y, s2]. The interval [at, s2]
has symmetric difference of cardinality 2 with A2 so it is not in P .
When t = 2, A1 = {s1, a1, a2, . . . , ar} and A2 = {s2, b1, b2, a3, . . . , ar}. The se-
quence a1, s1, a3, a2, . . . , ar−1, b1, ar, b2, s2 has only [s1, b1] and [s1, s2] as possible in-
admissible segments. If both are inadmissible, then interchanging a1 and a2 produces an
admissible sequence. If just [s1, b1] is inadmissible, then interchanging b1 and b2 does the
job. If just [s1, s2] is inadmissible, then interchange a1 and a2 making the new [s1, s2]
admissible. If the new [s1, b1] still is admissible, then we are done. However, if the new
[s1, b1] is inadmissible, then interchanging b1 and b2 finally achieves an admissible se-
quence.
The preceding argument works when r ≥ 2. When r = 1, the sequence a1, s1, a3,
b1, a2, b2, s2 has only the segments [s1, b1], [a2, s2] and [s1, s2] that may be inadmissible.
When the 6-segment is inadmissible, it either is a 6-atom or there is a unique partition into
two 3-atoms. If it is a 6-atom, there is an admissible sequence by Lemma 4.4. In the other
situation, it is a straightforward, though tedious, exercise to find an interchange of elements
that achieves an admissible sequence for the various partitions into two 3-atoms.
When the 6-segment is admissible, it is easy to fix any problems with the two 3-
segments. This completes this case.
We now consider the final case that both B2 and B3 are non-empty. We first provide
a general argument and then examine any special cases arising because certain Bi sets are
too small.
Label elements of B as: B3 = {b1, . . . , bq−`} and B4 = {bq−`+1, . . . , bq}. Consider
the sequence
π = a1, s1, a2, . . . , ar−1, b1, ar, b2, . . . , bq−`−1, bq−`+1, bq−`, . . . , bq, s2.
Proper segments beginning with a1 do not belong to P for the same reasons given earlier.
Segments of the form [s1, y], y 6∈ {b1, s2}, fail to be in P for the same reasons as before.
If the segment [s1, s2] ∈ P , then we may assume it is not an atom as Lemma 4.4 implies
P is sequenceable otherwise. Then [s1, s2] is not a disjoint union of three or more atoms
because this would give an atom properly contained in A. Hence, because the segment
contains A2, [s1, bq−`−1] ∪ {bq−`} must be an atom. But the cardinality of the latter set
of elements either has cardinality bigger than A1 or has symmetric difference with A1 of
cardinality 2. In either case we see that it cannot be an atom. Thus, [s1, s2] is not in P .
The segment [s1, b1] could belong to P and if it does, this is fixed by interchanging a1
and a2. This results in an admissible sequence. Now we consider situations for parameters
being too small to let π breathe.
The proof requires r ≥ 3 in order to make all segments beginning with a1 not be
members of P . Because r > 1, we are considering r = 2 which means A1 = {s1, a1, a2}.
So we begin a sequence with a1, s1 but now cannot use a2 as the next element. Because
B2 is non-empty and |A2| ≤ |A1|, we may assume A2 = {s1, a2} or {s2, a2, bq}.
We need to make certain the sequence does not end with the elements of A2. We know
that B3 6= ∅. If it has at least two elements, then we may choose b1 so that {s1, b1} is not
an atom. In this case, we start the sequence a1, s1, b1, a2,. The completion then depends
on q. When q ≥ 4, we complete the sequence so that it ends bq, bq−1, s2. The sequence is
admissible independent of whether or not bq ∈ A2.
10 Art Discrete Appl. Math. 3 (2020) #P1.02
There several other cases to check for 1 ≤ q ≤ 3 and |B3| = 1 and they can be checked
similarly. This completes the proof.
5 Small cardinality posets
There are several items worth mentioning before stating the main theorem. First, when
discussing a sequenceable poset, we tacitly assume the order relation is set inclusion for
subsets of a ground set Ω. Second, we now use lower case letters from the beginning of the
alphabet for the elements of Ω.
Third, it is clear that if a poset P is sequenceable, then any subposet is sequenceable
as well as any order-isomorphic poset which has arisen via a permutation of Ω. The latter
comment means we may relabel elements for some of the subsequent conclusions.
Theorem 5.1. If P is a diffuse poset whose ground set has cardinality at most 9, then P is
sequenceable.
Proof. It is easy to see that a diffuse poset whose ground set has cardinality 1 or 2 is
sequenceable. If the ground set is {a, b, c}, then a diffuse poset has either no elements, a
single 2-atom or a single 3-atom. The sequence a, c, b is admissible assuming the 2-atom
is {a, b} when there is a single 2-atom.
If the ground set is {a, b, c, d}, then P is sequenceable if {a, b, c, d} belongs to P by
Lemma 4.2, in particular, when there are two 2-atoms. If there is a 3-atom, then there is an
admissible sequence by Lemma 4.4. The situation is trivial if there is a single 2-atom or no
atoms at all.
So we see that diffuse posets with ground sets of cardinality at most 4 are sequenceable.
We next consider ground sets of cardinality 5.
If Ω = {a, b, c, d, e} and Ω belongs to the diffuse poset P , then P is sequenceable
by Lemma 4.2. If Ω 6∈ P but there is an atom of cardinality 4, then P is sequenceble
by Lemma 4.4. If there are no 4-atoms but there is a 3-atom, then P is sequenceable by
Lemma 4.5. If the only atoms are 2-atoms, then P is sequenceable by Lemma 4.1. Hence,
all diffuse posets with ground sets of cardinality 5 are sequenceable.
This takes us to ground sets of cardinality 6. Let Ω = {a, b, c, d, e, f}. As before,
Lemma 4.4 implies P is sequenceable if there is a 5-atom and Lemma 4.5 implies P is
sequenceable when there is a 4-atom. Lemma 4.2 implies that a diffuse poset P with
ground set Ω is sequenceable whenever Ω ∈ P . Thus, we may assume that every atom is
either a 2-atom or a 3-atom and there are neither two disjoint 3-atoms nor three 2-atoms.
It is not difficult to verify that to within order-isomorphism there is a unique maximal
diffuse poset on Ω with only 2- and 3-atoms, that is, every diffuse poset with this restric-
tion on the atoms is order-isomorphic to a subposet. The atoms of this unique poset are
{be, cd, abc, ade, bdf, cef} and an admissible sequence is d, b, c, f, a, e.
For ground sets of cardinalities 7 and 8, the results are displayed in Tables 1 and 2 in
the appendix. We have verified the result for ground sets of cardinality 9, but the number
of pages to display the table is about 30 and we have chosen to not include the table. This
concludes the proof.
Corollary 5.2. Let
−→
X be a Cayley digraph on an abelian group. If the connection S set for−→
X has at most nine elements, then
−→
X admits an orthogonal directed path when ΣS 6= 0 or
an orthogonal directed cycle when ΣS = 0.
B. Alspach and G. Liversidge: On strongly sequenceable abelian groups 11
.
ORCID iDs
Brian Alspach https://orcid.org/0000-0002-1034-3993
Georgina Liversidge https://orcid.org/0000-0002-4467-4328
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B. Alspach and G. Liversidge: On strongly sequenceable abelian groups 13
Appendix
Table 1 below provides admissible sequences for all diffuse posets with a ground set of
cardinality 7 and Table 2 does the same for ground sets of cardinality 8. However, two
conventions are the following:
(1) No poset having the ground set as an element is included because they are sequence-
able by Lemma 4.2; and
(2) only posets with k-atoms for k ∈ {2, 3, 4} and k ∈ {2, 3, 4, 5} are listed in Tables 1
and 2, respectively, as the others are sequenceable by the lemmas of Section 4.
included atoms excluded atoms sequence
ab,cd,ef,ace,adg cfg b,d,a,e,g,c,f
ab,cd,ef,ace adg,bdeg a,c,b,e,d,g,f
ab,cd,ef,acg ade,adf,adfg b,e,d,a,f,g,c
ab,cd,ef xyz,acfg a,c,g,f,d,e,b
ab cd,aef,ceg,bdf xy c,a,e,b,d,g,f
ab,cd,aef,ceg xy,bdf a,d,f,e,g,b,c
ab,cd,aef,bce xy,cfg,deg,dfg a,d,f,c,g,e,b
ab,cd,aef xy,ceg,bce,acfg e,b,d,a,f,c,g
ab,cd,ace xy,bef,afg,bfg,adg b,d,a,g,c,e,f
ab,cd xy,xyz,bcef e,b,f,c,a,d,g
ab,acd,cef,beg xy a,d,g,e,f,b,c
ab,acd,cef xy,beg,acfg b,d,a,f,c,g,e
ab,acd,bef xy,ceg,acfg a,c,g,f,d,e,b
ab,acd,bce xy,def,cfg,dfg,bfg,bdef a,f,b,e,d,c,g
ab,acd xy,cef,bef,bce,adef e,d,f,a,c,b,g
ab,cde xy,axy,bxy,acef,acdg b,f,e,a,c,d,g
ab,acde xy,xyz b,c,f,a,d,e,g
ab xy,xyz,axyz,bxyz,defg a,d,e,b,f,g,c
abc,ade,bdf xy,bdeg c,a,e,b,d,g,f
abc,ade,abdf xy,bef,cef,bdg,beg,ceg a,c,g,b,f,d,e
abc,ade xy,bdf,abdf f,b,d,a,c,e,g
abc,def xy,xyz,abeg,abfg,acdg a,c,g,d,b,e,f
bcdg,adeg,adfg,bdeg,bdfg
abc,abde xy,xyz c,a,f,b,d,e,g
abc xy,xyz,abxy,acxy,bcxy a,d,e,b,c,f,g
abcd xy,xyz,acef b,g,d,a,c,e,f
Table 1: Ground set of cardinality 7
We alter the notation for the tables in two ways. First, we use words rather than set
notation because it saves considerable space. Second, we use roman letters rather than
italics because the appearance of the words is better. In summary, the atom {a, b, c} in the
main body of the paper appears as abc in the tables.
The tables have been compacted to an extent that makes it necessary to describe how to
read them.
14 Art Discrete Appl. Math. 3 (2020) #P1.02
The column headed “included atoms” contains a list of atoms that definitely belong to
the poset under discussion . The only convention to keep in mind here is that an entry in
parentheses—such as in row 12 in Table 1—indicates that precisely one of the words is an
atom but not both. So in this situation, one of bf or cg is a 2-atom but it is not the case that
both 2-atoms are in the poset.
The column headed “excluded atoms” indicates which atoms are definitely not in the
poset, but there are some conventions being followed. These conventions are now listed.
included atoms excluded atoms sequence
ab,cd,ef,acg,beh - a,d,g,c,e,b,f,h
ab,cd,ef,acg,beg bfh,deh,dfh a,h,c,g,e,d,f,b
ab,cd,ef,acg,bgh deh,deg c,h,a,g,e,b,f,d
ab,cd,ef,acg,egh bfh,bfg,cfh a,g,e,c,h,f,d,b
ab,cd,ef,acg adf,beh,beg,bgh,egh b,d,a,f,c,g,h,e
ab,cd,ef,agh,ace gxy,hxy c,a,d,g,f,b,e,h
ab,cd,ef,agh xyz,acfg d,b,e,a,g,f,c,h
ab,cd,ef,ace,bdf xyz a,c,b,g,d,f,h,e
ab,cd,ef xyz,acfg b,d,c,a,f,d,e,h
ab,cd,aef,bcg,egh xy a,d,e,f,g,h,c,b
ab,cd,aef,bcg xy,egh g,e,b,c,f,d,a,h
ab,cd,aef,egh xy,bcg,bcf a,d,f,e,g,c,h,b
ab,cd,aef,ceg xy,bdg,bch e,h,b,g,c,a,d,f
bdh,fgh,beh
Table 2: Ground set of cardinality 8 (Continued)
(1) Any atom that violates the definition of a diffuse poset because of an included atom,
certainly is not in the poset and it is not listed as an excluded atom. For example,
if abc is an included atom, then no other 3-atom may contain ab, ac or bc so they
simply are not listed in the excluded atoms column. Similarly, if abcd is an included
atom, then neither abce nor abc can be atoms and they are not listed in the excluded
atoms column.
(2) If an excluded atom uses letters from the ground set, then that particular atom is not
in the poset.
(3) If an excluded atom uses letters from the end of the alphabet, then it means that
all atoms of that cardinality different from any included atoms are excluded. For
example, in line 7 of Table 1, xy indicates that there are no 2-atoms other than ae,
and xyzw indicates that abcd is the only 4-atom in the poset.
(4) If an excluded atom uses both letters from the ground set and the end of the alphabet,
it means all atoms of that form different from any included atoms are not in the poset.
For example, in poset 51 of Table 2, ab is an included atom and both axy and bxy are
excluded. This means there are no 3-atoms containing a and no 3-atoms containing
b. Of course, there are no 3-atoms containing both a and b because ab is a 2-atom.
(5) Excluded atoms in boldface indicate all atoms that can be formed via label changes
allowed because of the included atoms are excluded. The following example should
B. Alspach and G. Liversidge: On strongly sequenceable abelian groups 15
included atoms excluded atoms sequence
ab,cd,aef xy,bcg,egh,ceg,acfg e,g,c,f,a,d,b,h
ab,cd,ace,efg xy,afg,afh,bef,beh,bfg,bfh a,e,f,c,h,d,g,b
ab,cd,ace xy,afg,bef,bfg,efg,adh f,b,d,a,h,c,e,g
ab,cd,efg,aefh xy,xyz c,d,h,a,e,d,f,g
ab,cd,efg xy,xyz,aefh,adef,befh,cefh b,e,a,d,f,c,h,g
aegh,afgh,begh,bfgh,defh
cegh,cfgh,degh,dfgh,adef
ab,cd,aefg xy,xyz e,b,f,a,c,g,d,h
ab,cd xy,xyz,aefg e,b,g,a,c,h,d,f
ab,acd,bef,ceg,agh xy,dfg c,e,h,f,b,d,a,g
ab,acd,bef,ceg xy,agh a,d,g,c,f,e,h,b
ab,acd,bef,aeg xy,cfg,ceh,cfh b,g,f,a,e,c,d,h
ab,acd,bef,agh xy,ceg,bcg e,b,c,a,g,d,h,f
ab,acd,bef,cgh xy,deg,aeg,bdg a,e,d,c,g,f,h,b
ab,acd,bef xy,xyz,adgh a,c,g,d,e,f,h,b
ab,acd,cef,deg,bch xy,bfg a,d,f,c,h,e,b,g
ab,acd,cef,deg xy,beh,bfg d,a,h,c,b,e,f,g
bfh,bch,bceh
ab,acd,cef,egh xy,bfg,dfg,aceg b,g,a,e,c,d,f,h
ab,acd,cef,aeg xy,bfg,beh,bfh,bgh b,g,f,a,e,c,d,h
dfg,deh,dfh
ab,acd,cef,bdg xy,aeh,beh,deh,egh,aeg,bcfg d,a,g,c,b,f,e,h
ab,acd,cef,bde xy,bfg,bgh,dfg,egh g,a,c,b,d,f,e,h
fgh,aeg,afg,aeg,afh,bcdf
ab,acd,cef,aceg xy,beg,bfg,beh,bfh,bgh,deg,dfg b,d,a,f,c,g,e,h
deh,dfh,egh,fgh,afg,aeh
afh,bdg,bdh,bde,bdf
ab,acd,cef xy,beg,bgh,deg,egh,aeg,bdg d,a,g,c,f,b,e,h
bde,aceg,bcfg
ab,acd,bce,afg xy,bfg,bfh,cgh b,d,e,a,c,f,g,h
def,deh,dfg,dfh
ab,acd,bce,efg xy,afh,bfh,cgh,deh,dfh,adef c,b,d,e,f,g,h
ab,acd,bce,aef,adfg xy,bfg,bfh,bgh,cfg a,c,b,h,e,f,g,d
deh,dfh,cgh,dgh,agh
ab,acd,bce,aef xy,bfg,bgh,cfg,deg,deh b,g,a,d,f,c,e,h
dfg,cgh,dgh,agh,adfg
ab,acd,bce,fgh xy,def,aef,adfg d,c,f,e,b,g,a,h
Table 2: Ground set of cardinality 8 (Continued)
16 Art Discrete Appl. Math. 3 (2020) #P1.02
included atoms excluded atoms sequence
ab,acd,bce,adfg xy,xyz a,c,b,h,e,f,g,d
ab,acd,bce xy,xyz,adfg,bfgh e,f,b,c,a,g,d,h
ab,acd,bcdg xy,bef,bfg,cef,def,ceh b,d,a,g,e,f,h,c
deh,dfh,ceg,bce,adeg
ab,acd xy,bef,cef,def,ceg,cfg g,b,c,a,e,d,f,h
deg,dfg,ceh,deh,dfh,cgh
dgh,bce,bcdg,bcdh
ab,acd,efg,acef xy,xyz b,e,f,a,h,c,d,g
ab,acd,efg xy,xyz,acef,begh f,b,e,a,c,g,d,h
ab,acd,acef xy,xyz b,d,a,g,c,e,f,h
ab,acd,bcef xy,xyz,adef,aceg,adeg,acgh,adgh d,g,c,a,e,b,f,h
ab,acd,cefg xy,xyz,acef,adef,aceh,adeh b,f,a,d,e,c,g,h
bceh,bdef,bcef,bdeh,bcgh,begh
ab,acd xy,xyz,acef,bcef,cefg,befh,begh f,b,e,a,c,h,d,g
ab,cde,cfg,dfh,acef xy,axy,bxy b,d,a,e,f,h,c,g
ab,cde,cfg,dfh xy,axy,bxy,acef,bcdf e,a,d,b,f,c,h,g
ab,cde,cfg,acdf xy,xyz b,h,d,a,c,e,f,g
ab,cde,cfg xy,xyz,acdf,bfgh d,a,e,c,f,b,g,h
ab,cde,acdf xy,xyz,aceh b,e,a,h,c,d,f,g
ab,cde,acfg xy,xyz,adef,bcdf,bdef b,d,a,e,f,c,g,h
acdh,adeh,bcdh,bdeh
ab,cde xy,xyz,acdf,acfg,cefg f,a,g,c,b,e,d,h
ab,acde,bcdf xy,xyz a,e,g,c,d,f,h,b
ab,acde,acfg xy,xyz,bcdf,bdef b,d,a,e,f,c,g,h
bdeg,bcdh,bceh,bdeh
ab,acde xy,xyz,bcdf,acfg b,e,f,c,a,d,h,g
acefg,acdfh,acdgh
ab,cdef xy,xyz,axyz,bxyz,acdeh,acdfg b,g,f,a,c,d,e,h
ab,acdef xy,xyz,xyzw b,c,g,a,d,e,f,h
ab xy,xyz,xyzw,axyzw,defgh a,d,e,b,f,g,h,c
abc,ade,bdf,afg,beh xy,bcfg c,b,d,a,f,e,g,h
abc,ade,bdf,afg,ceh xy,bgh,dgh,cdef b,a,d,c,e,f,h,g
abc,ade,bdf xy,beh,cdh,efh,bgh a,c,g,f,b,h,d,e
afg,bcfh dgh,ceh,cgh,egh
abc,ade,bdf,afg xy,beh c,a,e,b,d,h,f,g
cgh,bcfh,bdeh,dfgh
Table 2: Ground set of cardinality 8 (Continued)
B. Alspach and G. Liversidge: On strongly sequenceable abelian groups 17
included atoms excluded atoms sequence
abc,ade,bdf,ceg,cfh xy,afg,bcgh a,f,d,e,g,h,c,b
abc,ade,bdf,ceg,bfgh xy,afg,afh,beh a,b,f,c,g,h,e,d
cdh,cfh,efh,dgh
abc,ade,bdf,ceg xy,afg,afh,beh b,f,c,g,a,e,h,d
cfh,bfgh,acfg
abc,ade,bdf,cgh,abeg xy,afg,afh,beh,efg,efh a,b,d,e,g,f,h,c
abc,ade,bdf,cgh,acdg xy,afg,beg,beh,afh,efg b,a,f,c,d,g,h,e
efh,abeg,abeh,abfh,abfg
abc,ade,bdf,cgh xy,afg,efg,abeg e,g,a,c,d,b,h,f
acdg,bcfg
abc,ade,bdf,agh,bcfg xy,beg,cdg,beh,cdh a,c,g,b,h,f,d,e
ceg,efg,ceh,cfh,efh
abc,ade,bdf,agh xy,beg,ceg,cfg c,g,a,b,d,e,f,h
bcfg,abdg
abc,ade,bdf,abeg xy,afg,cdg,afh,beh,cdh c,b,f,a,e,g,d,h
ceg,cfg,efg,ceh,cfh,efh
cgh,egh,fgh,agh,bgh,dgh
abc,ade,bdf,acfg xy,beg,cdg,afh,beh,cdh b,h,c,a,g,e,f,d
ceg,efg,ceh,cfh,efh,cgh
egh,fgh,agh,bgh,dgh,abeg
bcdg,bdeg,abeh,abfh
acdh,adfh,bcdh,bdeh
abc,ade,bdf,bcfg xy,beg,beh,cfh,egh,bgh a,c,g,b,h,f,d,e
bfg,afg,afh,ceg,ceh,cgh
agh,abeg,abfg,acdg
aefg,acfh,bceh,abeh
abfh,acdh,acfg,bceg,aefh
abc,ade,bdf xy,afg,ceg,cgh g,c,a,d,b,e,f,h
agh,abeg,acfg,bcfg,bcfh
abc,ade,bfg,dfh,cgh xy,xyz,adfg b,a,f,d,g,h,e,c
abc,ade,bfg xy,cef,cdg,ceg,beh,ceh c,a,d,b,e,f,h,g
abc,ade,bfg xy,bdf,bef,beg,bdgcdg b,c,g,a,f,e,d,h
dfh,acef ceg,adfg,abdg,bcdf
abc,ade,bfg,dfh xy,cef,cdg,ceg c,a,h,b,d,e,f,g
beh,ceh,abef,acef
Table 2: Ground set of cardinality 8 (Continued)
18 Art Discrete Appl. Math. 3 (2020) #P1.02
included atoms excluded atoms sequence
abc,ade,bfg,bcfh xy,cdf,cdg,bdh a,c,d,b,f,h,g,e
cdh, dfh,dgh
abc,ade,bfg,acdf xy,cef,cdg,ceg,bdh,beh b,c,f,a,g,d,e,h
cdh,ceh,dfh,efh,dgh,egh
bcfh,bcgh,dfgh,aceh
abc,ade,bfg xy,cdf,bdh,beh,cdh,ceh d,h,a,e,b,c,f,g
dfh bcfh,acdf,abeh
abc,ade,abdf xy,bef,cef,beg g,e,d,c,a,f,b,h
ceg,bdh,cdeg
abc,ade,bcdf xy,bef,bdg,beg g,b,c,d,a,f,e,h
abef,abdg,abeg,aefh
abc,ade,abfg xy,bdf,cdf,bdh d,f,g,a,e,b,c,h
cdh,acdf,abdh,acdh,bcdh
bcdf,bdef,cdef,bdeh,cdeh
abc,ade xy,bdf,abdf,bcdf,abfg f,b,c,d,a,h,e,g
bcfh,bcgh
abc,def,abdg xy,axy,bxy,cxy,acfh e,g,b,f,a,c,h
abc,def,adgh xy,axy,bxy,cxy,abeg b,a,g,d,f,h,e,c
bcdg,bceg,abdfg
abc,def,abde xy,axy,bxy,cxy,egh,abfg f,d,g,a,e,b,c,h
acdg,acfg,adgh,afgh,cfgh
abc,def,abgh xy,axy,bxy,cxy,xyzw,acdeg b,a,g,d,c,e,f,h
abc,def xy,axy,bxy,cxy,xyzw,abdfg c,a,g,d,b,f,e,h
abc,abde,adefg xy,xyz c,b,f,a,d,e,h,g
abc,abde xy,xyz,adefg,cdefg c,a,f,b,d,e,g,h
abc,adef xy,xyz,abxy,acxy,bcxy,abdfg g,d,f,a,b,e,c,h
abc,abdef xy,xyz,axyz,bxyz,cxyz c,a,g,b,d,e,f,h
abc xy,xyz,axyz,bxyz,cxyz,abxyz b,f,d,a,e,h,g,c
acxyz,bcxyz,adefh,adegh
bdefg,cdefg
abcd,abef,abceg xy,xyz c,d,g,a,b,e,h,f
abcd,abef xy,xyz,abceg c,g,d,a,b,e,h,f
abcd,abcef xy,xyz,abxy,acxy,adxy d,a,h,c,b,e,f,g
bcxy,bdxy,cdxy,befg
abcd xy,xyz,abef,abcef a,e,f,b,c,d,g
abcde xy,xyz,xyzw,abcfg d,h,e,a,b,c,f,g
Table 2: Ground set of cardinality 8
B. Alspach and G. Liversidge: On strongly sequenceable abelian groups 19
make this subtle concept clear. Start with the template obtained from the included
atoms and observe which label changes are allowed. For example, consider the six-
teenth entry from the end of Table 2. The included atoms are abc, ade so that the
template is two 3-atoms intersecting at a single point. The label a is fixed because
it is the only point belonging to both 3-atoms. The labels b and c may be switched
because they lie in the same 3-atom. The same holds for the labels d and e. Further-
more, the two sets {b,c} and {d,e} may be switched. Finally, the labels f,g,h may be
switched with each other as they belong to neither 3-atom. Thus, excluding the atom
bdf means that all of the 3-atoms bdf, bef, cdf, cef, bdg, beg, cdg, ceg, bdh, beh, cdh
and ceh are excluded. Similarly, excluding the atom abdf means all the atoms abdf,
abef, acdf, acef, abdg, abeg, acdg, aceg, abdh, abeh, acdh and aceh are excluded.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.03
https://doi.org/10.26493/2590-9770.1263.86e
(Also available at http://adam-journal.eu)
Self-dual, self-Petrie-dual and Möbius regular
maps on linear fractional groups*
Grahame Erskine
School of Mathematics and Statistics, Faculty of Science, Technology, Engineering and
Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, U.K.
Katarı́na Hriňáková†
Department of Mathematics, Faculty of Civil Engineering, Slovak University of
Technology, Radlinského 11, 810 05, Bratislava, Slovakia
Olivia Reade Jeans‡
School of Mathematics and Statistics, Faculty of Science, Technology, Engineering and
Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, U.K.
Received 13 August 2018, accepted 7 August 2019, published online 27 July 2020
Abstract
Regular maps on linear fractional groups PSL(2, q) and PGL(2, q) have been studied
for many years and the theory is well-developed, including generating sets for the associ-
ated groups. This paper studies the properties of self-duality, self-Petrie-duality and Möbius
regularity in this context, providing necessary and sufficient conditions for each case. We
also address the special case for regular maps of type (5, 5). The final section includes an
enumeration of the PSL(2, q) maps for q ≤ 81 and a list of all the PSL(2, q) maps which
have any of these special properties for q ≤ 49.
Keywords: Regular map, external symmetry, self-dual, self-Petrie-dual, Möbius regular.
Math. Subj. Class. (2020): 05C25, 05C10
*The authors would like to thank Jozef Širáň for many useful discussions during the preparation of this paper.
†The author acknowledges partial support by Slovak research grants VEGA 1/0026/16, VEGA 1/0142/17,
APVV-0136-12, APVV-15-0220 and APVV-17-0428.
‡Corresponding author.
E-mail addresses: grahame.erskine@open.ac.uk (Grahame Erskine), hrinakova@math.sk (Katarı́na
Hriňáková), olivia.jeans@open.ac.uk (Olivia Reade Jeans)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 3 (2020) #P1.03
1 Introduction
Regular maps always display inherent symmetry by virtue of their definition. A regular
map can have further symmetry properties which are called external symmetries. These
occur when a map is isomorphic to its image under a particular operation. The best known
example of this is the tetrahedron, a Platonic solid which is self-dual.
A map is a cellular embedding of a graph on a surface and is made up of vertices, edges
and faces. A flag is a triple incidence of edge-end, edge-side and face centre. Informally
we can visualise each flag as a triangle with its corners at the vertex, the centre of the face
and the midpoint of the edge. Thus there are four flags incident to any edge and the whole
surface is covered by flags. We consider the symmetries of a map by reference to its flags.
An automorphism of a map is an arbitrary permutation of its flags such that all adjacency
relationships of the flags are preserved. The map is regular if the group of automorphisms
acts regularly on the flags, that is the group is fixed-point-free and transitive. An implication
of this is that each vertex of a regular map has a given valency, say k, and the face lengths
are all equal, say to l. Henceforth we will refer to maps of type (k, l) where k is the vertex
degree and l is the face length of the regular map.
For further details about the theory of regular maps see [3, 9, 12, 15, 16].
Every regular map has an associated dual map which is also a regular map. Informally,
the dual map is created by forming a vertex at the centre of each original face and consid-
ering each of the original vertices as the centre of a face. Each edge of the dual map is
thereby formed by linking a pair of neighbouring vertices across one of the original edges.
A different type of dual, the Petrie dual of a map has the same edges and vertices as
the original map but the faces are different. That is, the underlying graph is the same, but
the embedding is different. The boundary walk of a face of the Petrie dual map can be
described informally as follows:
1. Starting from a vertex on the original map, trace along one side of an incident edge
until you get to the midpoint of that edge;
2. Cross over to the other side of the edge and continue tracing along the edge in the
same direction as before. When you approach a vertex, sweep the corner and con-
tinue along the next edge until you reach its midpoint;
3. Repeat step 2 until you rejoin the face boundary walk where you started.
When the associated dual or Petrie dual map is isomorphic to the original map, we
call the map self-dual or self-Petrie-dual respectively. This paper explores necessary and
sufficient conditions for a regular map with automorphism group PSL(2, q) or PGL(2, q),
where q is odd, to have each of these external symmetries.
Another property of interest in the theory of regular maps is Möbius regularity. This
concept was introduced by S. Wilson in [17] who originally named them cantankerous. A
regular map is Möbius regular if any two distinct adjacent vertices are joined by exactly
two edges and any open set supporting these edges contains a Möbius strip. Clearly such
a map must have even vertex degree k and we will establish the further conditions under
which a regular map on PSL(2, q) or PGL(2, q) is Möbius regular.
In Section 2 we state some of the background material and results which we will need.
Section 3 investigates regular maps of type (p, p), (k, p) and (p, l) when k and l are coprime
to p. Type (p, p) is self-dual but not self-Petrie-dual nor Möbius regular, and type (p, l) can
be self-Petrie-dual but not Möbius regular. Type (k, p) can be self-Petrie-dual or Möbius
G. Erskine et al.: Self-dual, self-Petrie-dual and Möbius regular maps 3
regular. In Section 4 we address the necessary and sufficient conditions for a map of type
(k, l) to be self-dual, self-Petrie-dual and Möbius regular respectively. Section 5 highlights
a special case, namely maps of type (5, 5) whose orientation-preserving automorphism
groups turn out to be isomorphic to A5 and Section 6 comments on and lists examples of
maps with some or all of these properties.
2 Background information, notes and notation
This paper is founded on work done by M. Conder, P. Potočnik and J. Širáň in [4] which
provides a detailed analysis of reflexible regular hypermaps for triples (k, l,m) on pro-
jective two-dimensional linear groups including explicit generating sets for the associated
groups. In particular this paper is concerned only with maps, not hypermaps, and so, with-
out loss of generality, we let m = 2.
The group of automorphisms of a regular map is generated by three involutions, two
of which commute, where the three involutions can be thought of as local reflections in
the boundary lines of a given flag which preserve all the adjacency relationships between
flags. As shown in Figure 1 the involutions act locally on the given flag as follows: X as
a reflection in the edge bisector; Y as a reflection across the edge; Z as a reflection in the
angle bisector at the vertex. The dots on the diagram indicate where there may be further
vertices, edges and faces while the dashed lines outline each of the flags of this part of the
map.
X
Y
Z
Figure 1: The action of automorphisms X , Y and Z on the shaded flag.
The study of regular maps is equivalent to the study of group presentations of the form
G ∼= 〈X,Y, Z | X2, Y 2, Z2, (Y Z)k, (ZX)l, (XY )2, . . .〉,
see [16]. The dots indicate the potential for further relations not listed, and we assume the
orders shown are indeed the true orders of those elements in the group.
The surface on which a regular map is embedded could be orientable or non-orientable.
If the regular map is on an orientable surface then G has a subgroup of index two which
corresponds to the orientation preserving automorphisms. Instead of the group generated
by these three involutionsX,Y and Z, we can consider the group of orientation-preserving
automorphisms which is generated by the two rotations R = Y Z and S = ZX . On a
4 Art Discrete Appl. Math. 3 (2020) #P1.03
non-orientable surface these two elements will still generate the full automorphism group
and we can say that studying these maps is equivalent to studying groups which have pre-
sentaions of the form 〈R,S | Rk, Sl, (RS)2, . . .〉.
We focus on regular maps of type (k, l) where the associated group G ∼= 〈X,Y, Z〉
is isomorphic to PSL(2, q) or PGL(2, q) where q is a power of a given odd prime p. By
[4], both k and l are either equal to p or divide q − 1 or q + 1. Following the convention
and notation of [4], in the latter two cases we let ξκ and/or ξλ be primitive 2kth or 2lth
roots of unity respectively. Note that in the case where k or l divides q − 1 then the
corresponding primitive root is in the field GF(q); otherwise it is in the unique quadratic
extension GF(q2). We also define ωi = ξi + ξ−1i for i ∈ {κ, λ}. Note that ωi is thus in the
field GF(q). We too assume that (k, l) is a hyperbolic pair, that is 1/k + 1/l < 1/2. This
implies that k ≥ 3 and l ≥ 3. The conditions in this paragraph are what we refer to as the
usual setup.
We can consider duality and Petrie duality as operators on a map. Since the dual of
a map is obtained by swapping the vertices for faces and vice versa, in terms of the invo-
lutions X,Y, Z the dual operator would fix Z and interchange X and Y . The Petrie dual
operator would replace X with XY and fix Y and Z. The automorphism associated with
any type of duality is an involution. This is because it acts on our map to produce the dual
map, and when this automorphism is repeated we get back to the original map. Self-duality
and self-Petrie-duality are therefore equivalent to the existence of precisely such involutory
automorphisms of G, the group associated with the regular map.
Our paper is devoted in large part to finding conditions for the existence of invo-
lutory automorphisms which imply self-duality and/or self-Petrie-duality. The automor-
phism group for G is PΓL(2, q), the semidirect product PGL(2, q) o Ce where q = pe,
[13]. Observe that, when e = 1, this group is essentially PGL(2, p) and so, in the case
where G ∼= PGL(2, p), all automorphisms are inner automorphisms. Elements (A, j) ∈
PΓL(2, q) act as follows: (A, j)(T ) = Aφj(T )A−1 where φj is the repeated Frobenius
field automorphism of the finite field, φj : x → xr with r = pj . The function φj acts
element-wise on a matrix and we use the general rule for composition in PΓL(2, q) which
is (B, j)(A, i) = (Bφj(A), i+ j).
When (A, j) ∈ PΓL(2, pe) is an involution, it must be such that (A, j)(A, j) =
(Aφj(A), 2j) is the identity, so 2j ≡ 0 (mod e). One case is when there is no field
automorphism involved, that is j = 0 and A2 = I . Alternatively e = 2j is even, and then
we need φj(A) = A−1. This is summarised in Lemma 2.1.
Lemma 2.1. (A, j) ∈ PΓL(2, pe) is an involution if and only if one of the following
conditions holds:
1. j = 0 and A2 = I
2. 2j = e and φj(A) = A−1.
Explicit generating sets are known for regular maps with automorphism group G iso-
morphic to PSL(2, q) or PGL(2, q), and for details we refer the interested reader to [4].
We present the results for maps of each type as required.
We will need to consider performing operations on the elements X , Y and Z of G. As
such we denote elements of the group PSL(2, q) or PGL(2, q), by a representative matrix
with square brackets. This allows us to perform the necessary calculations. We can then
determine whether or not two resulting matrices are equivalent within G, that is whether or
G. Erskine et al.: Self-dual, self-Petrie-dual and Möbius regular maps 5
not they correspond to the same element of the group G. A pair of matrices are in the same
equivalence class, that is they represent the same element of G, if one is a scalar multiple
of the other. We use curved brackets for matrix representatives for X , Y and Z.
Lemma 2.1 can then be used to find conditions for the elements of the matrix part A of
an involutory automorphism (A, j) as follows.
Lemma 2.2. Let A =
[
a b
c d
]
. The automorphism denoted (A, j) ∈ PΓL(2, pe) is an
involution, if and only if a, b, c, d satisfy the following equations, with r = pj for j = 0 or
2j = e.
1. ar+1 = dr+1,
2. bcr = cbr,
3. abr + bdr = 0,
4. car + dcr = 0.
Proof. By Lemma 2.1, and letting r = pj we have
Aφj(A) =
[
a b
c d
] [
ar br
cr dr
]
=
[
ar+1 + bcr abr + bdr
car + dcr cbr + dr+1
]
= I.
By comparing the leading diagonal entries we see that ar+1 + bcr = cbr +dr+1. Applying
the field automorphism φj yields a1+r+brc = crb+d1+r. Subtracting these two equations,
and remembering that q is odd, we get the first two equations, while looking at the off-
diagonal immediately gives rise to the final two equations.
When we are establishing the conditions under which a regular map is Möbius regular
we will rely on the following group-theoretic result, proved in [11] by Li and Širáň. Note
that implicit in this necessary and sufficient condition is that for a map of type (k, l) to be
Möbius regular k must be even.
Lemma 2.3. A regular map is Möbius regular if and only if XR k2X = R k2 Y where
R = Y Z.
3 Regular maps on linear fractional groups of type (p, p), (k, p) and
(p, l) where p is an odd prime
For odd prime p, by Proposition 3.1 in [4], maps of the type (p, p) have the following
representatives for X , Y and Z, where α2 = −1:
X1 = −α
(
1 0
2 −1
)
, Y1 = −α
(
1 −1
0 −1
)
and Z1 = α
(
1 0
0 −1
)
.
Proposition 3.1. With the usual setup, a map of type (p, p) is self-dual.
Proof. For self duality we need G ∼= 〈X,Y, Z〉 to admit an automorphism such that X
and Y are interchanged, and Z is fixed. So the question is: can we find an automorphism
(A, j) ∈ PΓL(2, q) such that Aφj(X)A−1 = Y , Aφj(Y )A−1 = X , and Aφj(Z)A−1 =
Z. It is easy to verify that (A, 0), whereA has the formA = [ 0 12 0 ] satisfies these conditions,
so this type of map is self-dual.
6 Art Discrete Appl. Math. 3 (2020) #P1.03
Proposition 3.2. With the usual setup, a map of type (p, p) is not self-Petrie-dual.
Proof. In order to be self-Petrie-dual, the group G needs to admit an involutory automor-
phism (B, j) which fixes Z and Y , and exchanges X with XY .
First notice that φj(Z) = Z and φj(Y ) = Y so if B exists, it must be of a form which
commutes with both Z and Y . To commute with Z, the necessarily non-identity element
B must be either B1 = [ 0 bc 0 ] or B2 = [
a 0
0 d ]. Note that 0 /∈ {a, b, c, d} and a 6= d. As
shown below, neither of these commute with Y .
B1Y = −α
[
0 −b
c −c
]
6= Y B1 = −α
[
−c b
−c 0
]
B2Y = −α
[
a −a
0 −d
]
6= Y B2 = −α
[
a −d
0 −d
]
Hence this type of map is not self-Petrie-dual.
Remark 3.3. Maps of type (k, p) and (p, l) where k and l are coprime to p clearly cannot
be self-dual since the vertex degree and face lengths differ.
Proposition 3.4. With the usual setup, and for k coprime to p, a map of type (k, p) is self-
Petrie-dual if and only if k | 2(r ± 1) and ±ω(r+1)κ = 4ξ(r±1)κ when the corresponding
signs in each (r ± 1) are read simultaneously, and where r = pj and j = 0 or 2j = e.
Proof. When k is coprime to p, [8] tells us that a map of type (k, p) has the following triple
of generating matrices corresponding to X , Y , and Z:
X2 = ηα
(
−ωκ −2ξκ
2ξ−1κ ωκ
)
, Y2 = −α
(
0 ξκ
ξ−1κ 0
)
, Z2 = α
(
0 1
1 0
)
,
where α2 = −1 and η = (ξκ − ξ−1κ )−1.
Suppose the map is self-Petrie-dual and (B, j) = (
[
a b
c d
]
, j) is the associated involutory
automorphism. In order to fix Z we must have a = d and b = c or a = −d and b = −c.
In order to fix Y we find that either a = 0 or c = 0 in which case we need k | 2(pj + 1)
or k | 2(pj − 1) respectively. When a = 0, the involution then interchanges X with XY
if and only if ±ω(r+1)κ = 4ξ(r+1)κ . When c = 0, the involution then interchanges X with
XY if and only if ±ω(r+1)κ = 4ξ(r−1)κ .
Proposition 3.5. Under the usual setup, and with l coprime to p, a regular map of type
(p, l) is self-Petrie-dual if and only if ω2λ = −ω2rλ where r = pj and 2j = e.
Proof. Using a similar argument to the above applied to the appropriate matrix triple from
[8], namely
X3 = α
(
0 ω−1λ
ωλ 0
)
, Y3 = −α
(
1 0
0 −1
)
, Z3 = α
(
1 1
0 −1
)
,
we find that the only allowable form for B is the identity. The necessary non-trivial field
automorphism applied to the X and XY interchange then yields the stated condition.
Remark 3.6. For odd p, a regular map of type (p, p) or (p, l) is not Möbius regular. This
is immediate from the fact that each pair of adjacent vertices in a Möbius regular map is
joined by exactly two edges, hence the vertex degree must be even.
G. Erskine et al.: Self-dual, self-Petrie-dual and Möbius regular maps 7
Proposition 3.7. Under the usual setup for even k, a map of type (k, p) is Möbius regular
if and only if ω2κ + 4 = 0
Proof. A regular map is Möbius regular if and only if the equation XR
k
2X = R
k
2 Y is
satisfied. We assume k is even, since if k is odd then the map is certainly not Möbius
regular. In this case
R = [Y2Z2] =
[
ξκ 0
0 ξ−1κ
]
so we have R
k
2 =
[
α 0
0 α−1
]
, where α2 = −1. Hence the map is Möbius regular if and only
if these matrices are equivalent:
XR
k
2X = −η2α
[
ω2κ + 4 4ωκξκ
−4ωκξ−1κ −(ω2κ + 4)
]
and R
k
2 Y =
[
0 ξκ
−ξ−1κ 0
]
.
These matrices are equivalent if and only if ω2κ + 4 = 0.
4 Regular maps on linear fractional groups of type (k, l) where both
k and l are coprime to p
In this case we have different generating triples for the group G. As per Proposition 3.2
in [4], the triple (X,Y, Z) has representatives as defined below where D = ω2κ + ω
2
λ − 4,
β = −1/
√
−D and η = (ξκ − ξ−1κ )−1.
X4 = ηβ
(
D Dωλξκ
−ωλξ−1κ −D
)
, Y4 = β
(
0 ξκD
ξκ
−1 0
)
, andZ4 = β
(
0 D
1 0
)
We will also consider the pair of matrices which representR and S, the rotations around
a vertex and a face respectively, which by Proposition 2.2 in [4] are:
R4 =
(
ξκ 0
0 ξ−1κ
)
and S4 = η
(
−ωλξ−1κ −D
1 ωλξκ
)
.
At this point we note that there is an exception for maps of type (5, 5), which is ad-
dressed in Section 5.
Theorem 4.1. Under the usual setup, a regular map of type (k, k) is self-dual if and only
if ωλ = ±ωrκ where r = pj , and j = 0 or 2j = e.
Proof. Suppose the map is self-dual.
There is an involutory automorphism of G which fixes Z and interchanges X and Y .
This is equivalent to interchanging the rotations R−1 = (Y Z)−1 = ZY and S = ZX
around a vertex and a face respectively. That is, there is an automorphism (A, j) which
interchanges ±R−1 with S. Here the ± takes into account both representative elements
for R. So A(±φj(R−1))A−1 = S. Remembering that conjugation preserves traces this
implies ±φjtr(R−1) = tr(S) which immediately yields the condition ±ωrκ = ωλ.
Conversely suppose ±ωrκ = ωλ.
We note that ω2rκ = ω
2
λ ⇐⇒ ω2κ = ω2rλ and soDr = (ω2κ+ω2λ−4)r = ω2rκ +ω2rλ −4 =
ω2λ + ω
2
κ − 4 = D.
We aim to find an involutory automorphism (A, j) which demonstrates this map is self-
dual. Consider A =
[
a D
−1 −a
]
which, by Lemma 2.2, so long as ar = a, satisfies all
the equations necessary for the element (A, j) to be involutory. Notice that (A, j) also
8 Art Discrete Appl. Math. 3 (2020) #P1.03
fixes Z. We also need X and Y to be interchanged by the automorphism in which case the
following matrices are equivalent.
Aφj(X) = η
rβr
[
D(a− ωrλξ−rκ ) D(aωrλξrκ −D)
ωrλξ
−r
κ a−D D(a− ωrλξrκ)
]
and Y A = β
[
−ξκD −aξκD
aξ−1κ Dξ
−1
κ
]
Ratio of elements in the leading diagonal: −ξ2κ = (a− ωrλξ−rκ )/(a− ωrλξrκ)
Ratio of elements in the left column: −Dξ2κ/a = D(a− ωrλξ−rκ )/(ωrλξ−rκ a−D)
Ratio of elements in the top row: 1/a = (a− ωrλξ−rκ )/(aωrλξrκ −D)
The last ratio listed yields the following quadratic in a: 0 = a2 − aωrλ(ξ−rκ + ξrκ) +D,
that is 0 = a2 − aωrλωrκ + D. This is consistent with all the necessary ratios. All that
remains is to check that a value of a satisfying this quadratic is invariant under the repeated
Frobenius field automorphism. The discriminant ∆ = ω2κω
2r
κ − 4(ω2κ + (ωrκ)2 − 4) =
((ωrκ)
2 − 4)(ω2κ − 4). Furthermore the expression for a = (ωrλωrκ ±
√
∆)/2 is invariant
under the transformation x→ xr as required. Hence the map is self-dual.
Theorem 4.2. With the usual setup, where k, l are coprime to p, a map of type (k, l) is
self-Petrie-dual if and only if one of the following conditions is fulfilled:
1. ω2λ = −D
2. q = r2 = p2j , ω2rλ = −D and k|(r ± 1).
Proof. First suppose the map is self-Petrie-dual. So there exists (B, j) ∈ PΓL(2, q) such
that Bφj(X)B−1 = XY , Bφj(Y )B−1 = Y , and Bφj(Z)B−1 = Z. By comparing the
traces of φj(ZX) and ZXY we get the necessary conditon: ω2rλ = −D.
For the rest of the proof we split the situation into two cases: the first when j = 0 and we
do not consider any field automorphism, and the second case where a field automorphism
is included.
Case 1: j = 0.
Suppose ω2λ = −D. Notice that B =
[
1 0
0 −1
]
fixes both Y and Z. The map is
self-Petrie-dual if BX = ηβ
[
D Dωλξκ
ωλξ
−1
κ D
]
and XY B = ηβ2
[
Dωλ −D2ξκ
−Dξ−1κ Dωλ
]
are also equivalent. Comparing these and applying our assumption that ω2λ = −D we
conclude this map is self-Petrie-dual.
Case 2: 2j = e. By Lemma 2.1 we include the repeated Frobenius automorphism.
Suppose ω2rλ = −D and k|(r ± 1).
The map is self-Petrie-dual if there is an involutory automorphism which not only fixes
Z but also fixes Y and interchanges XY with X. We hope to find (B, j) = (
[
a b
c d
]
, j),
the associated element of PΓL. In addition to the conditions for a, b, c, d established in
Lemma 2.2, we require Bφj(Z)B−1 = Z and Bφj(Y )B−1 = Y . In order to fix Z we
must have bd = acDr and D(d2 − c2Dr) = a2Dr − b2. Fixing Y yields two further
equations: bd = acξ2rκ D
r and ξ2κD(d
2ξ−rκ − c2ξrκDr) = a2ξrκDr − b2ξ−rκ . Since ξ2rκ 6= 1
and D 6= 0, notice that bd = acξ2rκ Dr = acDr tells us that either a = 0 or c = 0.
If a = 0, we immediately see d = 0 too, and so we can assume b = 1 without loss of
generality. The equations for a, b, c, d tell us that to fix Z we have c2 = 1Dr+1 and to fix Y
we have c2ξ2r+2 = 1Dr+1 . So this automorphism exists only if ξ
2r+2
κ = 1. By definition
ξκ is a primitive 2kth root of unity and ξ2r+2κ = 1 ⇐⇒ 2k|(2r + 2) ⇐⇒ k|(r + 1),
which is the case by our assumption.
G. Erskine et al.: Self-dual, self-Petrie-dual and Möbius regular maps 9
Bφj(XY ) = η
rβ2r
[
−Drξ−r ∓Dr
√
−D
±cDr
√
−D cD2rξrκ
]
andXB = ηβ
[
cDωλξκ D
−cD −ωλξ−1κ
]
are also equivalent if the map is self-Petrie-dual so we compare the ratios of the elements in
turn. This yields±c = 1
ωλξ
r+1
κ
√
−D =
ωλ
√
−D
Dr+1ξr+1κ
which is true only if Dr = −ω2λ, which is
again the case by our assumption. These conditions are consistent with our other require-
ments for the value of c, (namely that cr = c) so we have an automorphism demonstrating
that this map is self-Petrie-dual.
If on the other hand c = 0 then we have b = 0 and we assume a = 1 without loss of
generality. Fixing Z yields d2D = Dr. Fixing Y yields d2D = ξ2r−2κ D
r. So the map is
self-Petrie-dual only if ξ2r−2κ = 1, which is the case by our assumption.
Now Bφj(XY ) = ηrβ2rDr
[
ωrλ D
rξrκ
−dξ−rκ −dωrλ
]
and XB = ηβ
[
D dDωλξκ
−ωλξ−1κ −dD
]
.
Again, the map is self-Petrie-dual if these two elements are equivalent, that is if both
Drξr−1κ = dω
r+1
λ and ω
r+1
λ ξ
r−1
κ = dD. Applying our assumption ω
2r
κ = −D, we con-
clude the map is self-Petrie-dual.
Using the fact that when q = p the conditions are often much simpler to state, the
preceding two results, Theorem 4.1 and Theorem 4.2, indicate a sufficient condition for a
regular map of type (k, k) to be both self-dual and self-Petrie-dual, namely ω2κ = ω
2
λ =
−D. Corollary 4.3 shows this becomes a tractable sufficient condition for both self-duality
and self-Petrie-duality.
Corollary 4.3. If ω = ωκ = ωλ and 3ω2 = 4 then the associated map is both self-dual
and self-Petrie-dual.
We now turn our attention to the conditions for Möbius regularity.
Proposition 4.4. With the usual setup, a regular map of type (k, l) is Möbius regular if and
only if k is even and ω2κ + 2ω
2
λ = 4.
Proof. By Lemma 2.3, a regular map is Möbius regular if and only if the equationXR
k
2X =
R
k
2 Y is satisfied. In this case R =
[
ξκ 0
0 ξ−1κ
]
. So R
k
2 =
[
α 0
0 −α
]
, where α2 = −1.
Then XR
k
2X = R
k
2 Y is satisfied if and only if
η2β2
[
αD2 + αDω2λ 2αD
2ωλξκ
−2αDωλξ−1κ −αDω2λ − αD2
]
= β
[
0 αξκD
−αξ−1κ 0
]
.
The elements on the leading diagonal must be zero, which yields just one equation:
η2β2αD(D + ω2λ) = 0. The ratio between the non-zero entries is the same for both
matrices and so no further conditions arise.
We conclude that for even k, the map is Möbius regular if and only if D = −ω2λ, which
is equivalent to ω2κ + 2ω
2
λ = 4.
It is not surprising to see some similarity between conditions for self-Petrie-duality
and Möbius regularity since we know all Möbius regular maps (with any automorphism
group) are also self-Petrie-dual [17]. However, since there are alternative conditions which
imply self-Petrie-duality, the converse is not true – not all self-Petrie-dual regular maps are
Möbius regular.
10 Art Discrete Appl. Math. 3 (2020) #P1.03
5 Regular maps of type (5, 5) whose orientation-preserving automor-
phism group 〈R,S〉 is isomorphic to A5
Adrianov’s [1] enumeration of regular hypermaps on PSL(2, q) includes a constant which
deals with the special case which occurs for maps of type (5, 5).
For us to be considering a map of type (k, l) we must have 2k|(q ± 1) and 2l|(q ± 1),
and it is known, see [10], that PSL(2, q) has subgroup A5 when q ≡ ±1 (mod 10). The
constant in Adrianov’s enumeration, which is 2 for maps of type (5, 5) and zero otherwise,
is subtracted to account for the cases when the group 〈R,S〉 collapses into the subgroup
A5 ≤ PSL(2, q).
The following result, with the usual definitions for ωκ and ωλ, indicates when the
orientation-preserving automorphism group of a type (5, 5) map is not the linear fractional
group that we might expect, and as such addresses an omission in [4].
Proposition 5.1. The group 〈R,S〉 of a regular map of type (5, 5), generated by the repre-
sentative matrices R4 and S4, is isomorphic to A5 if and only if ωλ 6= ωκ.
Proof. From [14] we know a presentation of the group A5 is: 〈a, b|a5, b5, (ab)2, (a4b)3〉.
Considering the group 〈R,S|R5, S5, (RS)2, . . . 〉, it is clear that this will be isomorphic
to A5 if and only if the condition (R4S)3 = I is also satified. This is the case if and only
if R−1S has order 3.
R−1S = η
[
ξ−1κ 0
0 ξκ
] [
−ωλξ−1κ −D
1 ωλξκ
]
= η
[
−ωλξ−2κ −ξ−1κ D
ξκ ωλξ
2
κ
]
(R−1S)3 = η3
[
ωλ(2Dξ
−2
κ − ω2λξ−6κ −Dξ2κ) Dξ−2κ (Dξκ + ω2λ(ξκ − ξ5κ − ξ−3κ ))
−(Dξκ + ω2λ(ξκ − ξ5κ − ξ−3κ )) ωλ(Dξ−2κ + ω2λξ6κ − 2Dξ2κ)
]
The off diagonal elements are both zero if and only if Dξκ + ω2λ(ξκ − ξ5κ − ξ−3κ ) = 0.
This condition is equivalent to (ωκ + 2)(ωκ− 2)(1 +ωκωλ)(1−ωκωλ) = 0 and we know
that ωκ 6= ±2 so long as k 6= p.
The leading diagonal entries are equal if and only if D(ξ−2κ + ξ
2
κ) = ω
2
λ(ξ
6
κ + ξ
−6
κ ).
Applying ξ10κ = 1 and eliminating D shows this is equivalent to
(ω2κ − 4)(ω2κ − ω2κω2λ + ω2λ − 2) = 0.
Assume ω2κω
2
λ = 1. The off-diagonals are clearly zero, and the leading diagonal entries
are equal since (ω2κ − ω2κω2λ + ω2λ − 2) = ω−2κ (ω4κ − 3ω2κ + 1) = 0 is always the case
since the expression inside the bracket is the sum of powers of ξ2κ, a 5th root of unity. Then
R−1S has order 3.
Conversely, assume (R−1S)3 = I . Then we instantly have ω2κω
2
λ = 1 since p 6= 5.
We conclude that (R−1S)3 = I if and only if ωκωλ = ±1. By considering the two
possible values for ωκ and ωλ we see that this will happen if and only if ωκ 6= ωλ.
6 Tables of results and comments
In the following tables, produced using the computer package GAP [7], we list for given
q ≤ 49, all the PSL(2, q) maps which have one or more of the properties we have addressed
in the paper, the ticks indicating when the map has each property. The tables are ordered by
G. Erskine et al.: Self-dual, self-Petrie-dual and Möbius regular maps 11
the characteristic of the field, and the elements ξκ, ξλ, ωκ and ωλ are expressed as powers
of a primitive element ξ in the field GF(q2). For a given k, l, only one map is shown in
each equivalence class under the action of the automorphism group. Extended tables of
results detailing the PSL(2, q) regular maps for q ≤ 81 are available in the ancillary file
to [5].
For interest we also include an enumeration in Table 2 which shows how many
PSL(2, q) maps there are with each of these combinations of properties for q ≤ 81.
Table 1: All the PSL(2, q) maps which have one or more of the properties we have ad-
dressed in the paper.
q k l logξ ξκ logξ ξλ logξ ωκ logξ ωλ SD SPD MR
32 5 5 8 8 30 30 3
33 7 7 52 52 420 420 3
33 13 7 140 156 28 532 3
33 13 13 28 28 560 560 3
33 13 13 28 252 560 672 3
33 13 13 140 140 28 28 3
33 14 14 26 26 476 476 3
5 5 5 3
52 3 13 104 72 0 494 3
52 4 13 78 168 390 26 3
52 6 13 52 24 546 260 3
52 12 12 26 26 416 416 3
52 12 13 26 216 416 130 3 3
52 13 13 24 24 260 260 3
52 13 13 24 120 260 52 3
52 13 13 72 72 494 494 3
52 13 13 72 264 494 598 3
52 13 13 168 168 26 26 3
52 13 13 168 216 26 130 3
7 7 7 3
72 4 25 300 144 400 1250 3
72 5 5 240 240 1350 1350 3
72 6 24 200 250 1400 2050 3
72 7 24 50 500 3
72 7 25 432 1500 3
72 7 25 816 100 3
72 8 25 150 48 2200 1450 3
72 8 25 450 528 600 950 3
12 Art Discrete Appl. Math. 3 (2020) #P1.03
q k l logξ ξκ logξ ξλ logξ ωκ logξ ωλ SD SPD MR
72 12 12 100 100 750 750 3
72 12 25 100 144 750 1250 3 3
72 24 12 50 100 500 750 3 3
72 24 24 50 50 500 500 3
72 24 24 50 350 500 1100 3
72 24 24 250 250 2050 2050 3
72 24 24 250 550 2050 1150 3
72 24 25 250 912 2050 700 3 3
72 25 25 48 48 1450 1450 3
72 25 25 48 336 1450 550 3
72 25 25 144 144 1250 1250 3
72 25 25 144 1008 1250 1550 3
72 25 25 432 432 1500 1500 3
72 25 25 432 624 1500 900 3
72 25 25 528 528 950 950 3
72 25 25 528 1104 950 1850 3
72 25 25 816 816 100 100 3
72 25 25 816 912 100 700 3
11 5 5 12 12 36 36 3
11 5 5 36 36 24 24 3 3
11 5 6 12 10 36 108 3
11 6 6 10 10 108 108 3
11 11 11 3
13 6 6 14 14 112 112 3
13 7 7 12 12 126 126 3
13 7 7 12 60 126 56 3
13 7 7 36 36 70 70 3 3
13 7 7 60 60 56 56 3
13 7 13 60 56 3
13 13 13 3
17 8 8 18 18 36 36 3
17 8 8 54 54 90 90 3
17 8 9 54 80 90 54 3 3
17 8 17 18 36 3 3
17 9 9 16 16 216 216 3
17 9 9 80 80 54 54 3
G. Erskine et al.: Self-dual, self-Petrie-dual and Möbius regular maps 13
q k l logξ ξκ logξ ξλ logξ ωκ logξ ωλ SD SPD MR
17 9 9 112 112 18 18 3
17 17 17 3
19 3 9 60 20 0 300 3
19 5 5 36 36 320 320 3
19 5 5 108 108 220 220 3
19 9 5 20 36 300 320 3
19 9 9 20 20 300 300 3
19 9 9 100 100 80 80 3
19 9 9 140 100 340 80 3
19 9 9 140 140 340 340 3
19 9 10 100 18 80 60 3
19 10 10 18 18 60 60 3
19 10 10 54 54 100 100 3
19 19 19 3
23 3 11 88 72 0 168 3
23 6 6 44 44 192 192 3
23 6 11 44 216 192 240 3 3
23 11 11 24 24 360 360 3
23 11 11 72 72 168 168 3
23 11 11 120 120 480 480 3
23 11 11 168 168 336 336 3
23 11 11 216 216 240 240 3
23 12 11 22 24 144 360 3 3
23 12 12 22 22 144 144 3
23 12 12 110 110 384 384 3 3 3
23 23 23 3
29 3 15 140 28 0 480 3
29 5 5 84 84 180 180 3
29 5 5 252 252 240 240 3
29 5 14 84 90 180 270 3
29 5 29 252 240 3
29 7 7 60 60 570 570 3
29 7 7 180 180 750 750 3
29 7 7 300 300 780 780 3
29 14 14 30 30 630 630 3
29 14 14 90 90 270 270 3
29 14 14 150 150 540 540 3
14 Art Discrete Appl. Math. 3 (2020) #P1.03
q k l logξ ξκ logξ ξλ logξ ωκ logξ ωλ SD SPD MR
29 15 7 308 180 300 750 3
29 15 7 364 300 810 780 3
29 15 14 196 30 90 630 3
29 15 15 28 28 480 480 3
29 15 15 28 308 480 300 3
29 15 15 196 196 90 90 3
29 15 15 308 308 300 300 3
29 15 15 364 364 810 810 3
29 29 29 3
31 5 5 96 96 128 128 3
31 5 5 288 288 352 352 3
31 8 8 60 60 160 160 3
31 8 8 180 180 704 704 3
31 8 15 60 352 160 320 3 3
31 8 16 180 30 704 256 3 3
31 15 15 32 32 416 416 3
31 15 15 224 224 512 512 3
31 15 15 352 352 320 320 3
31 15 15 416 416 672 672 3
31 16 5 210 288 448 352 3 3
31 16 8 90 60 576 160 3 3
31 16 15 150 416 544 672 3 3
31 16 16 30 30 256 256 3
31 16 16 30 150 256 544 3 3
31 16 16 90 90 576 576 3
31 16 16 150 150 544 544 3
31 16 16 210 210 448 448 3
31 31 31 3
37 6 6 114 114 1178 1178 3
37 9 9 76 76 190 190 3
37 9 9 380 380 1292 1292 3
37 9 9 532 532 1254 1254 3
37 18 18 38 38 836 836 3
37 18 18 190 190 266 266 3
37 18 18 266 266 76 76 3
37 19 9 180 380 798 1292 3
37 19 9 468 76 646 190 3
G. Erskine et al.: Self-dual, self-Petrie-dual and Möbius regular maps 15
q k l logξ ξκ logξ ξλ logξ ωκ logξ ωλ SD SPD MR
37 19 18 252 38 1140 836 3
37 19 18 612 266 988 76 3
37 19 19 36 36 1026 1026 3
37 19 19 36 108 1026 418 3
37 19 19 108 108 418 418 3
37 19 19 108 252 418 1140 3
37 19 19 180 180 798 798 3
37 19 19 252 252 1140 1140 3
37 19 19 324 324 1064 1064 3
37 19 19 396 396 228 228 3 3
37 19 19 468 468 646 646 3
37 19 19 540 324 532 1064 3
37 19 19 540 540 532 532 3
37 19 19 612 612 988 988 3
37 19 37 324 1064 3
37 37 37 3
41 5 5 168 168 1638 1638 3
41 5 5 504 504 882 882 3
41 5 7 168 600 1638 126 3
41 5 20 504 294 882 924 3
41 7 7 120 120 210 210 3
41 7 7 360 360 504 504 3
41 7 7 600 600 126 126 3
41 10 10 84 84 630 630 3
41 10 10 252 252 672 672 3
41 10 21 84 520 630 336 3 3
41 10 41 252 672 3 3
41 20 20 42 42 1302 1302 3
41 20 20 42 126 1302 714 3 3
41 20 20 126 126 714 714 3
41 20 20 294 294 924 924 3
41 20 20 378 378 420 420 3
41 20 21 126 680 714 1218 3 3
41 20 21 294 200 924 168 3 3
41 20 21 378 440 420 756 3 3
41 21 21 40 40 1428 1428 3
41 21 21 200 200 168 168 3
16 Art Discrete Appl. Math. 3 (2020) #P1.03
q k l logξ ξκ logξ ξλ logξ ωκ logξ ωλ SD SPD MR
41 21 21 440 440 756 756 3
41 21 21 520 520 336 336 3
41 21 21 680 680 1218 1218 3
41 21 21 760 760 1134 1134 3
41 41 41 3
43 3 22 308 294 0 1276 3
43 7 7 132 132 792 792 3
43 7 7 396 396 220 220 3
43 7 7 660 660 1760 1760 3
43 7 21 132 484 792 1628 3
43 7 21 396 572 220 484 3
43 7 21 660 748 1760 1496 3
43 11 11 84 84 1452 1452 3
43 11 11 252 252 1012 1012 3
43 11 11 420 420 1540 1540 3
43 11 11 588 588 1584 1584 3
43 11 11 756 756 1804 1804 3
43 21 7 484 660 1628 1760 3
43 21 11 44 420 396 1540 3
43 21 11 220 588 1100 1584 3
43 21 11 748 84 1496 1452 3
43 21 11 836 252 440 1012 3
43 21 21 44 44 396 396 3
43 21 21 220 220 1100 1100 3
43 21 21 484 484 1628 1628 3
43 21 21 572 220 484 1100 3
43 21 21 572 572 484 484 3
43 21 21 748 748 1496 1496 3
43 21 21 836 836 440 440 3
43 22 22 42 42 132 132 3
43 22 22 126 126 308 308 3
43 22 22 210 210 968 968 3
43 22 22 294 294 1276 1276 3
43 22 22 378 378 1672 1672 3
43 43 43 3
47 3 23 368 528 0 1152 3
47 6 6 184 184 480 480 3
G. Erskine et al.: Self-dual, self-Petrie-dual and Möbius regular maps 17
q k l logξ ξκ logξ ξλ logξ ωκ logξ ωλ SD SPD MR
47 6 24 184 46 480 672 3 3
47 8 8 138 138 960 960 3
47 8 8 414 414 576 576 3
47 8 23 138 624 960 144 3 3
47 8 23 414 432 576 528 3 3
47 12 8 92 414 1200 576 3 3
47 12 12 92 92 1200 1200 3
47 12 12 460 460 1008 1008 3
47 12 23 460 816 1008 768 3 3
47 23 23 48 48 1344 1344 3
47 23 23 144 144 1056 1056 3
47 23 23 240 240 288 288 3
47 23 23 336 336 720 720 3
47 23 23 432 432 528 528 3
47 23 23 528 528 1152 1152 3
47 23 23 624 624 144 144 3
47 23 23 720 720 1296 1296 3
47 23 23 816 816 768 768 3
47 23 23 912 912 624 624 3
47 23 23 1008 1008 2016 2016 3
47 24 12 46 460 672 1008 3 3
47 24 23 322 144 1920 1056 3 3
47 24 23 506 48 336 1344 3 3
47 24 24 46 46 672 672 3
47 24 24 230 230 1488 1488 3 3 3
47 24 24 322 322 1920 1920 3
47 24 24 506 506 336 336 3
47 47 47 3
18 Art Discrete Appl. Math. 3 (2020) #P1.03
Table 2: External symmetries of regular maps on PSL(2, q).
q Maps None SD only SP only SD+SP SP+MR SD+SP+MR
32 3 2 1 0 0 0 0
33 54 48 4 2 0 0 0
34 381 356 15 7 0 3 0
5 1 0 1 0 0 0 0
52 63 52 7 3 0 1 0
7 5 4 1 0 0 0 0
72 264 238 16 7 0 3 0
11 16 11 3 1 1 0 0
13 33 26 4 2 1 0 0
17 58 50 6 0 0 2 0
19 70 58 8 4 0 0 0
23 113 101 8 1 0 2 1
29 183 163 13 7 0 0 0
31 209 190 13 0 0 6 0
37 315 290 16 8 1 0 0
41 382 356 18 2 0 6 0
43 430 400 20 10 0 0 0
47 515 485 20 1 0 8 1
53 663 625 25 13 0 0 0
59 820 779 27 13 1 0 0
61 879 836 28 14 1 0 0
67 1072 1024 32 16 0 0 0
71 1199 1151 32 4 0 11 1
73 1276 1227 33 3 1 12 0
79 1493 1438 37 2 0 16 0
The work in this paper has only addressed regular maps on linear fractional groups
where the associated finite field has odd characteristic. Many of the calculations would
look quite different if we were to consider the case when p = 2.
It has been an open problem for some time as to whether there exists a self-dual and self-
Petrie-dual regular map for any given vertex degree k on some surface. In [2], Archdeacon,
Conder and Širáň proved the existence of such a map for any even valency. The results in
this paper allow Fraser, Jeans and Širáň [6] to prove the existence of a self-dual, self-Petrie-
dual regular map for any given odd valency k ≥ 5.
G. Erskine et al.: Self-dual, self-Petrie-dual and Möbius regular maps 19
ORCID iDs
Grahame Erskine https://orcid.org/0000-0001-7067-6004
Katarı́na Hriňáková https://orcid.org/0000-0001-6137-2607
Olivia Reade Jeans https://orcid.org/0000-0002-7598-9680
References
[1] N. M. Adrianov, Regular maps with the automorphism group PSL2(q), Uspekhi Mat. Nauk 52
(1997), 195–196, doi:10.1070/rm1997v052n04abeh002061.
[2] D. Archdeacon, M. Conder and J. Širáň, Trinity symmetry and kaleidoscopic regular maps,
Trans. Amer. Math. Soc. 366 (2014), 4491–4512, doi:10.1090/s0002-9947-2013-06079-5.
[3] H. R. Brahana, Regular Maps and Their Groups, Amer. J. Math. 49 (1927), 268–284, doi:
10.2307/2370756.
[4] M. Conder, P. Potočnik and J. Širáň, Regular hypermaps over projective linear groups, J. Aust.
Math. Soc. 85 (2008), 155–175, doi:10.1017/s1446788708000827.
[5] G. Erskine, K. Hriňáková and O. Jeans, Self-dual, self-petrie-dual and Mobius regular maps on
linear fractional groups, arXiv (2018), https://arxiv.org/abs/1807.11307.
[6] J. Fraser, O. Jeans and J. Širáň, Regular self-dual and self-Petrie-dual maps of arbitrary valency,
Ars Math. Contemp. 16 (2019), 403–410, doi:10.26493/1855-3974.1749.84e.
[7] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.9.1, 05 2018,
https://www.gap-system.org.
[8] K. Hriňáková, Regular maps on linear fractional groups, Ph.D. thesis, Slovak University of
Technology, Bratislava, Slovakia, SK, 2016, https://is.stuba.sk/lide/clovek.
pl?zalozka=7;id=62155;studium=94093;zp=29315;dinfo_jazyk=3;
interni_vzorek=62155;lang=en.
[9] G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math.
Soc. (3) 37 (1978), 273–307, doi:10.1112/plms/s3-37.2.273.
[10] O. H. King, The subgroup structure of finite classical groups in terms of geometric configura-
tions, in: Surveys in combinatorics 2005, Cambridge Univ. Press, Cambridge, volume 327 of
London Math. Soc. Lecture Note Ser., pp. 29–56, 2005, doi:10.1017/cbo9780511734885.003.
[11] C. H. Li and J. Širáň, Mobius regular maps, J. Combin. Theory Ser. B 97 (2007), 57–73, doi:
10.1016/j.jctb.2006.04.001.
[12] A. M. Macbeath, Generators of the linear fractional groups, in: Number Theory (Proc. Sympos.
Pure Math., Vol. XII, Houston, Tex., 1967), Amer. Math. Soc., Providence, R.I., 1969 pp. 14–
32, http://www.ams.org/books/pspum/012/.
[13] Q. Mushtaq, Some remarks on coset diagrams for the modular group, Math. Chronicle 16
(1987), 69–77.
[14] H. E. Rose, A course on finite groups, Universitext, Springer-Verlag London, Ltd., London,
2009, doi:10.1007/978-1-84882-889-6.
[15] C.-h. Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13–42, doi:
10.1007/bf02392383.
[16] J. Širáň, How symmetric can maps on surfaces be?, in: Surveys in combinatorics 2013, Cam-
bridge Univ. Press, Cambridge, volume 409 of London Math. Soc. Lecture Note Ser., pp. 161–
238, 2013.
[17] S. Wilson, Cantankerous maps and rotary embeddings of Kn, J. Combin. Theory Ser. B 47
(1989), 262–279, doi:10.1016/0095-8956(89)90028-2.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.04
https://doi.org/10.26493/2590-9770.1279.02c
(Also available at http://adam-journal.eu)
A new generalization of generalized Petersen
graphs*
Katarı́na Jasenčáková
Faculty of Management Science and Informatics, University of Žilina, Žilina, Slovakia
Robert Jajcay†
Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava,
Slovakia; also affiliated with Faculty of Mathematics, Natural Sciences and Information
Technology, University of Primorska, Koper, Slovenia
Tomaž Pisanski‡
Faculty of Mathematics, Natural Sciences and Information Technology,
University of Primorska, Koper, Slovenia; also affiliated with
FMF and IMFM, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
Received 30 November 2018, accepted 13 August 2019, published online 27 July 2020
Abstract
We discuss a new family of cubic graphs, which we call group divisible generalised Pe-
tersen graphs (GDGP -graphs), that bears a close resemblance to the family of generalised
Petersen graphs, both in definition and properties. The focus of our paper is on determin-
ing the algebraic properties of graphs from our new family. We look for highly symmetric
graphs, e.g., graphs with large automorphism groups, and vertex- or arc-transitive graphs.
In particular, we present arithmetic conditions for the defining parameters that guarantee
that graphs with these parameters are vertex-transitive or Cayley, and we find one arc-
transitive GDGP -graph which is neither a CQ graph of Feng and Wang, nor a generalised
Petersen graph.
Keywords: Generalised Petersen graph, arc-transitive graph, vertex-transitive graph, Cayley graph,
automorphism group.
Math. Subj. Class. (2020): 05C25
*We thank the anonymous referee for all the helpful and insightful comments.
†Author acknowledges support by the projects VEGA 1/0596/17, VEGA 1/0719/18, VEGA 1/0423/20,
APVV-15-0220, and by the Slovenian Research Agency (research projects N1-0038, N1-0062, J1-9108).
‡The work is supported in part by the Slovenian Research Agency (research program P1-0294 and research
projects N1-0032, J1-9187, J1-1690, N1-0140), and in part by H2020 Teaming InnoRenew CoE.
E-mail addresses: katarina.jasencakova@fri.uniza.sk (Katarı́na Jasenčáková), robert.jajcay@fmph.uniba.sk
(Robert Jajcay), pisanski@upr.si (Tomaž Pisanski)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 3 (2020) #P1.04
1 Introduction
Generalised Petersen graphs GP (n, k) (the name and notation coined in 1969 by Watkins
[18], with the subclass with n and k relatively prime considered already by Coxeter in
1950 [6]) constitute one of the central families of algebraic graph theory. While there are
many reasons for the interest in this family, with a bit of oversimplification one could say
that among the most important is the simplicity of their description (requiring just two
parameters n and k) combined with the richness of the family that includes the well-known
Petersen and dodecahedron graphs, as well as large families of vertex-transitive graphs and
seven symmetric (arc-transitive) graphs.
Our motivation for studying the new family of group divisible generalised Petersen
graphs (GDGP -graphs; introduced in [11] under the name SGP -graphs) lies in the fact
that they share the above characteristics with the generalised Petersen graphs. They include
all vertex-transitive generalised Petersen graphs but the dodecahedron as a proper subclass,
are easily defined via a sequence of integral parameters, and contain graphs of various
levels of symmetry.
Historically, ours is certainly not the first attempt at generalising generalised Petersen
graphs. In 1988, the family of I-graphs was introduced in the Foster Census [2]. This
family differs from the generalised Petersen graphs in allowing the span on the outer rim
to be different from 1: The I-graph I(n, j, k) is the cubic graph with vertex set {ui, vi | i ∈
Zn} and edge set {{ui, ui+j}, {ui, vi}, {vi, vi+k} | i ∈ Zn}. However, the only I-graphs
that are vertex-transitive are the original generalised Petersen graphs which are the graphs
I(n, 1, k) [1, 14].
The family of GI-graphs introduced in [5] by Conder, Pisanski and Žitnik in 2014 is a
further generalisation of I-graphs. For positive integers n ≥ 3,m ≥ 1, and a sequenceK of
elements in Zn−{0, n2 },K = (k0, k1, . . . , km−1), theGI-graphGI(n; k0, k1, . . . , km−1)
is the graph with vertex set Zm × Zn and edges of two types:
(i) an edge from (u, v) to (u′, v), for all distinct u, u′ ∈ Zm and all v ∈ Zn,
(ii) edges from (u, v) to (u, v ± ku), for all u ∈ Zm and all v ∈ Zn.
The GI-graphs are (m+1)-regular, thus cubic when m = 2, which is the case that covers
the I-graphs, with the subclass of the GI(n; 1, k) graphs covering the generalised Petersen
graphs.
Another generalisation is due to Lovrečič-Saražin, Pacco and Previtali, who extended
the class of generalised Petersen graphs to the so-called supergeneralised Petersen graphs
[15]. Let n ≥ 3 and m ≥ 2 be integers and k0, k1, . . . , km−1 ∈ Zn − {0}. The vertex-set
of the supergeneralised Petersen graph P (m,n; k0, . . . , km−1) is Zm × Zn and its edges
are of two types:
(i) an edge from (u, v) to (u+ 1, v), for all u ∈ Zm and all v ∈ Zn,
(ii) edges from (u, v) to (u, v ± ku), for all u ∈ Zm and all v ∈ Zn.
Note that GP (n, k) is isomorphic to P (2, n; 1, k).
Finally, in 2012, Zhou and Feng [20] modified the class of generalised Petersen graphs
in order to classify cubic vertex-transitive non-Cayley graphs of order 8p, for any prime p
[20]. In their definition, the subgraph induced by the outer edges is a union of two n-cycles.
Let n ≥ 3 and k ∈ Zn − {0}. The double generalised Petersen graph DP (n, k) is defined
to have the vertex set {xi, yi, ui, vi | i ∈ Zn} and the edge set equal to the union of the outer
K. Jasenčáková et al.: A new generalisation of generalised Petersen graphs 3
Figure 1: GI(5; 1, 1, 1, 2) and P (4, 5; 1, 1, 1, 2).
x0 y0u0 v0
x1 y1u1 v1
x2 y2u2 v2
x3 y3u3 v3
x4 y4u4 v4
Figure 2: DP (5, 2).
edges {{xi, xi+1}, {yi, yi+1} | i ∈ Zn}, the inner edges {{ui, vi+k}, {vi, ui+k} | i ∈ Zn},
and the spokes {{xi, ui}, {yi, vi} | i ∈ Zn}.
Even though non-empty intersections exist between the above classes and the class of
group divisible generalised Petersen graphs considered in our paper, none of these is sig-
nificant, and we believe that ours is, in a way, the most natural generalisation of generalised
Petersen graphs.
2 Generalised Petersen graphs
Let us review the basic properties of generalised Petersen graphs. A generalised Petersen
graph GP (n, k) is determined by integers n and k, n ≥ 3 and n2 > k ≥ 1. The vertex set
V (GP (n, k)) = {ui, vi | i ∈ Zn} is of order 2n and the edge set E(GP (n, k)) of size 3n
consists of edges of the form
{ui, ui+1}, {ui, vi}, {vi, vi+k}, (2.1)
where i ∈ Zn. Thus, GP (n, k) is always a trivalent graph, the Petersen graph is the graph
GP (5, 2), the dodecahedron is GP (10, 2), and the ADAM graph is GP (24, 5).
We will call the ui vertices the outer vertices, the vi vertices the inner vertices, and
the three distinct forms of edges displayed in (2.1) outer edges, spokes, and inner edges,
respectively. Graphs introduced in this paper will also contain vertices and edges of these
types. We will use the symbols Ω,Σ and I , respectively, to denote the three n-sets of
edges. The n-circuit induced in GP (n, k) by Ω will be called the outer rim. If d denotes
the greatest common divisor of n and k, then I induces a subgraph which is the union of
d pairwise-disjoint nd -circuits, called inner rims. The parameter k also denotes the span of
4 Art Discrete Appl. Math. 3 (2020) #P1.04
1 k
0
Zn
Figure 3: Voltage graph for the generalised Petersen graph GP (n, k).
the inner rims (which is the distance, as measured on the outer rim, between the outer rim
neighbors of two vertices adjacent on an inner rim).
The class of generalised Petersen graphs is well understood and has been studied by
many authors. In 1971, Frucht, Graver and Watkins [9] determined their automorphism
groups. They proved that GP (n, k) is vertex-transitive if and only if k2 ≡ ±1 mod n or
(n, k) = (10, 2). Later, Nedela and Škoviera [16], and (independently) Lovrečič-Saražin
[13] proved that a generalised Petersen graph GP (n, k) is Cayley if and only if k2 ≡ 1
mod n. Recall that a Cayley graph Cay(G,X), where G is a group generated by the set
X which does not contain the identity 1G and is closed under taking inverses, is the graph
whose vertices are the elements of G and edges are the pairs {g, xg}, g ∈ G, x ∈ X .
3 Polycirculants and voltage graphs
A non-identity automorphism of a graph is (m,n)-semiregular if its cycle decomposition
consists of m cycles of length n. Graphs admitting (m,n)-semiregular automorphisms
are called m-circulants (if one chooses to suppress the parameter m, they are sometimes
called polycirculants). If m = 1, 2, 3, or 4, an m-circulant is said to be a circulant, a bi-
circulant, a tricirculant, or a tetracirculant, respectively. It is easy to see that generalised
Petersen graphs are bicirculants; the corresponding automorphism consists of the two cy-
cles (u0, u1, u2, . . . , un−1), (v0, v1, v2, . . . , vn−1).
The reader is also most likely familiar with the fact that generalised Petersen graphs
can be defined in a nice and compact way using the language of voltage graphs (a more
detailed treatment may be found for example in [10]):
If Γ is an undirected graph, we associate each edge of Γ with a pair of opposite arcs
and denote the set of all such arcs by D(Γ). A voltage assignment on Γ is any mapping α
from D(Γ) into a group G that satisfies the condition α(e−1) = (α(e))−1 for all e ∈ D(Γ)
(with e−1 being the opposite arc of e, and (α(e))−1 being the inverse of α(e) in G). The
lift (sometimes called the derived regular cover) of Γ with respect to a voltage assignment
α on Γ is a graph denoted by Γα. The vertex set V (Γα) consists of |V (Γ)| · |G| vertices
ug = (u, g), (u, g) ∈ V (Γ) × G. Two vertices ug and vf are adjacent in Γα if e = (u, v)
is an arc of Γ and f = g · α(e) in G.
All generalised Petersen graphs GP (n, k) are lifts of the dumbbell graph D which
consists of two vertices joined by an edge and loops attached to them. The corresponding
voltage assignment α : D(D) −→ Zn assigns 0 to the arcs connecting the two vertices, 1
and −1 to the arcs of the loop at one of the vertices, and k and −k to the arcs of the loop at
the other vertex (Figure 3).
Similarly, the I-graph I(n, j, k) is a derived regular cover of the dumbbell graph with
0 assigned to the ‘handle’, and the values j,−j and k,−k ∈ Zn assigned to the loops
(Figure 4).
The GI-graph GI(n; a, b, c, d) is a lift of the complete graph K4, and so is the super-
generalised Petersen graph P (4, n; a, b, c, d) (see Figures 5 and 6, respectively).
K. Jasenčáková et al.: A new generalisation of generalised Petersen graphs 5
j k
0
Zn
Figure 4: Voltage graph for the I-graph I(n, j, k).
a b
c d
Zn
0
0
0 0
0 0
Figure 5: Voltage graph for the GI-graph GI(n; a, b, c, d).
Recently, Conder, Estélyi, and Pisanski in [4] considered more general voltage assign-
ments, and thus generalised the double generalised Petersen graphs even further.
There is another family of polycirculants that will prove useful later in our paper, in-
troduced by Feng and Wang in 2003 [7]. Their graphs are called CQ graphs, and were
originally introduced as octacirculants (for their voltage graph description see Figure 7).
The definition of CQ(k, n) used in [7] makes sense for any k, n such that gcd(k, n) = 1,
which is equivalent to k ∈ Z∗n. Frelih and Kutnar [8] later correctly showed that each
CQ(k, n) is in fact a tetracirculant. However, their voltage graph depiction is not correct.
One needs two different voltage graphs, depending on the parity of m. The correct voltage
graphs are depicted in Figures 8 and 9.
Moreover, in the definition of CQ(k,m) used by Frelih and Kutnar the inverse k−1 is
a b
c d
Zn
0
0
0 0
Figure 6: Voltage graph for the supergeneralised Petersen graph P (4, n; a, b, c, d).
6 Art Discrete Appl. Math. 3 (2020) #P1.04
Zn
0
1
0 0
0
0
0
0
kk
k−1
k−1−1
Figure 7: Original voltage graph for the octacirculant graph CQ(k, n), gcd(k, n) = 1,
which appeared in [7].
0 0
2k (mod 2m)
m−2k−1 (mod 2m)
0
1
Z2m
Figure 8: Corrected voltage graph for the tetracirculant graph CQ(k,m), k odd, m even.
0 0
2k (mod 2m)
2m−2k−1(mod 2m)
0
1
Z2m
Figure 9: Corrected voltage graph for the tetracirculant graph CQ(k,m), k odd, m odd.
K. Jasenčáková et al.: A new generalisation of generalised Petersen graphs 7
not needed. Hence, their voltage graphs define a family of graphs that is more general than
that of Feng and Wang [7]. In our paper, we shall use this more general definition.
4 GDGP -graphs
The graphs we shall focus on in this paper are defined as follows.
Definition 4.1. Let n ≥ 3 and m ≥ 2 be positive integers such that m divides n, let a be a
non-zero element of Zm, and let K = (k0, k1, . . . , km−1) be a sequence of elements from
Zn all of which are congruent to a modulo m and satisfy the requirement kj + kj−a 6≡ 0
(mod n), for all j ∈ Zm.
The graph GDGPm(n;K), or alternatively GDGPm(n; k0, k1, . . . , km−1), has the
vertex set {ui, vi | i ∈ Zn} of order 2n and the edge set of size 3n:
{ui, ui+1}, {ui, vi}, {vmi+j , vmi+j+kj}, (4.1)
where i ∈ Z n
m
, j ∈ Zm, and the arithmetic operations are performed modulo n.
While the GDGP -graphs defined above share the outer rim edges and the spokes with
the generalised Petersen graphs (and are therefore all connected), the inner edges are de-
termined by the more complicated rule {{vmi+j , vmi+j+kj} | i ∈ Z nm , j ∈ Zm} applied in
groups of size m ≥ 2.
The choices made in our definition guarantee that the GDGP -graphs are cubic. This
claim is obviously true for the outer vertices ui. To prove the claim for the inner ver-
tices vmi+j , i ∈ Z nm , j ∈ Zm, it is enough to show that each inner vertex vmi+j of
GDGPm(n;K) is incident with exactly two edges of the type determined by the third rule
of (4.1). Thus, assume that {vmi′+j′ , vmi′+j′+kj′} is incident with vmi+j . Then, either
vmi+j = vmi′+j′ or vmi+j = vmi′+j′+kj′ . If vmi+j = vmi′+j′ , then i = i
′, j = j′, and
kj = kj′ is uniquely determined; there is exactly one such edge. If vmi+j = vmi′+j′+kj′ ,
then mi + j = mi′ + j′ + kj′ , thus j ≡ j′ + kj′ ≡ j′ + a (mod m), which uniquely
determines j′ = j − a ∈ Zm as well as kj′ . The equation mi + j = mi′ + j′ + kj′
then uniquely determines i′, and therefore there is exactly one edge {vmi′+j′ , vmi′+j′+kj′}
for which vmi+j = vmi′+j′+kj′ . To complete the argument, note that the two edges
{vmi+j , vmi+j+kj} and {vmi+j−kj−a , vmi+j} are necessarily different, since we assume
that kj + kj−a 6≡ 0 (mod n), for all j ∈ Zm. Let us observe for future reference that the
three neighbors of vmi+j are the vertices umi+j , vmi+j+kj and vmi+j−kj−a .
Being 2-regular, the graph induced by the inner vertices vi, i ∈ Zn consists of disjoint
cycles. If we denote the order of a in Zm by om(a), it is easy to see that the length of
the inner cycle containing vmi+j is the product of om(a) with the order on(kj + kj+a +
kj+2a + . . . + kj+(om(a)−1)a) of the element kj + kj+a + kj+2a + . . . + kj+(om(a)−1)a
in Zn (with the indices calculated modulo m). Thus, if a is chosen to be a generator for
Zm (i.e., om(a) = m), all inner cycles in GDGPm(n; k0, k1, . . . , km−1) are of the same
length m · on(k0 + k1 + . . .+ km−1). In particular, if m = 2, a is by definition necessarily
congruent to 1 (mod 2), and is therefore a generator for Z2, hence all inner cycles of the
graphs GDGP2(n; k0, k1) are of length 2 · on(k0 + k1).
Example 4.2. Consider the graph GDGP2(8; 1, 3) in Figure 10. Both 1 and 3 are congru-
ent to 1 (mod 2), which is a generator for Z2. The order of the sum 1 + 3 = 4 is 2 in Z8,
and hence the inner edges of this graph form two disjoint 4-cycles.
8 Art Discrete Appl. Math. 3 (2020) #P1.04
Figure 10: GDGP2(8; 1, 3).
It is easy to see that, for even n and odd k,GP (n, k) is isomorphic toGDGP2(n; k, k),
and for n divisible by 3 and k 6≡ 0 (mod 3),GP (n, k) is isomorphic toGDGP3(n; k, k, k).
We generalise this observation in the following lemma. A sequence k0, . . . , km−1 is said
to be periodic if there exists an integer 0 < p < m that divides m and ki = ki+p, for all
i ∈ Zm. The smallest p with this property is the period of the sequence. The proof of the
following lemma is now obvious.
Lemma 4.3. Let GDGPm(n;K) be a graph such that K = (k0, . . . , km−1) is a periodic
sequence with a period p. If p = 1, the graph GDGPm(n;K) is isomorphic to the graph
GP (n, k), and if p > 1, GDGPm(n;K) is isomorphic to the graph GDGPp(n;K ′),
where K ′ = (k0, . . . , kp−1).
Consequently, GDGPm(n; k, k, . . . , k) ∼= GP (n, k), for all divisors m of n and all
k 6≡ 0 (mod m). To simplify our notation and arguments, we will assume from now on
that the sequence K used in the notation GDGPm(n;K) is aperiodic.
Most importantly, not all GDGP -graphs are generalised Petersen graphs. Consider,
for example, the graph GDGP2(8; 1, 3) constructed in Example 4.2. The subgraph in-
duced by the inner vertices consists of two disjoint 4-cycles. While the same is true for
GP (8, 2), nevertheless, GDGP2(8; 1, 3) is not isomorphic to any generalised Petersen
graph. To see this, note that each of the outer vertices of this graph lies on exactly one
4-cycle, while each of the inner vertices lies on two 4-cycles. Thus, no automorphism of
GDGP2(8; 1, 3) interchanges the outer and inner vertices (and GDGP2(8; 1, 3) is not a
vertex-transitive graph). If GDGP2(8; 1, 3) were to be isomorphic to a generalised Pe-
tersen graph, it would have to be a bicirculant and would have to admit a (2, 8)-semiregular
automorphism. The two 8-orbits would thus necessarily consist of the outer and inner ver-
tices. The automorphism group of the 8-cycle induced by the outer vertices is equal to
the dihedral group D8 which contains only two 8-cycles: (u0, u1, . . . , u7), and its inverse.
Thus, the action of any (2, 8)-semiregular automorphism of GDGP2(8; 1, 3) on the outer
vertices would have to be equal to one of these cycles. Since automorphisms must pre-
serve adjacency, this would necessarily force the action of this semiregular automorphism
on the inner vertices to be the cycle (v0, v1, . . . , v7) or its inverse. However, neither the
permutation (u0, u1, . . . , u7)(v0, v1, . . . , v7) nor its inverse are graph automorphisms of
GDGP2(8; 1, 3). Hence, GDGP2(8; 1, 3) is not a bicirculant, and is therefore not isomor-
phic to any generalised Petersen graph.
K. Jasenčáková et al.: A new generalisation of generalised Petersen graphs 9
Figure 11: GDGP2(8; 1, 5).
On the other hand, it is not hard to see that the (4, 4)-semiregular permutation (u0, u2,
u4, u6)(u1, u3, u5, u7)(v0, v2, v4, v6)(v1, v3, v5, v7) is an automorphism ofGDGP2(8; 1, 3),
and that the following theorem holds in general.
Theorem 4.4. For all m ≥ 2, the permutation
α : V (GDGPm(n;K))→ V (GDGPm(n;K)), ui 7→ ui+m, vi 7→ vi+m, i ∈ Zn,
is a (2m, nm )-semiregular automorphism of GDGPm(n;K).
Thus, every GDGPm(n;K)-graph is a 2m-circulant.
We conclude the section with an easy but useful graph-theoretical property of the
GDGP -graphs.
Lemma 4.5. The GDGPm(n;K) is a bipartite graph if and only if n is even and all
elements in K are odd.
5 Automorphisms of GDGP -graphs
Many of the ideas of this and the forthcoming sections can be demonstrated with the use of
the following family of GDGP -graphs.
Example 5.1. For each even n ≥ 4, consider the graph GDGP2(n; 1, n − 3). The
graphs in this family are also known as the crossed prism graphs [19]. In particular,
GDGP2(4; 1, 1) ∼= GP (4, 1), and GDGP2(6; 1, 3) is the Franklin graph. The graph
GDGP2(8; 1, 5) is depicted in Figure 11. All the graphs GDGP2(n; 1, n− 3) are vertex-
transitive, which means, in particular, that they admit an automorphism mapping an outer
vertex to an inner vertex.
Our first result follows already from our discussion of GDGP2(8; 1, 3).
Lemma 5.2. If γ ∈ Aut(GDGPm(n;K)) fixes set-wise any of the sets Ω,Σ or I , then it
either fixes all three sets or fixes Σ set-wise and interchanges Ω and I .
We have observed already that the action on Ω of an Ω-preserving automorphism must
belong to Dn. Assume that an automorphism σ preserves Ω and acts on Ω as a reflection.
Then one of the following occurs:
10 Art Discrete Appl. Math. 3 (2020) #P1.04
1. σ has no fixed points, in which case n is necessarily even and there exists an s ∈ Zn
such that σ swaps us and us+1, and, consequently, σ swaps all the pairs us−i and
us+1+i, i ∈ Zn;
2. σ fixes at least one vertex, say us, and consequently swaps the pairs us+i and us−i,
i ∈ Zn.
The same is necessarily true for the inner vertices, and in case (1) σ swaps vs−i and vs+1+i,
i ∈ Zn, while in case (2) σ swaps the pairs vs+i, vs−i, i ∈ Zn.
In either case, σ is a bijection on the vertices and it preserves the outer edges and the
spokes. Thus the image of an inner edge must again be an inner edge. Assume first that σ is
of type (1), and consider an arbitrary inner edge {vmi+j , vmi+j+kj}. Its image under σ is
the pair {v2s−mi−j+1, v2s−mi−j−kj+1}, which is an edge of GDGPm(n;K) if and only
if 2s−mi−j−kj+1 ≡ 2s−mi−j+1+kj′ (mod n), i.e.,−kj ≡ kj′ (mod n), where
j′ ≡ 2s−mi−j+1 ≡ 2s−j+1 (mod m), or, 2s−mi−j+1 ≡ 2s−mi−j−kj+1+kj′′
(mod n), i.e., kj ≡ kj′′ (mod n), where j′′ ≡ 2s −mi − j − kj + 1 ≡ 2s − j − a + 1
(mod m). Therefore, σ is a graph automorphism ofGDGPm(n;K) if and only if (at least)
one the above equalities holds for each i ∈ Z n
m
, j ∈ Zm. In the special case when m = 2,
a is necessarily 1, j′′ ≡ 2s+ j + 1 + 1 ≡ j (mod 2), and hence:
Lemma 5.3. For every GDGP2(n;K), and for every s ∈ Zn, the reflection σs swapping
the pairs us−i and us+1+i, and the pairs vs−i and vs+1+i, for all i ∈ Zn, is a graph
automorphism of GDGP2(n;K). Consequently, Aut(GDGP2(n;K)) acts transitively
on the two sets of outer and inner vertices of GDGP2(n;K).
Proof. The graph automorphism α : V (GDGP2(n;K))→ V (GDGP2(n;K)) defined in
Theorem 4.4 and sending ui 7→ ui+2 and vi 7→ vi+2, for all i ∈ Z2, has two orbits on the
outer and two orbits on the inner vertices. The reflection automorphisms σs mix these two
orbits.
Both of our examples, GDGP2(8; 1, 3) and GDGP2(8; 1, 5), can be easily seen to be
symmetric with respect to reflections about axes passing through the centers of a pair of
opposing outer edges.
Next, let us consider σ of type (2). The image of an arbitrary edge {vmi+j , vmi+j+kj}
is the pair {v2s−mi−j , v2s−mi−j−kj}, which is an edge if and only if 2s−mi− j − kj ≡
2s −mi − j + kj′ (mod n), i.e., −kj ≡ kj′ (mod n), for j′ ≡ 2s −mi − j ≡ 2s − j
(mod m), or 2s−mi− j ≡ 2s−mi− j − kj + kj′′ (mod n), i.e., kj ≡ kj′′ (mod n),
for j′′ ≡ 2s − mi − j − kj ≡ 2s − j − a (mod m). Comparing this result to that of
Lemma 5.3, assuming m = 2 would require −kj ≡ kj′ (mod n), for j′ ≡ j (mod 2),
or kj ≡ kj′′ (mod n), for j′′ ≡ j + a (mod 2). The first is impossible as that would
require k0 = k1 = n2 , which would violate our agreement that we do not consider periodic
sequences, while the second possibility is explicitely prohibited in the definition of the
GDGP-graphs. Hence, no GDGP2(n;K) that is not a generalised Petersen graph admits
automorphisms of type (2). Specifically, it is easy to see that neither GDGP2(8; 1, 3) nor
GDGP2(8; 1, 5) admits such automorphisms. There are, on the other hand, infinitely many
graphs that do admit at least one such automorphism. The proof of the next lemma follows
from the above discussion.
Lemma 5.4. Let n ≥ 3, and let K have the property that −kj ≡ kj′ (mod n), for
j′ ≡ 2s− j (mod m), or let K have the property kj ≡ kj′′ (mod n), for j′′ ≡ 2s− j−a
K. Jasenčáková et al.: A new generalisation of generalised Petersen graphs 11
Figure 12: GDGP3(12; 1, 4, 1).
1
ab
c
0 0
0
0 0
Z4
Figure 13: The voltage graph for GDGP3(12; 1, 4, 1), a = 0, b = 0, c = 2.
(mod m). Then GDGPm(n,K) admits an automorphism σ that fixes us and vs, and
swaps the pairs us+i and us−i, vs+i and vs−i, i ∈ Zn.
Example 5.5. The graphsGDGP3(n; a, 1, n−1) as well as the graphsGDGP3(n; 1, a, 1),
a ∈ Zn, all admit an automorphism fixing the vertex u0. In particular, the graph
GDGP3(12; 1, 4, 1) pictured in Figure 12 is symmetric with respect to the axes passing
through the vertices u0 and u6, and through the vertices u3 and u9.
All the automorphisms considered so far preserve the outer and inner rim as well as the
spokes. We conclude this section by considering the graphs we started the section with,
namely with the graphs GDGP2(n; 1, n − 3). As stated at the beginning of the section,
all of them are vertex-transitive and admit an automorphism mapping an outer vertex to an
inner vertex. Computational evidence collected in [12] suggests that the order of the full
automorphism group of an GDGP2(n; 1, n − 3) is n · 2
n
2 , and it is easy to see that these
graphs are neither edge- nor arc-transitive. In what follows, we present automorphisms that
do not preserve the set of spokes.
Lemma 5.6. Let n ≥ 6. Then GDGP2(n; 1, n − 3) admits at least one automorphism
which does not fix its set of spokes.
Proof. The desired automorphism δ ∈ Aut(GDGP2(n; 1, n − 3)) consists of just two
2-cycles: δ = (u1v0)(u2v3). Since δ moves only four vertices, to show that it is indeed
12 Art Discrete Appl. Math. 3 (2020) #P1.04
1
ab
c
0 0
0
0 0
Zm
x
ab
c
y z
0
0 0
Zm
Figure 14: The voltage graphs forGDGP3(m; k1, k2, k3) and SI3(m; l1, l2, l3, k1, k2, k3).
a graph automorphism, it suffices to show that it maps edges incident with the vertices
u1, v0, u2, v3 to edges incident to the vertices u1, v0, u2, v3. This easy exercise is left to the
reader. It is also easy to see that (ui+1vi)(ui+2vi+3) belongs to Aut(GDGP2(n; 1, n−3))
for all positive integers i divisible by 4 and smaller than n− 3.
Lemma 5.3 together with Lemma 5.6 yield that the graphs GDGP2(n; 1, n − 3) are
indeed vertex-transitive for all n ≥ 6.
6 Vertex-transitive and Cayley GDGP2-graphs
We continue searching for vertex-transitive graphs. Lemma 5.3 asserts that
Aut(GDGP2(n; k0, k1)) acts transitively on the set of the outer and the set of the in-
ner vertices of every GDGP2(n; k0, k1). Thus, an GDGP2(n; k0, k1) graph is vertex-
transitive if and only if it admits a graph automorphism mapping at least one outer vertex
to an inner vertex. In this section, we present some sufficient conditions for this to happen.
Note that the graphs GDGP2(n; k0, k1), GDGP2(n; k1, k0), GDGP2(n;−k0,−k1) and
GDGP2(n;−k1,−k0) are all isomorphic.
Since we only seek sufficient conditions, we will focus on the special case of graphs that
admit automorphisms preserving the set of spokes and swapping the entire sets of outer and
inner vertices. Obviously, these must be those GDGP -graphs in which the graphs induced
by the inner vertices form a single cycle. As observed already in the discussion following
the definition of the GDGP -graphs, this is the case if and only if the order of the element
k0 + k1 in Zn is equal to n2 . Thus, we shall assume from now on that on(k0 + k1) =
n
2 .
Suppose γ ∈ Aut(GDGP2(n;K)) swaps the outer and the inner rim and preserves the
spokes. Because of Lemma 5.3, we may assume that γ maps u0 to v0. The outer rim can
be mapped onto the inner rim in either the clockwise or in the counterclockwise direction.
Hence, there might be two automorphisms which swap the outer and inner cycles and map
u0 to v0.
Let γ be an automorphism which maps the outer cycle to the inner cycle in the same
direction. Thus,
γ(u2i) = vi(k0+k1), γ(u2i+1) = vi(k0+k1)+k0 , (6.1)
for all i ∈ Zn
2
. Since γ is assumed to preserve the set of spokes, the image of a spoke must
K. Jasenčáková et al.: A new generalisation of generalised Petersen graphs 13
be a spoke again and thus it must be the case that
γ(v2i) = ui(k0+k1), γ(v2i+1) = ui(k0+k1)+k0 , (6.2)
for all i ∈ Zn
2
. On the other hand, γ must map the inner cycle to the outer cycle, and, in
particular, γ must map inner edges to outer edges. For any i ∈ {0, 1, . . . , n − 1}, because
of (6.2), the vertex v2i, which is adjacent to vertices v2i+k0 and v2i−k1 , is mapped to the
vertex ui(k0+k1), adjacent to the vertices ui(k0+k1)+1 and ui(k0+k1)−1. Thus, γ maps the
2-set {v2i+k0 , v2i−k1} onto the 2-set {ui(k0+k1)+1, ui(k0+k1)−1}. However, invoking (6.2)
again,
γ(v2i+k0) = u(i+ k0−12 )(k0+k1)+k0
, while γ(v2i−k1) = u(i− k1+12 )(k0+k1)+k0
.
This gives us the congruences:
(i+
k0 − 1
2
)(k0 + k1) + k1 ≡ i(k0 + k1)± 1 (mod n),
(i− k1 + 1
2
)(k0 + k1) + k1 ≡ i(k0 + k1)∓ 1 (mod n),
which are equivalent to the system of congruencies:
k0−1
2 (k0 + k1) + k1 ≡ ±1 (mod n),
−k1+12 (k0 + k1) + k1 ≡ ∓1 (mod n).
(6.3)
Moreover, applying the same ideas to the vertices v2i+1 yields conditions equivalent to the
conditions (6.3).
Similarly, one can define an automorphism γ̄ which maps the outer rim onto the inner
rim in the opposite direction. In this case:
γ̄(u2i) = v−i(k0+k1), γ̄(v2i) = u−i(k0+k1),
γ̄(u2i+1) = v−i(k0+k1)−k1 , γ̄(v2i+1) = u−i(k0+k1)−k1 ,
(6.4)
for all i ∈ Zn
2
. Inner edges are preserved if and only if the system of congruencies
1−k0
2 (k0 + k1) − k1 ≡ ±1 (mod n),
k1+1
2 (k0 + k1) − k1 ≡ ∓1 (mod n)
(6.5)
is satisfied.
Note that k0, k1 that satisfy the system (6.3) also necessarily satisfy the congruence
(k0 + k1)
2 ≡ ±4 (mod n), (6.6)
and parameters k0, k1 that satisfy (6.5) also satisfy
(k0 + k1)
2 ≡ ∓4 (mod n). (6.7)
Recall that a necessary and sufficient condition for a vertex-transitivity of the generalised
Petersen graphs GP (n, k) is k2 ≡ ±1 (mod n) (except for GP (10, 2))[9], which implies
the congruence (k + k)2 ≡ ±4 (mod n). In this sense, the conditions (6.6) and (6.7) are
generalisations of the well-known characterization of vertex-transitive generalised Petersen
graphs.
We have proved the following:
14 Art Discrete Appl. Math. 3 (2020) #P1.04
Figure 15: Graph GDGP2(16; 3, 11) which admits the automorphism γ.
Lemma 6.1. The graphGDGP2(n; k0, k1) admits an automorphism that preserves the set
of spokes and swaps the outer and inner vertices if and only if the order on(k0 + k1) = n2
in Zn and the parameters n, k0, k1 satisfy one of the systems of congruencies (6.3) or (6.5).
Proof. We have proved in detail that at least one of the conditions is necessary. Their suf-
ficiency follows from the fact that the vertex map of GDGP2(n; k0, k1) whose parameters
satisfy (6.3) and which is defined by equations (6.1) and (6.2) fixes the spokes and swaps
the outer and inner edges. The same holds true for the vertex map whose parameters satisfy
(6.5) and which is defined via (6.4).
It is easy to observe that the parameters of a fixed GDGP2(n; k0, k1) can satisfy at
most one of the systems (6.3) or (6.5). Therefore, graphs GDGP2(n; k0, k1) admit at most
one of the automorphisms γ or γ̄.
The following theorem provides the sufficient condition promised at the beginning of
the section. Its proof follows from Lemma 5.3 and Lemma 6.1.
Theorem 6.2. Let GDGP2(n; k0, k1) be a graph whose parameters satisfy on(k0 +k1) =
n
2 and one of the systems (6.3) or (6.5). Then GDGP2(n; k0, k1) is a vertex-transitive
graph.
Example 6.3. The parameters of the graphs GDGP2(n; a, n−a+ 2) satisfy the condition
on(a+ n− a+ 2) = on(2) = n2 as well as the system of congruencies (6.3). Hence, they
all admit the automorphism γ defined by formulas (6.1) and (6.2).
Example 6.4. The parameters of the graphs GDGP2(n; 1, n − 3) satisfy neither of the
systems (6.3) or (6.5). They are nevertheless vertex-transitive and their inner edges form a
single cycle. Thus, conditions (6.3) or (6.5) are sufficient but not necessary.
Example 6.5. The thesis [11] contains yet another family of graphs whose parameters
do not satisfy (6.3) or (6.5), but nevertheless includes vertex-transitive graphs. These are
the graphs GDGP2(8a + 4; 1, 4a − 1), with a being a positive integer. The inner rim
of these graphs does not form a single cycle. The smallest graph in this family is the
graph GDGP2(12; 1, 3) isomorphic to the truncated octahedral graph. It is known that
the truncated octahedral graph is the Cayley graph Cay(G,X), where G = S4 and X =
{(1234), (1432), (12)}, and the group of automorphisms has order 48. Another member of
the family is the graph GDGP2(20; 1, 7).
K. Jasenčáková et al.: A new generalisation of generalised Petersen graphs 15
Figure 16: Graphs GDGP2(12; 1, 3) and GDGP2(20; 1, 7).
Example 6.6. Based on computational evidence, there are other families of vertex-transitive
GDGP2-graphs with parameters that do not satisfy (6.3) or (6.5). In the thesis [12], relying
on exhaustive search of all GDGP2(n; k0, k1) with n ≤ 300, six more graphs have been
found whose parameters do not satisfy (6.3) or (6.5) but are vertex-transitive. These are the
graphs
GDGP2(96, 21, 49), GDGP2(96, 27, 47), GDGP2(192, 45, 97),
GDGP2(192, 51, 95), GDGP2(288, 69, 145), GDGP2(288, 75, 143).
Since all their orders are multiples of 96, their existence suggests a possible infinite family.
Once again referring to the characterization of vertex-transitive generalised Petersen
graphs, we observe GP (n, k) is a Cayley graph if and only if k2 ≡ 1 (mod n), and
thus vertex-transitive generalised Petersen graphs whose parameters satisfy the congruence
relation k2 ≡ −1 (mod n) are not Cayley [9]. We show that this is not the case for the
graphs considered in this section. Namely, we show that all graphs GDGP2(n; k0, k1)
whose parameters satisfy (6.3) or (6.5) are Cayley graphs. We will somewhat abbreviate
our arguments. Detailed proofs of the claims made in this part can be found in [11].
Let GDGP2(n; k0, k1) be a graph admitting one of the automorphisms γ or γ̄, α be
the automorphism from Theorem 4.4, and β be the automorphism of GDGP2(n; k0, k1)
that maps ui to u1−i and vi to v1−i, 0 ≤ i ≤ n − 1. The groups GΣ = 〈α, β, γ〉 and
ḠΣ = 〈α, β, γ̄〉 are subgroups of Aut(GDGP2(n; k0, k1)) that preserve the set of spokes.
It is easy to verify that if (k0 + k1)2 ≡ 4 (mod n), then
GΣ = 〈α, β, γ |α
n
2 = β2 = γ2 = 1, βαβ = α−1, γαγ = α
k0+k1
2 , βγ = γβα
k0−1
2 〉,
and
ḠΣ = 〈α, β, γ̄ |α
n
2 = β2 = γ̄2 = 1, βαβ = α−1, γ̄αγ̄ = α−
k0+k1
2 , βγ̄ = γ̄βα−
k1+1
2 〉.
On the other hand, if (k0+k1)2 ≡ −4 (mod n), thenGDGP2(n; k0, k1) is isomorphic
to the generalised Petersen graph GP (n, k0).
Theorem 6.7. The following statements are true for all GDGP2(n; k0, k1):
16 Art Discrete Appl. Math. 3 (2020) #P1.04
1. If γ is an automorphism of the graph GDGP2(n; k0, k1), then GDGP2(n; k0, k1) is
isomorphic to the Cayley graph Cay(GΣ, {β, αβ, γ}).
2. If γ̄ is an automorphism of the graph GDGP2(n; k0, k1), then GDGP2(n; k0, k1) is
isomorphic to the Cayley graph Cay(ḠΣ, {β, αβ, γ̄}).
Proof. Leaving out the technical details, we claim that the map
ϕ : GDGP2(n; k0, k1) −→ Cay(GΣ, {β, αβ, γ}),
defined on the vertices of GDGP2(n; k0, k1) via the formulas
u2i 7→ αi, u2i+1 7→ βαi,
v2i 7→ γαi, v2i+1 7→ γβαi,
is an isomorphism between GDGP2(n;K) and Cay(GΣ, {β, αβ, γ}).
Similarly, the map
ϕ̄ : GDGP2(n; k0, k1) −→ Cay(ḠΣ, {β, αβ, γ̄}),
defined via
u2i 7→ αi, u2i+1 7→ βαi,
v2i 7→ γ̄αi, v2i+1 7→ γ̄βαi,
is an isomorphism between the graphs GDGP2(n; k0, k1) and Cay(ḠΣ, {β, αβ, γ̄}).
7 Symmetric GDGP2-graphs
One of the main goals of our paper is to determine which of the GDGP -graphs are highly
symmetric. In the previous section, we have presented sufficient conditions for GDGP2-
graphs being vertex-transitive. For the rest of our paper, we are going to consider even a
higher level of symmetry, namely, we are going to address the question which GDGP2-
graphs are symmetric (arc-transitive), i.e., whichGDGP2-graphs possess enough automor-
phisms to map any arc of the graph to any other arc.
We have already established in Theorem 4.4 that all GDGP2(n; k0, k1) are tetracircu-
lants. Since all cubic symmetric tetracirculants have been classified by Frelih and Kutnar
in [8], in order to classify the symmetric GDGP2-graphs (which are cubic tetracirculants),
it is enough to determine which of the cubic symmetric tetracirculants listed in [8] are
GDGP2-graphs. Since the symmetric graphs in [8] are described in the form of the lifts,
we shall achieve this goal by viewing the GDGP2-graphs as lifts as well.
Recall that generalised Petersen graphs are the lifts of the dumbbell graphs from Fig-
ure 3. Note also that the dumbbell graph may be viewed as mono-gonal prism (which
makes the generalised Petersen graphs bicirculant). Since the GDGP2-graphs are tetracir-
culants, in order to view them as lifts, we need to consider base graphs of order 4. Consider
the voltage graph in Figure 17 which is a di-gonal prism. In both Figures 3 and 17, the
voltages along one basis cycle add up to 1. Since the voltages on the other edges must be
integers, both k0 and k1 in Figure 17 must be odd. If we recall that the definition of the
GDGP2(n; k0, k1)-graphs also requires that the parameters k0 and k1 be odd, it is not hard
to see that every GDGP2(n; k0, k1) is isomorphic to the lift described in Figure 17.
Furthermore, the voltage graph in Figure 17 is a rather special case of the more general
voltage graph of Figure 7. As is well-known, the voltages along any spanning tree of the
K. Jasenčáková et al.: A new generalisation of generalised Petersen graphs 17
0 0
k0−1
2
k1+1
2
0
1
Zm
2
Figure 17: Voltage graph for GDGP2(m; k0, k1).
0 0
a
b
c
d
Zm
Figure 18: Voltage graph for the tetracirculant graph C1(m; a, b, c, d).
base graph may be chosen to be equal to 0, and still, after appropriate changes of the other
voltages, produce the same graph. Hence, c may always be chosen to be 0. Such graphs
have been studied before, for instance in [1], under the name of C-graphs. However, the
spanning tree used in [1] differs from our choice here. Nevertheless, each GDGP2 graph
is a C-graph, while the converse is not true.
Let us now continue with the task of classifying symmetric GDGP2-graphs. As men-
tioned above, the paper [8] contains a complete classification of cubic symmetric (i.e. arc-
transitive) tetracirculants.
Theorem 7.1 ([8, Theorem 1.1]). A connected cubic symmetric graph is a tetracirculant if
and only if it is isomorphic to one of the following graphs:
(i) F008A,F020A,F020B,F024A,F028A,F032A,F040A,
(ii) F016A,F048A,F056C,F060A,F080A,F096A,F112B,F120B,F224C,
F240C,
(iii) CQ(t,m) for 2 ≤ t ≤ m− 3 satisfying m|(t2 + t+ 1),
18 Art Discrete Appl. Math. 3 (2020) #P1.04
Figure 19: The Dyck graph GDGP2(16; 3, 7).
(iv) CQ(2t− 1, 2m) for 2 ≤ t ≤ m− 1 satisfying m|(4t2 − 2t+ 1).
The notation FnA,FnB, etc., refers to the corresponding graphs in the Foster census
[2], [3]. The graphs CQ(t,m) are the lifts from Figures 8 and 9 introduced in [7].
Since all GDGP2-graphs are bipartite (Lemma 4.5), we can easily rule out the tetracir-
culants that are not bipartite, such as GP (5, 2) or GP (10, 2). Graphs F008A, F016A,
F020B, F024A, and F048A are the generalised Petersen graphs GP (4, 1), GP (8, 3),
GP (10, 3), GP (12, 5), and GP (24, 5), respectively. Therefore, these graphs are not iso-
morphic to a non-periodic GDGP2(n; k0, k1). Furthermore, we checked all the sporadic
cases in (i) and (ii) by our program in SAGE-math. That showed that the two graphs F040A
and F080A are also not isomorphic to any non-periodic GDGP2(n; k0, k1).
Finally, the graph F032A is isomorphic to GDGP2(16; 3, 7). It is also known under
the name of the Dyck graph; Figure 19.
Summing up the above observations yields the following.
Theorem 7.2. The only symmetric GDGP2(n; k0, k1) graph not isomorphic to a gen-
eralised Petersen graph GP (n, k) or one of the graphs CQ(t,m), 2 ≤ t ≤ m − 3,
m|(t2 + t + 1) or CQ(2t − 1, 2m), 2 ≤ t ≤ m − 1, m|(4t2 − 2t + 1), is the Dyck
graph GDGP2(16; 3, 7) = F032A.
Note that our computer program indicates that the only arc-transitive C-graph that is
not a GDGP2 graph is the graph F040A. It can alternately be described as an SI2-graph,
i.e., a further generalisation in which one allows for spans other than 1 in both rims; see
Figure 20.
Our computer experiments also indicate the following:
Conjecture 7.3.
1. The girth of CQ(t,m), m odd, gcd(t,m) = 1, is equal to 6. For m even, the girth
may be 6,8 or 10.
2. EveryCQ(t,m)-graph is aGDGP2 graph. Every graphCQ(t,m) with gcd(t,m) =
1 is vertex transitive.
We close our paper with two open questions:
K. Jasenčáková et al.: A new generalisation of generalised Petersen graphs 19
Figure 20: F040A as SI2(10; 2, 2, 1, 11).
1. Which of the graphs CQ(t,m), 2 ≤ t ≤ m−3,m|(t2 +t+1), and CQ(2t−1, 2m),
2 ≤ t ≤ m− 1, m|(4t2 − 2t+ 1), are isomorphic to a GDGP2(n; k0, k1)?
2. Which of the graphs GDGPm(n;K), m > 2, are symmetric?
ORCID iDs
Katarı́na Jasenčáková https://orcid.org/0000-0001-7615-0038
Robert Jajcay https://orcid.org/0000-0002-2166-2092
Tomaž Pisanski https://orcid.org/0000-0002-1257-5376
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.05
https://doi.org/10.26493/2590-9770.1290.f73
(Also available at http://adam-journal.eu)
Regular balanced Cayley maps on nonabelian
metacyclic groups of odd order*
Kai Yuan , Yan Wang†
School of Mathematics and Information Science, Yan Tai University, Yan Tai, P.R.C.
Haipeng Qu
School of Mathematics and Computer Science, Shan Xi Normal University,
Shan Xi, P.R.C.
Received 30 January 2019, accepted 2 September 2019, published online 27 July 2020
Abstract
In this paper, we show that nonabelian metacyclic groups of odd order do not have
regular balanced Cayley maps.
Keywords: Regular balanced Cayley map, metacyclic group.
Math. Subj. Class. (2020): 05C25, 05C30
1 Introduction
Groups are often studied in terms of their action on the elements of a set or on particular
objects within a structure. The aim of this article is to study metacyclic groups of odd
order acting on maps. A Cayley graph Γ = Cay(G,X) will be a graph based on a
group G and a generating set X = {x1, x2, . . . , xk} which does not contain 1G, is closed
under the operation of taking inverses. In this paper, we call X a Cayley subset of G.
The vertices of the Cayley graph Γ are the elements of G, and two vertices g and h are
joined by an edge if and only if g = hxi for some xi ∈ X . The ordered pairs (h, hx) for
h ∈ G and x ∈ X are called the darts of Γ. For a cyclic permutation ρ of the set X , the
Cayley mapM = CM(G,X, ρ) is the 2-cell embedding of the Cayley graph Cay(G,X)
*The authors want to thank the reviewer and editors for their patience on revising our paper and giving us
valuable suggestions.
†Corresponding author. Supported by NSFC (No. 11671276, 11671347, 61771019, 11771258), NSFS (No.
ZR2017MA022) and J16LI02.
E-mail address: pktide@163.com (Kai Yuan), wang−yan@pku.org.cn (Yan Wang), orcawhale@163.com
(Haipeng Qu)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 3 (2020) #P1.05
in some orientable surface such that the local rotation of darts emanating from every vertex
is induced by the same cyclic permutation ρ of X .
An (orientation preserving) automorphism of a Cayley mapM is a permutation of the
set of darts ofM which preserves the incidence relation of the vertices, edges, faces, and
the orientation of the map. The full automorphism group ofM, denoted by Aut(M), is
the group of all such automorphisms ofM under the operation of composition. This group
always acts semi-regularly on the set of darts of M, that is, the stabilizer in Aut(M)
of each dart of M is trivial. If the action of Aut(M) on the darts of M is transitive
(and therefore regular), we say that the Cayley map M is a regular Cayley map. As the
left regular multiplication action of the underlying group G lifts naturally into the full
automorphism group of any Cayley map CM(G,X, ρ), Cayley maps are a very good source
of regular maps. There are many papers on the topic of regular Cayley maps, we refer
the readers to [2], [5] and [6] and the references therein. Furthermore, A Cayley map
CM(G,X, ρ) is called balanced if ρ(x)−1 = ρ(x−1) for every x ∈ X . In [6], the authors
showed that a Cayley map CM(G,X, ρ) is regular and balanced if and only if there exists a
group automorphism σ such that σ|X = ρ, where σ|X denotes the restricted action of σ on
X . Therefore, determining all the regular balanced Cayley maps of a group is equivalent
to determining all the orbits of its automorphisms that can be Cayley subsets.
In [2], it was shown that all odd order abelian groups possess at least one regular bal-
anced Cayley map [2]. Wang and Feng [7] classified all regular balanced Cayley maps
for cyclic, dihedral and generalized quaternion groups. In [4], the author proved the non-
existence of regular balanced Cayley maps with semi-dihedral groups. In [8], Yuan, Wang
and Qu proved that a nonabelian metacyclic p-group for an odd prime number p does not
have regular balanced Cayley maps. This was the first work on regular balanced Cayley
map of nonabelian groups of odd order. We will take a step further and show that non-
abelian metacyclic groups of odd order do not have regular balanced Cayley maps.
2 Preliminaries
We use the standard notation for group theory; see [3]. We denote by (r, s) the greatest
common divisor of two positive integers r and s. By |x|, |H|, we denote the order of
element x and subgroup H of a group G, respectively. By N : H , we denote a semidirect
product of the group N by the group H . Set [x, y] = x−1y−1xy, the commutator of x and
y and set [H,K] = 〈[h, k]
∣∣ h ∈ H, k ∈ K〉, where H,K ≤ G. For α ∈ Aut(G) and
g ∈ G, denote the orbit of g under 〈α〉 by g〈α〉.
Define G1 = G, and, proceeding recursively, define Gn = [Gn−1, G] for 1 < n ∈ Z.
Then G is said to be nilpotent if Gn = 1 for some 1 ≤ n ∈ Z. The following lemma is
basic for nilpotent groups.
Lemma 2.1 ([3, Kapitel III.2.3]). A finite group is nilpotent if and only if it is the direct
product of its Sylow groups.
Lemma 2.2 ([3, Kapitel III.7.2]). Let G be a p-group, N E G and |N | = p. Then N ≤
Z(G).
Define Mp,q(m, r) = 〈a, b | ap = 1, bq
m
= 1, b−1ab = ar〉, where p and q are distinct
prime numbers, m is a positive integer and r 6≡ 1 (mod p) but rq ≡ 1 (mod p). The
following lemma is about the automorphism group of Mp,q(m, r).
Kai Yuan, Yan Wang and Haipeng Qu: Regular balanced Cayley maps 3
Lemma 2.3 ([8, Lemma 2.2]). The automorphism group of Mp,q(m, r) is
Aut(Mp,q(m, r)) = {σ | aσ = ai, bσ = bjak, 1 6 i 6 p−1, 1 6 j 6 qm−1, q | (j−1) }.
The following lemma shows that G/N has a regular balanced Cayley map whenever G
has.
Lemma 2.4 ([8, Lemma 2.5]). LetG be a finite group andN be a nontrivial characteristic
subgroup of G. Take α ∈ Aut(G) and g ∈ G. If X = g〈α〉 is a Cayley subset of G, then
X = g〈α〉 = ḡ〈ᾱ〉 is a Cayley subset of G = G/N .
Proposition 2.5 ([8, Corollary 4.7]). For any odd prime number p, the nonabelian meta-
cyclic p-group does not have regular balanced Cayley maps.
3 Regular balanced Cayley maps on nonabelian metacyclic group of
odd order
A metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group
G having a cyclic normal subgroup 〈a〉 such that the quotientG/〈a〉 is also cyclic. We now
prove our main theorem.
Theorem 3.1. Nonabelian metacyclic groups of odd order do not have regular balanced
Cayley maps.
Proof. Let G be a counterexample of minimal order. Then G is a metacyclic group of odd
order which has regular balanced Cayley maps. Set G = 〈a〉〈b〉 where 〈a〉 E G. We will
get a contradiction through the following seven steps.
Step 1. The order of the derived subgroup G′ of G is a prime number p:
Clearly, the derived subgroup G′ is a subgroup of 〈a〉. Thus G′ is cyclic. If the order of
G′ is not a prime number, then there is a proper subgroup N of G′ which has prime order.
SinceN is characteristic inG′ andG′ is characteristic inG, we find thatN is characteristic
in G. Now consider the quotient group G/N . On one hand, by Lemma 2.4, G/N has a
regular balanced Cayley map. On the other hand, from the choice ofG,G/N does not have
regular balanced Cayley maps; a contradiction. Thus |G′| is a prime number p.
Step 2. G does not have a nontrivial normal p′-subgroup. In particular, Z(G) = 1 or
Z(G) is a p-group:
If not, let N be a nontrivial normal p′-subgroup of G. We consider G/N . Since |G′| =
p, we know that G/N is still nonabelian. By the minimality of G, it follows that G/N
does not have regular balanced Cayley maps. However, by Lemma 2.4, G/N has a regular
balanced Cayley map. This is a contradiction.
Step 3. The subgroup 〈a〉 is a p-group:
Otherwise, suppose 〈a〉 = P ×Q, where P is a p-group andQ is a non-trivial p′-group.
Since Q is a characteristic subgroup of 〈a〉, it is normal in G. This contradicts Step 2.
Step 4. G′  Z(G):
Otherwise, assume G′ ≤ Z(G). Then G is nilpotent. By Lemma 2.1, we have G =
P1 × Q1, where P1 is a p-group and Q1 is a p′-group. By Step 2, Q1 = 1. Thus G is a
p-group. This contradicts Proposition 2.5.
Now, we assume 〈b〉 = P2×Q2, where P2 is a p-group andQ2 is a non-trivial p′-group.
We will show P2 = 1, so that 〈b〉 is a p′-group.
4 Art Discrete Appl. Math. 3 (2020) #P1.05
Step 5. The order of 〈a〉 is p:
Set |a| = pn and Q2 = 〈c〉. By Step 1, [a, c] = aip
n−1
. Hence ac = a1+ip
n−1
. Notice
that (1 + ipn−1)p ≡ 1 mod pn when n > 1. If n > 1, then acp = a. Hence [cp, a] = 1.
Since (|c|, p) = 1, we find that c ∈ 〈cp〉. Thus [a, c] = 1. This gives Q2 ≤ Z(G),
contradicting Step 2. Thus n = 1.
Step 6. Z(G) = 1 and P2 = 1:
Since G is nonabelian, 〈a〉  Z(G). By Step 5, 〈a〉 ∩Z(G) = 1. Thus G/Z(G) is still
nonabelian. By the choice of G, we find that Z(G) = 1.
Since 〈a〉 is a normal subgroup of 〈a〉P2 and the order of 〈a〉 is p, from Lemma 2.2, we
have 〈a〉 ≤ Z(〈a〉P2). It follows that P2 ≤ Z(G). So P2 = 1.
Step 7. G does not have regular balanced Cayley maps:
We can assume G = 〈a〉 : 〈b〉, where 〈a〉 ∼= Zp and 〈b〉 is a p′-group. Since Z(G) = 1,
we have C〈b〉(a) = 1. By the Normalizer-Centralizer (N/C) Theorem, N〈b〉(a)/C〈b〉(a) .
Aut(〈a〉), and so 〈b〉 . Aut(〈a〉). Take a prime factor q of |b|. If G has a regular balanced
Cayley map, then we may take a corresponding Cayley subset X of G. Without loss of
generality, we assume b ∈ X . Then there is some σ ∈ Aut(G) such that bσ = b−1. Let
b1 = b
|b|/q . Then bσ1 = b
−1
1 . Set K = 〈a〉 : 〈b1〉 and let σ1 be the restricted action of σ
on K. Then σ1 ∈ Aut(K). However, from Lemma 2.3, the metacyclic group K of order
pq does not have any automorphism that can reverse b1, a contradiction. Hence nonabelian
metacyclic groups of odd order do not have regular balanced Cayley maps.
From Lemma 2.4 and Theorem 3.1, we get the following:
Corollary 3.2. Let H be a characteristic subgroup of G. If the quotient group G/H is
isomorphic to a nonabelian metacyclic group of odd order, then G does not admit regular
balanced Cayley maps. In particular, a direct product of a nonabelian metacyclic group of
odd order and a 2-group does not admit regular balanced Cayley maps.
Remark 3.1. Since the only nonabelian group with odd order less than 26 is metacyclic,
we know from Corollary 3.2 that the minimal odd order of a nonabelian group that admits a
regular balanced Cayley map is at least 27. The only non-metacyclic and nonabelian group
of order 27 is M3(1, 1, 1) = 〈a, b, c
∣∣ a3 = b3 = c3 = 1, [a, b] = c, [c, a] = [c, b] = 1〉.
With the help of MAGMA [1], we can easily see that M3(1, 1, 1) has regular balanced
Cayley maps, and the corresponding Cayley graphs have valency 4, 6 and 8, respectively.
ORCID iDs
Kai Yuan https://orcid.org/0000-0003-1858-3083
Yan Wang https://orcid.org/0000-0002-0148-2932
Haipeng Qu https://orcid.org/0000-0002-3858-5767
References
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senschaften, Springer-Verlag, Berlin, 1967, doi:10.1007/978-3-642-64981-3.
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[4] J.-M. Oh, Regular t-balanced Cayley maps on semi-dihedral groups, J. Comb. Theory Ser. B 99
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1043.9d5.
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.06
https://doi.org/10.26493/2590-9770.1275.e77
(Also available at http://adam-journal.eu)
Automorphism groups of maps,
hypermaps and dessins
Gareth Aneurin Jones*
School of Mathematical Sciences, University of Southampton,
Southampton SO17 1BJ, UK
Received 5 November 2018, accepted 2 September 2019, published online 5 August 2020
Abstract
A detailed proof is given of a theorem describing the centraliser of a transitive permu-
tation group, with applications to automorphism groups of objects in various categories of
maps, hypermaps, dessins, polytopes and covering spaces, where the automorphism group
of any object is the centraliser of its monodromy group. An alternative form of the theo-
rem, valid for finite objects, is discussed, with counterexamples based on Baumslag–Solitar
groups to show how it can fail in the infinite case. The automorphism groups of objects
with primitive monodromy groups are described, as are those of non-connected objects.
Keywords: Permutation group, centraliser, automorphism group, map, hypermap, dessin d’enfant.
Math. Subj. Class. (2020): 05C10, 14H57, 20B25, 20B27, 52B15, 57M10
1 Introduction
In certain categories C, such as those consisting of maps or hypermaps, oriented or un-
oriented, or of dessins d’enfants (regarded as finite oriented hypermaps), each object O
can be identified with a permutation representation θ : Γ → S := Sym(Ω) of a ‘parent
group’ Γ = ΓC on some set Ω, and the morphisms O1 → O2 can be identified with the
functions Ω1 → Ω2 which commute with the actions of Γ on the corresponding sets Ω1
and Ω2. These are the ‘permutational categories’ defined and discussed in [9] (see also
§2). The automorphism group AutC(O) of an object O within C is then identified with
the centraliser C := CS(G) in S of the monodromy group G := θ(Γ) of O. Now O is
connected if and only ifG is transitive on Ω, as we will assume unless otherwise stated (see
§6). In this situation, an important result is the following, where N denotes the normaliser
of a subgroup:
*The author is grateful to Ernesto Girondo, Gabino González-Diez and Rubén Hidalgo for discussions about
dessins d’enfants which motivated this work.
E-mail address: g.a.jones@maths.soton.ac.uk (Gareth Aneurin Jones)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 3 (2020) #P1.06
Theorem 1.1. If O is connected then
AutC(O) ∼= NG(H)/H ∼= NΓ(M)/M,
where H and M are the stabilisers in G and Γ of some α ∈ Ω, and NG(H) and NΓ(M)
are their normalisers.
For instance, this theorem has recently been used in [10] to show that in various cate-
gories of maps and hypermaps, every countable group can be realised as the automorphism
group of uncountably many non-isomorphic objects, infinitely many of which can be cho-
sen to be finite if the group is. This is an analogue of the well-known theorems of Frucht [5]
and Sabidussi [20] for graphs and their automorphism groups.
There are analogues of Theorem 1.1 in other contexts, ranging from abstract polytopes
to covering spaces. Proofs of Theorem 1.1 for particular categories can be found in the
literature: for instance, in [11] it is deduced for oriented maps from a more general re-
sult about morphisms in that category; in [12, Theorem 2.2 and Corollary 2.1] a proof
for dessins is briefly outlined; analogous results for covering spaces are proved in [15,
Appendix] and [17, Theorem 81.2], and for abstract polytopes in [16, Propositions 2D8
and 2E23(a)]. In §3 we give a detailed proof of the following ‘folklore’ theorem about
permutation groups (see also Theorem 3.2 in the recent [19]), which immediately implies
Theorem 1.1 for all permutational categories, including the special cases listed above.
Theorem 1.2. Let G be a transitive permutation group on a set Ω, with H the stabiliser
of some α ∈ Ω, and let C := CS(G) be the centraliser of G in the symmetric group
S := Sym(Ω). Then
1. C ∼= NG(H)/H ,
2. C acts regularly on the set Φ of elements of Ω with stabiliser H .
One sometimes finds proofs or statements of particular cases of Theorem 1.2 which
include the assertion that C acts regularly on the set Φ of fixed points of H in Ω; while
this is valid if H is finite, in §4 we give counter-examples, based on Baumslag–Solitar
groups [1], to show that if H is infinite then Φ must be redefined more precisely as in (2).
It follows from Theorem 1.1 that if the monodromy groupG of an objectO acts primitively
on Ω, then either AutC(O) is trivial, or G is a cyclic group of prime order, acting regularly
on Ω; in §5 we describe the objects with the latter property in various categories C. In §6 we
briefly consider the structure and cardinality of the automorphism groups of non-connected
objects in permutational categories, and in §7 we extend Theorem 1.1 to cover morphisms
between connected objects.
2 Permutational categories
A permutational category C is defined in [9] to be a category in which the objects O can
be identified with the permutation representations θ : Γ → S := Sym(Ω) of a parent
group Γ = ΓC, and the morphisms O1 → O2 with the Γ-invariant functions Ω1 → Ω2,
those commuting with the actions of Γ on Ω1 and Ω2. The automorphism group Aut(O) =
AutC(O) ofO in the category C is then the group of all permutations of Ω commuting with
Γ, that is, the centraliser CS(G) of the monodromy group G = θ(Γ) ofO in the symmetric
group S. Here we will restrict our attention to the connected objects, those for which Γ
acts transitively on Ω.
G. A. Jones: Automorphism groups of maps, hypermaps and dessins 3
We will concentrate mainly on five particular examples of permutational categories,
outlined briefly below (for full details, and other examples, see [9]). In each case, the
parent group Γ is either an extended triangle group
∆[p, q, r] = 〈R0, R1, R2 | R2i = (R1R2)p = (R2R0)q = (R0R1)r = 1〉,
or its orientation-preserving subgroup of index 2, the triangle group
∆(p, q, r) = 〈X,Y, Z | Xp = Y q = Zr = XY Z = 1〉,
where X = R1R2, Y = R2R0 and Z = R0R1. Here p, q, r ∈ N ∪ {∞}, and we ignore
any relations of the form W∞ = 1. In what follows, ∗ denotes a free product, Cn denotes
a cyclic group of order n ∈ N ∪ {∞}, while Vn is an elementary abelian group of order n,
and Fr is a free group of rank r.
1. The category M of maps on surfaces (possibly non-orientable or with boundary), that
is, embeddings of graphs with simply connected faces, has parent group
Γ = ΓM = ∆[∞, 2,∞] ∼= V4 ∗ C2.
This permutes the set Ω of incident vertex-edge-face flags of a map (equivalently, the
faces of its barycentric subdivision), with each involution Ri (i = 0, 1, 2) changing the
i-dimensional component of each flag (whenever possible) while preserving the other two.
2. The category M+ of oriented maps, those in which the underlying surface is oriented
and without boundary, has parent group
Γ = ΓM+ = ∆(∞, 2,∞) ∼= C∞ ∗ C2.
This group permutes directed edges, with X usng the local orientation to rotate them about
their target vertices, and Y reversing their direction, so that Z rotates them around incident
faces.
3. There are several ways of defining or representing hypermaps. For our purposes, the
most convenient is the Walsh representation as a bipartite map [21], in which the black and
white vertices of the embedded graph correspond to the hypervertices and hyperedges of the
hypermap, the edges correspond to incidences between them, and the faces correspond to
its hyperfaces. The category H of all hypermaps, where the underlying surface is unoriented
and possibly with boundary, has parent group
Γ = ΓH = ∆[∞,∞,∞] ∼= C2 ∗ C2 ∗ C2.
This permutes incident edge-face pairs of the bipartite map, with involutions R0 and R1
preserving the face and the incident white and black vertex respectively, whileR2 preserves
the edge.
4. The category H+ of oriented hypermaps, those in which the underlying surface is ori-
ented and without boundary, has parent group
Γ = ΓH+ = ∆(∞,∞,∞) ∼= C∞ ∗ C∞ ∼= F2.
This permutes the edges of the embedded graph, with X and Y using the local orientation
to rotate them around their incident black and white vertices, so that Z rotates them around
incident faces.
4 Art Discrete Appl. Math. 3 (2020) #P1.06
5. The category D of dessins d’enfants can be identified with the subcategory of H+ con-
sisting of its finite objects, those in which the embedded bipartite graph is finite and the
surface is compact. It has the same parent group Γ = ∆(∞,∞,∞) ∼= F2 as H+, permut-
ing edges as before.
If we wish to restrict any of these categories to the subcategory of objects of a particular
type (p, q, r), we replace the parent group given above with the corresponding triangle or
extended triangle group of that type.
Mn Hn
Figure 1: The mapMn and the hypermapHn
Example 2.1. The planar mapMn ∈M+ (n ≥ 2), shown on the left in Figure 1, has one
vertex, of valency n, incident with one loop and n − 2 half edges. It corresponds to the
epimorphism θ : ∆(∞, 2,∞)→ Sn given by
X 7→ x = (1, 2, . . . , n), Y 7→ y = (1, 2), Z 7→ z = (n, n− 1, . . . , 2).
This map can be regarded as a hypermap Hn, shown on the right in Figure 1, by adding a
white vertex to each edge; this corresponds to composing θ with the natural epimorphism
∆(∞,∞,∞) → ∆(∞, 2,∞). The hypermap Hn has type (n, 2, n − 1), and it can be
regarded as a member of the category of oriented hypermaps of this type by factoring θ
through ∆(n, 2, n − 1). In all cases the monodromy group G is Sn, in its natural repre-
sentation, and the automorphism group is trivial. However, if we regardMn or Hn as an
unoriented map or hypermap, then its monodromy group is Sn × S2 in its natural product
action of degree 2n, and the automorphism group has order 2, generated by the obvious
reflection.
We briefly mention two other classes of permutational categories in which Theorem 1.1
applies. The first concerns abstract polytopes [16], regarded as higher-dimensional gener-
alisations of maps. Those n-polytopes of a particular type, associated with the Schläfli
symbol {p1, . . . , pn−1}, can be identified with transitive permutation representations of
the Coxeter group Γ with presentation
〈R0, . . . , Rn | R2i = (Ri−1Ri)pi = (RiRj)2 = 1 (|i− j| > 1)〉.
For instance, ΓM is associated with the Schläfli symbol {∞,∞}. However, in higher
dimensions, not all transitive representations correspond to abstract polytopes, since the
monodromy groups must satisfy the intersection property [16, Proposition 2B10].
G. A. Jones: Automorphism groups of maps, hypermaps and dessins 5
The second class of examples concerns covering spaces [15, 17]. Under suitable con-
nectedness assumptions, the (connected, unbranched) coverings Y → X of a topological
space X can be identified with the transitive permutation representations θ : Γ → S =
Sym(Ω) of its fundamental group Γ = π1X , acting by unique path-lifting on the fibre Ω
over a base-point in X . In this case the automorphism group of an object Y → X in this
category is the group of covering transformations, the centraliser in S of the monodromy
group θ(Γ) of the covering.
This last example helps to explain the importance of the fifth category listed above, the
category D of dessins d’enfants. By Belyi’s Theorem [2], a compact Riemann surface R is
defined (as a projective algebraic curve) over the field Q of algebraic numbers if and only
if it admits a non-constant meromorphic function β branched over at most three points of
the complex projective line (the Riemann sphere) P1(C) = Ĉ = C ∪ {∞}. Composing
β with a Möbius transformation if necessary, we may assume that its critical values are
contained in {0, 1,∞}. Such Belyi functions β correspond to unbranched finite coverings
R \ β−1({0, 1,∞}) → X of the thrice-punctured sphere X = Ĉ \ {0, 1,∞}, and hence
to transitive finite permutation representations of its fundamental group Γ = π1X; this is a
free group of rank 2, freely generated by the homotopy classes of small loops around 0 and
1. The unit interval [0, 1] ⊂ Ĉ lifts to a bipartite graph embedded inR, with black and white
vertices over 0 and 1, and face-centres over∞. Conversely, any finite oriented hypermap,
after suitable uniformisation, yields a Riemann surface R defined over Q; see [6, 7, 12, 13]
for details of these connections, and of the action of the absolute Galois group Gal(Q/Q)
on dessins.
3 Proof of Theorems 1.1 and 1.2
Let O be a connected object in a permutational category C, identified with a transitive
permutation representation Γ → G ≤ S := Sym(Ω), so that its automorphism group
AutC(O) is identified with the centraliser CS(G) ofG in S. Then Theorem 1.1 asserts that
AutC(O) ∼= NG(H)/H ∼= NΓ(M)/M,
where G is the monodromy group of O, and H and M are point-stabilisers in G and Γ.
The second isomorphism follows immediately from the first, and this in turn follows from
part (1) of Theorem 1.2, both parts of which we will now prove.
Proof. The centraliser C of G acts semi-regularly (i.e. freely) on Ω. To see this, suppose
that c ∈ C and βc = β for some β ∈ Ω. Given any ω ∈ Ω, there is some g ∈ G such that
ω = βg, since G is transitive on Ω. Then ωc = (βg)c = (βc)g = βg = ω. Thus c = 1, as
required.
Let Φ = {β ∈ Ω | Gβ = H}, so in particular α ∈ Φ. Then C leaves Φ invariant, since
if β ∈ Φ and c ∈ C then for all h ∈ H we have (βc)h = (βh)c = βc, so that βc ∈ Φ.
Let us identify Ω with the set of cosets of H in G in the usual way, identifying each
ω ∈ Ω with the unique coset Hx such that x ∈ G and αx = ω. Thus α is identified with
H itself, and G acts on the cosets by g : Hx 7→ Hxg.
Then Φ is identified with the set of cosets of H in N := NG(H). To see this, let
ω ∈ Ω correspond to a coset Hx of H in G. First suppose that x = n ∈ N . Then
(Hn)h = (nH)h = n(Hh) = nH = Hn for all h ∈ H , so the coset Hn is fixed by
H , giving H ≤ Gω , while if g ∈ Gω then Hng = Hn, so H = Hn = Hng = Hg and
6 Art Discrete Appl. Math. 3 (2020) #P1.06
hence g ∈ H , giving Gω ≤ H . Thus Gω = H and hence ω ∈ Φ. Conversely, suppose that
ω ∈ Φ. Then Gω = H , so Hxg = Hx if and only if g ∈ H , that is, Hxg = Hx if and
only if g ∈ H , so Hx = H and hence x ∈ N .
Let us define a new action of N on Ω (now regarded as the set of cosets of H in G) by
n : Hx 7→ n−1Hx = Hn−1x for all n ∈ N and x ∈ G. If n1, n2 ∈ N then n1, followed
by n2, sends Hx to n−12 n
−1
1 Hx = (n1n2)
−1Hx, as does n1n2, so this is indeed a group
action of N . It commutes with the action of G on Ω, since n−1(Hxg) = (n−1Hx)g for all
n ∈ N and x, g ∈ G, so it defines a homomorphism θ : N → C. In particular, this action
preserves Φ, since C does.
The induced action of N on Φ is transitive, since if n′, n′′ ∈ N then the element
n = n′(n′′)−1 ∈ N sends Hn′ to n−1Hn′ = Hn−1n′ = Hn′′. Thus θ(N) acts on Φ as
a transitive subgroup of C. But C acts semi-regularly on Φ, so it has no transitive proper
subgroups. Therefore θ is an epimorphism, and C acts regularly on Φ, giving (2).
In this action of N , we have n−1Hx = Hx if and only if n ∈ H , so the subgroup
stabilising any cosetHx isH , which is therefore the kernel ker(θ) of this action ofN . The
First Isomorphism Theorem therefore gives N/H ∼= C, so (1) is proved.
Remark 3.1. The most symmetric objects in C are the regular objects, those for which
AutC(O) acts transitively on Ω. By Theorem 1.2(2) this is equivalent to Φ = Ω, that is, to
H = 1, meaning that G acts regularly on Ω. This is also equivalent to M being a normal
subgroup of Γ. Then G ∼= C ∼= Γ/M , and G and C can be identified with the right and left
regular representations of the same group. (In fact, if G is abelian then C = G.)
Remark 3.2. A variety of groups is a class of groups defined by identical relations between
their elements (see [18]); simple examples include groups of exponent dividing n, solvable
groups of derived length at most l, and nilpotent groups of class at most c. The fact that the
centraliser C of a transitive permutation group G can be realised within G as NG(H)/H
means that if G is a member of a variety V, then so is C. This may seem surprising
since, as subgroups of the symmetric group S, the groups C and G could have a very small
intersection: for instance, C ∩ G = 1 if, as in many cases, G has a trivial centre. Of
course, this fact applies to permutational categories: if the parent group Γ is in V then
the automorphism group of each connected object is also in V. However, for most of the
examples we are interested in, Γ generates the variety of all groups, and the only restrictions
on automorphism groups are the obvious ones imposed by cardinality, as shown in [10].
4 An alternative form of Theorem 1.2(2)
One sometimes finds part (2) of Theorem 1.2 stated in the following alternative form:
(2′) C acts regularly on the set of fixed points of H in Ω.
This is equivalent to (2) in cases where Ω is finite, or more generally where H is finite,
so that an inclusion H = Hα ≤ Hβ between conjugate subgroups is equivalent to their
equality. However, (2′) can be false if H is infinite, as shown by the following example.
Example 4.1. Although our aim here is mainly group-theoretic, to construct a permutation
group G on a set Ω such that condition (2′) fails, the original motivation, as in much of this
paper, is combinatorial. It may therefore be useful, throughout this construction, to think
of G as the monodromy group of an oriented hypermap H, regarded as a bipartite map,
with elements of Ω as edges, and cycles of the generators a and b of G as black and white
vertices (see Remark 4.2).
G. A. Jones: Automorphism groups of maps, hypermaps and dessins 7
Let G be the Baumslag–Solitar group [1]
G = BS(1, 2) = 〈a, b | ab = a2〉.
This is a semidirect product G = A o B, where B = 〈b〉 ∼= C∞, and A is the normal
closure of a in G, an abelian group of countably infinite rank, generated by the conjugates
ai := a
2i = ab
i
(i ∈ Z)
of a, with a2i = ai+1 for all i ∈ Z. This subgroup A can be identified with the additive
group of the ring Z[ 12 ], with each finite product
∏
i a
ei
i in A corresponding to
∑
i ei2
i ∈
Z[ 12 ]. Thus a = a0 corresponds to the element 1 ∈ Z[
1
2 ], and b, acting by conjugation on
A as the automorphism ai 7→ ai+1, acts on Z[ 12 ] by t 7→ 2t. In particular, the subgroup
H := 〈a〉 has conjugate subgroups Hi := Hb
−i
= 〈a2−i〉 for all i ∈ Z, with a chain of
index 2 inclusions
· · · < H−2 < H−1 < H0 (= H) < H1 < H2 < · · · .
Now let T be the Sylow 2-subgroup of Q/Z, that is,
T := Z[ 12 ]/Z ∼= C2∞ :=
⋃
e≥0
C2e
where C2e corresponds to the subgroup generated by the image of 2−e in Z[ 12 ]/Z. Let
Ω = T× Z =
⋃
i∈Z
Ωi,
the disjoint union of countably many copies Ωi = T×{i} of T. Let the generators a and b
of G act on Ω by
a : (t, i) 7→ (t+ 2i, i) and b : (t, i) 7→ (t, i− 1)
for t ∈ T and i ∈ Z, where we interpret t+ 2i as an element of T in the obvious way. Thus
a preserves each set Ωi, fixing it pointwise for i ≥ 0, and with cycles of length 2−i on it
for i < 0, while b induces the obvious bijection Ωi → Ωi−1. Then the element ab = b−1ab
acts on Ω by a composition of three permutations
ab : (t, i) 7→ (t, i+ 1) 7→ (t+ 2i+1, i+ 1) 7→ (t+ 2i+1, i).
This has the same effect as
a2 : (t, i) 7→ (t+ 2i+1, i),
so we have a group action of G on Ω.
It is easy to see from the decomposition G = A o B that G acts faithfully and tran-
sitively on Ω, that the stabiliser of the element α = (0, 0) ∈ Ω is H = 〈a〉, and that
NG(H) = A, so that
NG(H)/H = A/H ∼= T.
We now calculate
C := CS(G) = CS(A) ∩ CS(B),
8 Art Discrete Appl. Math. 3 (2020) #P1.06
where S := Sym(Ω). First note that CS(A) must permute the orbits of A, which are the
sets Ωi, and must do so trivially since A has a different representation, with kernel Hi =
〈a2−i〉, on each Ωi. Since A induces the regular representation of the abelian group A/Hi
on Ωi, it follows thatCS(A) must be the cartesian product
∏
i∈ZA/Hi of the groupsA/Hi,
each factor A/Hi acting regularly on Ωi and fixing Ω \Ωi. Even though the subgroups Hi
are all distinct, we have A/Hi ∼= T for all i ∈ Z, so CS(A) ∼= TZ. The only elements of
CS(A) commuting with b are those corresponding to elements of the diagonal subgroup of
TZ, inducing the same permutation on each subset Ωi. These form a group isomorphic to
T, proving that
C ∼= NG(H)/H.
Thus G satisfies Theorem 1.2(1), but what about statements (2) and (2′)? In this exam-
ple, the orbits of C are the sets Ωi, each permuted regularly by C, while the subset of Ω
fixed by H is the disjoint union ⋃
i≥0
Ωi
of infinitely many of these orbits. Thus G does not satisfy statement (2′). However, it does
satisfy (2) since the points with stabiliser H are those in Ω0, forming a regular orbit of C;
the points in Ωi for i > 0, although they are also fixed by H , have stabiliser Hi properly
containing H .
Remark 4.2. This example also gives a construction of the oriented hypermap H with
monodromy group G, corresponding to the epimorphism θ : F2 → G, X 7→ a, Y 7→ b.
We can identify the edges with the elements (t, i) of Ω = T × Z = ∪iΩi. The white
vertices, corresponding to the cycles of b, can be identified with the elements t of T, each
incident with the edges (t, i) for i ∈ Z in decreasing order of i as one follows the local
orientation. The black vertices correspond to the cycles of a; those in Ωi for i ≥ 0 are
fixed points (t, i) of a, so they have valency 1, each of them connected by the edge (t, i)
to the white vertex t; those in Ωi for i < 0, also denoted by (t, i), correspond to cycles
{(t + k2i, i) | k = 0, 1, . . . , 2−i − 1} of length 2−i, so they have valency 2−i and are
connected by edges (t + k2i, i) to the white vertices t + k2i in cyclic order of increasing
k. The two faces incident with an edge (t, i) have vertices, following the orientation and in
alternate colours, given by
. . . , (t+ 2i, i− 1), t+ 2i, (t, i), t, (t, i+ 1), t− 2i+1, . . .
and
. . . , t+ 2i−1, (t, i− 1), t, (t, i), t− 2i, (t− 2i, i+ 1), . . . .
The group AutH+(H) can be identified with T: each t0 ∈ T acts on edges of H by
(t, i) 7→ (t+ t0, i), with the obvious induced actions on black and white vertices and faces.
Thus it acts transitively on white vertices and on faces, whereas its orbits on black vertices
and on edges consist of those in each subset Ωi for i ∈ Z. In particular, it acts regularly on
edges (and on black vertices) in Ω0, those with stabiliser H in G, but it is intransitive on
those with i ≥ 0, fixed by H .
It is not easy to visualise this hypermap H. An alternative approach is to regard it
as a regular branched covering, with covering group T, of the quotient hypermap H =
H/AutH+(H) corresponding to the epimorphism F2 → Gab = G/A ∼= Z. As a bipartite
map this is planar, with one face and one white vertex incident with 1-valent black vertices
G. A. Jones: Automorphism groups of maps, hypermaps and dessins 9
v0
v−1
v1
v−2
v2
v−3
v3
Figure 2: The hypermapH
vi in decreasing order of i ∈ Z (see Figure 2). The coveringH → H is branched only at the
vertices vi for i < 0, where the local monodromy permutation has infinitely many cycles
of length 2−i, corresponding to the cycles of b on Ωi. The automorphism group of H can
be identified with the group of covering transformations, acting regularly on the sheets of
the covering.
Remark 4.3. There is an obvious generalisation of this example based on the Baumslag–
Solitar group G = BS(1, q) = 〈a, b | ab = aq〉 for an arbitrary integer q 6= 0,±1. See [8]
for a discussion of the oriented hypermaps associated with the Baumslag–Solitar groups
BS(p, q) = 〈a, b | (ap)b = aq〉 for arbitrary p, q 6= 0.
5 Primitive monodromy groups
If G is a permutation group on a set Ω, then the relation Gα = Gβ , appearing in Theo-
rem 1.2 via the definition of Φ, is a G-invariant equivalence relation on Ω, and its equiv-
alence classes are the orbits of the centraliser C of G. Recall that a permutation group is
primitive if it preserves no non-trivial equivalence relation; equivalently, the point-stabilisers
are maximal subgroups. As an immediate consequence of Theorem 1.2, we hav
Corollary 5.1. IfG is a primitive permutation group, then eitherG ∼= Cp, acting regularly,
for some prime p, with centraliser C = G, or the centraliser C of G is the trivial group.
Proof. The equivalence relation Gα = Gβ on Ω must be either the identity or the universal
relation. In the first case the equivalence classes are singletons, so |C| = 1. In the second
case Gα = {1}; this is a maximal subgroup of G, so G ∼= Cp for some prime p, with
C = G.
Corollary 5.2. In a permutational category C, if the monodromy group G of an object O
is a primitive permutation group, then either O is regular, with AutC(O) = G ∼= Cp for
some prime p, or AutC(O) is the trivial group.
Of course, there are many examples of primitive permutation groups, either sporadic or
members of infinite families: just represent a group on the cosets of a maximal subgroup.
On the other hand, Cameron, Neumann and Teague [3] have shown that for a set of integers
10 Art Discrete Appl. Math. 3 (2020) #P1.06
n ∈ N of asymptotic density 1 the only primitive groups of degree n are the alternating and
symmetric groups An and Sn.
Example 5.3. The symmetric and alternating groups, in their natural representationas, arise
quite frequently as monodromy groups in various categories. For instance, let C = H+, the
category of oriented hypermaps, with parent group Γ = F2 = 〈X,Y | −〉. A theorem
of Dixon [4] states that a randomly chosen pair of permutations x, y ∈ Sn generate Sn
or An with probability approaching 3/4 or 1/4 as n → ∞, so in that sense ‘most’ finite
objects in this category have a symmetric or alternating monodromy group, and a trivial
automorphism group.
In most of the permutational categories of current interest, it is simple to describe the
regular objects with automorphism groupCp for each prime p; apart from these exceptions,
objects with a primitive monodromy group have a trivial automorphism group. The excep-
tions correspond to the normal subgroups of index p in the parent group Γ, or equivalently
to the maximal subgroups in the elementary abelian p-group Γ/Γ′Γp, where Γ′ and ΓP are
the subgroups of Γ generated by the commutators and p-th powers. In the categories listed
in §2, they are as follows.
If C = H+ or D then Γ = F2, so Γ/Γ′Γp ∼= Cp × Cp, with p+ 1 maximal subgroups.
Of the corresponding oriented hypermaps, three have type a permutation of (p, p, 1) and
are planar, while the remaining p− 2 have type (p, p, p) and genus (p− 1)/2. As dessins,
the former are on the Riemann sphere, with Belyi functions β : z 7→ zp, 1/(1 − zp) and
1 − z−p, and automorphisms z 7→ ζz where ζp = 1. The latter are on Lefschetz curves
yp = xu(x − 1) for u = 1, . . . , p − 2, each with a Belyi function β : (x, y) 7→ x and
automorphisms (x, y) 7→ (x, ζy) where ζp = 1 (see [12, Example 5.6]). The four dessins
for p = 3 are shown in Figure 3; in the dessin on the right, opposite sides of the hexagon
are identified to form a torus.
Figure 3: The four dessins with primitive monodromy group Cp, p = 3
If C = M+ then Γ = C∞ ∗ C2, so Γ/Γ′Γp ∼= V4 or Cp as p = 2 or p > 2, giving
three oriented maps or one, all planar. Their types are the three permutations of (2, 2, 1),
together with (p, 1, p). They are shown, for p = 2 and 3, in Figure 4.
Figure 4: The four oriented maps with primitive monodromy group Cp, p = 2, 3
G. A. Jones: Automorphism groups of maps, hypermaps and dessins 11
Figure 5: The seven hypermaps with primitive monodromy group C2
If C = H or M then Γ = C2 ∗ C2 ∗ C2 or V4 ∗ C2, so in either case Γ/Γ′Γp ∼= V8 or 1
as p = 2 or p > 2. If p = 2 there are seven hypermaps and seven maps; if p > 2 there are
none. The hypermaps are shown in Figure 5. The hypermap on the left is planar, while the
other six are on the closed disc, shown by a broken line. The seven maps can be obtained
from these hypermaps by ignoring all the white vertices.
If X is a compact orientable surface of genus g then there are (p2g−1)/(p−1) regular
coverings Y → X with monodromy group Cp, corresponding to the normal subgroups of
index p in π1X = 〈Ai, Bi (i = 1, . . . , g) |
∏
i[Ai, Bi] = 1〉; the surfaces Y are orientable,
of genus 1 + p(g − 1). In the non-orientable case, with π1X = 〈Ri (i = 1, . . . , g) |∏
iR
2
i = 1〉 there are (pg−1 − 1)/(p− 1) or 2g − 1 such coverings as p > 2 or p = 2; the
surfaces Y are all non-orientable, of genus 2 + p(g − 2), apart from the orientable double
cover of X for p = 2, which has genus g − 1.
6 Automorphism groups of non-connected objects
It is often convenient to restrict attention to the connected objects in a category, as we have
done so far. Here we will briefly show how Theorem 1.1 extends to non-connected objects.
If C is a permutational category with parent group Γ, then the connected components
Oi (i ∈ I) of an object O in C correspond bijectively to the orbits Ωi (i ∈ I) of Γ on
the set Ω associated with O. As before, AutC(O) is isomorphic to the centraliser C of Γ
in S = Sym(Ω). In order to describe the structure of C in general, we first consider two
extreme cases.
Suppose first that the components Oi are mutually non-isomorphic. This is equivalent
to the point stabilisers Mi ≤ Γ for different orbits Ωi being mutually non-conjugate in Γ.
Then C is the cartesian product of the centralisers Ci ≤ Sym(Ωi) of Γ on the sets Ωi. By
the transitivity of Γ on Ωi, we have Ci ∼= NGi(Hi) ∼= NΓ(Mi)/Mi for each i ∈ I , where
Gi is the permutation group induced by Γ on Ωi, and Hi is a point stabiliser in Gi for this
action.
At the other extreme, suppose that the components Oi are all isomorphic, or equiva-
lently the point stabilisers Mi are conjugate to each other. Then C is the wreath product
Ci oSym(I) ofCi by Sym(I). This is a semidirect product, in which the normal subgroup is
the cartesian product of the mutually isomorphic groups Ci (∼= NGi(Hi) ∼= NΓ(Mi)/Mi)
for i ∈ I , and the complement is Sym(I), acting on this normal subgroup by permuting
the factors Ci via isomorphisms between them.
We can now describe the general form of C by combining these two constructions.
12 Art Discrete Appl. Math. 3 (2020) #P1.06
We first partition the set of components of O into maximal sets {Oij |i ∈ Ij} (j ∈ J) of
mutually isomorphic objects Oij , each subset indexed by a set Ij . We then define Cij (∼=
NGij (Hij)
∼= NΓ(Mij)/Mij with obvious notation) to be the centraliser of Γ in Sym(Ωij).
Then C, and hence also AutC(O), is the cartesian product over all j ∈ J of the wreath
products Cij o Sym(Ij) where i ∈ Ij . This is again a semidirect product, where the normal
subgroup is the cartesian product of all the groups Cij (i ∈ Ij , j ∈ J), and the complement
is the cartesian product of the groups Sym(Ij) (j ∈ J), each factor Sym(Ij) of the latter
acting on the normal subgroup by permuting the factors Cij for i ∈ Ij while fixing all other
factors.
This description can be used to determine the cardinality of AutC(O). We will re-
strict attention to categories where the parent group Γ is countable (for instance, where
it is finitely generated), since this condition is satisfied by most of the examples studied;
the modifications required for an uncountable parent group are straightforward. By Theo-
rem 1.1 this implies that AutC(O) is also countable for each connected object O. Since a
cartesian product of infinitely many non-trivial groups is uncountable, as is the symmetric
group on any infinite set, the following is clear:
Theorem 6.1. Let C be a permutational category for which the parent group Γ is countable,
and let O be an object in C with connected components Oij , indexed by sets Ij (j ∈ J) as
above. Then
1. |AutC(O)| > ℵ0 if and only if either O has infinitely many components Oij with
|AutC(Oij)| > 1, or at least one set Ij is infinite;
2. |AutC(O)| = ℵ0 if and only if O has only finitely many components Oij with
|AutC(Oij)| > 1, each set Ij is finite, and AutC(Oij) is infinite for some component
Oij;
3. |AutC(O)| < ℵ0 if and only if O has only finitely many components Oij with
|AutC(Oij)| > 1, each set Ij is finite, and AutC(Oij) is finite for each component
Oij .
Corollary 6.2. In Case (3), where AutC(O) is finite, it has order∏
j∈J
|AutC(Oij)||Ij ||Ij |! .
Example 6.3. Let C = M+, the category of oriented maps, which has parent group Γ =
∆(∞, 2,∞) = 〈X,Y | Y 2 = 1〉. For each integer n ≥ 2 let M̃n be the minimal regular
cover of the mapMn ∈M+ in Figure 1. This is a regular oriented map with automorphism
and monodromy group Sn (in its regular representation) corresponding to the epimorphism
Γ → Sn, X 7→ (1, 2, . . . , n), Y 7→ (1, 2). If we takeM to be the disjoint union of these
maps M̃n, then AutM+(M) is the cartesian product
∏
n≥2 Sn. This uncountable group is
very rich in subgroups: for instance, every finitely generated residually finite group (such
as every finitely generated linear group, by Mal’cev’s Theorem [14]) can be embedded in a
cartesian product of finite groups of distinct orders, and hence (by Cayley’s Theorem) can
be embedded in AutM+(M).
7 Morphisms
Although this paper has concentrated on automorphisms, Theorem 1.1 can be extended to
describe the morphisms O1 → O2 between connected objects. If each Oi corresponds to
G. A. Jones: Automorphism groups of maps, hypermaps and dessins 13
an action of the parent group Γ on Ωi, with stabiliser Mi, then such a morphism exists if
and only if M1 is conjugate to a subgroup of M2. The set B := {b ∈ Γ | M b1 ≤ M2}
is a union of cosets bM2 (b ∈ B), and these correspond bijectively to the morphisms
O1 → O2. Specifically, if we identify elements of Ωi with cosets Mig (g ∈ Γ), then each
b ∈ B corresponds to the morphism φb : M1g 7→ M2b−1g, where φb = φb′ if and only
if b′ ∈ bM2. The proof of this in [11, Theorem 3.5] for oriented maps extends easily to
all permutational categories. This shows that morphisms between connected objects are
always surjective, but Example 5 shows that endomorphisms O → O of infinite objects
need not be automorphisms. This also shows that the number of morphisms O1 → O2
is bounded above by |Ω2| = |Γ : M2|, attained if (and only if, when O2 is finite) M1 is
contained in the core of M2.
Example 7.1. If Oi is regular for i = 1 or 2, so that Mi is a normal subgroup of Γ, then
B = Γ or ∅ as M1 ≤ M2 or not, and there are respectively |Ω2| morphisms O1 → O2 or
none.
There is an action of AutC(O1) × AutC(O2) on the set MorC(O1,O2) of morphisms
O1 → O2, given by (θ1, θ2) : φ 7→ θ−11 ◦ φ ◦ θ2. To realise this action concretely, note that
the action (n1, n2) : b 7→ n−11 bn2 ofNΓ(M1)×NΓ(M2) onB yields an induced action on
the set of orbits bM2 ⊆ B of its normal subgroup 1×M2, with M1×M2 in the kernel; the
resulting action of (NΓ(M1)×NΓ(M2))/(M1×M2) ∼= NΓ(M1)/M1×NΓ(M2)/M2 on
these orbits is equivalent to the action of AutC(O1)×AutC(O2) on MorC(O1,O2). Thus
if θi is given by Mig 7→ n−1i Mig for i = 1, 2, then (θ1, θ2) : φb 7→ θ
−1
1 ◦ φb ◦ θ2 = φb′ ,
where b′ = n−11 bn2 ∈ B.
The subgroup of AutC(O1) × AutC(O2) fixing a morphism φ consists of those pairs
(θ1, θ2) such that θ1 ◦ φ = φ ◦ θ2, that is, φ lifts θ2 to θ1. In particular, the subgroup of
AutC(O1) fixing a morphism φ consists of the covering transformations of φ. Similarly,
the action of AutC(O2) on MorC(O1,O2) is equivalent to its action on Ω2: it is always
semi-regular, and regular if and only if O2 is regular.
ORCID iDs
Gareth Aneurin Jones https://orcid.org/0000-0002-7082-7025
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.07
https://doi.org/10.26493/2590-9770.1284.3ad
(Also available at http://adam-journal.eu)
Reflexible complete regular dessins and
antibalanced skew morphisms of cyclic groups
Kan Hu∗
Department of Mathematics, Zhejiang Ocean University,
Zhoushan, Zhejiang 316022, P.R. China
Young Soo Kwon†
Department of Mathematics, Yeungnam University,
Kyeongsan, 712-749, Republic of Korea
Received 27 December 2018, accepted 19 September 2019, published online 3 August 2020
Abstract
A skew morphism of a finite group A is a bijection ϕ on A fixing the identity element
of A and for which there exists an integer-valued function π on A such that ϕ(ab) =
ϕ(a)ϕπ(a)(b), for all a, b ∈ A. In addition, if ϕ−1(a) = ϕ(a−1)−1, for all a ∈ A,
then ϕ is called antibalanced. In this paper we develop a general theory of antibalanced
skew morphisms and establish a one-to-one correspondence between reciprocal pairs of
antibalanced skew morphisms of the cyclic additive groups and isomorphism classes of
reflexible regular dessins with complete bipartite underlying graphs. As an application,
reflexible complete regular dessins are classified.
Keywords: Graph embedding, antibalanced skew morphism, reciprocal pair.
Math. Subj. Class. (2020): 20B25, 05C10, 14H57
1 Introduction
A mapM is a 2-cell embedding i : Γ ↪→ S of a connected graph Γ, possibly with loops
or multiple edges, into a closed surface S such that each component of S \ i(Γ) is home-
omorphic to an open disc. A map is orientable if its supporting surface S is orientable,
∗Author was supported by Natural Science Foundation of Zhejiang Province (LY16A010010, LQ17A010003)
and National Natural Science Foundation of China (11801507).
†Author is supported by the Basic Science Research Program through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education (2018R1D1A1B05048450).
E-mail addresses: hukan@zjou.edu.cn (Kan Hu), ysookwon@ynu.ac.kr (Young Soo Kwon)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 3 (2020) #P1.07
otherwise it is non-orientable. Throughout the paper, maps considered are all orientable. A
map with a 2-coloured bipartite underlying graph is called a dessin. An automorphism of
a dessin D is a permutation of the edges of the underlying bipartite graph which preserves
the graph structure and vertex colouring, and extends to an orientation-preserving self-
homeomorphism of the supporting surface. It is well known that the automorphism group
of a dessin acts semi-regularly on its edges. In the case where this action is transitive, and
hence regular, the dessin is called regular as well.
A regular dessin is reflexible if it is isomorphic to its mirror image, otherwise it is called
chiral. Moreover, a regular dessin is symmetric if it has an external symmetry transposing
the vertex colors. Thus, a symmetric regular dessin may be viewed as a regular map, that
is, a map whose orientation-preserving automorphism group acts transitively on the arcs.
A regular dessin is complete if its underlying graph is the complete bipartite graph
Km,n. Due to its important connection to generalized Fermat curves, the classification
problem of complete regular dessins has attracted much attention. A full classification of
the symmetric complete regular dessins was obtained in a series of papers [8, 9, 17, 18,
19, 22]. For the general case, complete bipartite graphs which underly a unique regular
dessin were determined by Fan and Li [10], and complete regular dessins of odd prime
power order have been recently classified by Hu, Nedela and Wang [13]. These results
were proved by group-theoretic methods through a translation of complete regular dessins
to exact bicyclic groups with two distinguished generators.
Recently, Feng et al discovered an alternative approach to this problem by establishing
a surprising correspondence between complete regular dessins and reciprocal pairs of skew
morphisms of cyclic groups [12]. A skew morphism of a finite group A is a bijection ϕ on
A fixing the identity element of A and for which there exists an integer function π on A
such that ϕ(ab) = ϕ(a)ϕπ(a)(b), for all a, b ∈ A. Suppose that ϕ and ϕ̃ are a pair of skew
morphisms of the cyclic additive groups Zn and Zm, and π and π̃ are associated power
functions, respectively. The skew morphism pair (ϕ, ϕ̃) is called reciprocal if they satisfy
the following conditions:
(a) the order of ϕ divides m and the order of ϕ̃ divides n,
(b) π(x) ≡ ϕ̃x(1) (mod |ϕ|) and π̃(y) ≡ ϕy(1) (mod |ϕ̃|) for all x ∈ Zn and y ∈ Zm.
In [12, Theorem 5] the authors proved that the isomorphism classes of complete regular
dessins with underlying graphs Km,n are in one-to-one correspondence with the reciprocal
pairs of skew morphisms of the cyclic groups Zn and Zm.
The aim of this paper is to classify the reflexible complete regular dessins. Employ-
ing methods used in [12] we are led naturally to introduce a new concept of antibalanced
skew morphism. More precisely, a skew morphism of a finite group A is antibalanced
if ϕ−1(a) = ϕ(a−1)−1, for all a ∈ A. We note that there is a big difference between
antibalanced skew morphisms and skew morphisms arising from antibanlanced regular
Cayley maps, because the latter have a generating orbit which is closed under taking in-
verses [4, 16], while the former do not have such a restriction.
In Section 3 we develop a general theory of antibalanced skew morphisms, and present
a classification and enumeration of antibalanced skew morphisms of cyclic groups, extend-
ing the results obtained by Conder, Jajcay and Tucker [4, Theorem 7.1]. In Section 4, we
establish a one-to-one correspondence between reflexible complete regular dessins and re-
ciprocal pairs of antibalanced skew morphisms of cyclic groups. In Section 5 all reciprocal
pairs of antibalanced skew morphisms of cyclic groups are completely determined.
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 3
2 Preliminaries
The theory of skew morphisms has been developed and expanded by various authors. In
this section we summarize and prove some preliminary results for future reference.
Let ϕ be a skew morphism of a finite group A, and let π be a power function of ϕ. In
general, the function π is not uniquely determined by ϕ. However, if ϕ has order k, then π
may be regarded as a function π : A → Zk, which is unique. In this case we will refer to
π as the power function of ϕ. A subgroup N of A is ϕ-invariant if ϕ(N) = N , in which
case the restriction of ϕ to N is a skew morphism of N . Moreover, it is well known [16]
that Kerϕ and Fixϕ defined by
Kerϕ = {a ∈ A | π(a) = 1} and Fixϕ = {a ∈ A | ϕ(a) = a}
are subgroups of A, and in particular, Fixϕ is ϕ-invariant. Note that, for any two elements
a, b ∈ A, π(a) = π(b) if and only if ab−1 ∈ Kerϕ, so the power function π of ϕ takes
exactly |A : Kerϕ| distinct values in Zk. The index |A : Kerϕ| will be called the skew
type of ϕ. It follows that a skew morphism of A is an automorphism if and only if it has
skew type 1. A skew morphism which is not an automorphism will be called a proper skew
morphism.
Moreover, define
Coreϕ =
k⋂
i=1
ϕi(Kerϕ).
Then Coreϕ is a ϕ-invariant normal subgroup of A, and it it is the largest ϕ-invariant
subgroup of A contained in Kerϕ. In particular, if A is abelian, then Coreϕ = Kerϕ [4,
Lemma 5.1].
The following properties of skew morphisms are fundamental.
Proposition 2.1 ([16]). Let ϕ be a skew morphism of a finite group A, let π be the power
function of ϕ, and let k be the order of ϕ. Then the following hold:
(a) for any positive integer ` and for any a, b ∈ A, ϕ`(ab) = ϕ`(a)ϕσ(a,`)(b), where
σ(a, `) =
∑̀
i=1
π(ϕi−1(a));
(b) for all a, b ∈ A, π(ab) ≡
π(a)∑
i=1
π(ϕi−1(b)) (mod k).
Proposition 2.2 ([1]). Let ϕ be a skew morphism of a finite group A, let π be the power
function of ϕ, and let k be the order of ϕ. Then µ = ϕ` is a skew morphism of A if and
only if the congruence `x ≡ σ(a, `) (mod k) is soluble for every a ∈ A, in which case
πµ(a) is the solution
Proposition 2.3 ([15]). If ϕ is a skew morphism of a finite group A, then O−1a = Oa−1 for
any a ∈ A, where Oa and Oa−1 denote the orbits of ϕ containing a and a−1, respectively.
Proposition 2.4 ([1, 25]). Let ϕ be a skew morphism of a finite group A, and let π be the
power function of ϕ. Then for any automorphism θ of A, µ = θ−1ϕθ is a skew morphism
of A with power function πµ = πθ.
4 Art Discrete Appl. Math. 3 (2020) #P1.07
Proposition 2.5 ([26, 25]). Let ϕ be a skew morphism of a finite group A, and let π be
the power function of ϕ. If A = 〈a1, a2, . . . , ar〉, then |ϕ| = lcm(|Oa1 |, |Oa2 |, . . . , |Oar |).
Moreover, the skew morphism ϕ and its power function π are completely determined by the
action of ϕ and the values of π on the generating orbits Oa1 , Oa2 , . . . , Oar , respectively.
Proposition 2.6 ([26]). Let ϕ be a skew morphism of a finite group A, and let π be the
power function of ϕ. IfN is a ϕ-invariant normal subgroup ofA, then ϕ̄ defined by ϕ̄(x̄) =
ϕ(x) is a skew morphism of Ā := A/N and the power function π̄ of ϕ̄ is determined by
π̄(x̄) ≡ π(x) (mod |ϕ̄|).
Since Coreϕ is a ϕ-invariant normal subgroup of A, by Proposition 2.6, ϕ induces a
skew morphism ϕ̄ of Ā = A/Coreϕ. It is shown in [25] that the ϕ̄-invariant subgroup
Fix ϕ̄ of Ā lifts to a ϕ-invariant subgroup Smoothϕ of A, namely,
Smoothϕ = {a ∈ A | ϕ̄(ā) = ā}.
In particular, if Smoothϕ = A, then the skew morphism ϕ is called a smooth skew mor-
phism.
Proposition 2.7 ([25]). Let ϕ be a skew morphism of a finite group A, and let π be the
power function of ϕ. Then ϕ is smooth if and only if π(ϕ(a)) = π(a) for all a ∈ A.
The most important properties of smooth skew morphisms are summarized as follows.
Proposition 2.8 ([25]). Let ϕ be a smooth skew morphism of A, |ϕ| = k, and let π be the
power function of ϕ. Then the following hold:
(a) ϕ(Kerϕ) = Kerϕ;
(b) π : A → Zk is a group homomorphism of A into the multiplicative group Z∗k with
Kerπ = Kerϕ;
(c) for any ϕ-invariant normal subgroupN ofA, the induced skew morphism ϕ̄ onA/N
is also smooth, and in particular, if N = Kerϕ then ϕ̄ is the identity permutation;
(d) for any positive integer `, µ = ϕ` is a smooth skew morphism of A;
(e) for any automorphism θ of A, µ = θ−1ϕθ is a smooth skew morphism of A.
Lemma 2.9. Let ϕ be a skew morphism of a finite group A. Then ϕ is smooth if and only
if there exists a ϕ-invariant normal subgroup N of A contained in Kerϕ such that the
induced skew morphism ϕ̄ of Ā = A/N is the identity permutation.
Proof. If ϕ is smooth, then by Proposition 2.8(a), ϕ is kernel-preserving, and so Kerϕ =
Coreϕ. Take N = Kerϕ, then by Proposition 2.8(c) the induced skew morphism ϕ̄ of
A/N is the identity permutation.
Conversely, suppose that there exists a ϕ-invariant normal subgroup N of A contained
in Kerϕ such that the induced skew morphism ϕ̄ of Ā = A/N is the identity permutation.
Then, for each a ∈ A, there is an element u ∈ N ≤ Kerϕ such that ϕ(a) = ua. Thus,
π(ϕ(a)) = π(a), and therefore ϕ is smooth by Proposition 2.7.
There is a fundamental correspondence between skew morphisms and groups with
cyclic complements.
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 5
Proposition 2.10 ([5]). If G = AC is a factorisation of a finite group G into a product
of a subgroup A and a cyclic subgroup C = 〈c〉 with A ∩ C = 1, then c induces a skew
morphism ϕ of A via the commuting rule ca = ϕ(a)cπ(a), for all a ∈ A; in particular
|ϕ| = |C : CG|, where CG = ∩g∈GCg .
Conversely, if ϕ is a skew morphism of a finite groupA, thenG = LA〈ϕ〉 is a transitive
permutation group on A with LA ∩ 〈ϕ〉 = 1 and 〈ϕ〉 core-free in G, where LA is the left
regular representation of A.
3 Antibalanced skew morphisms
In this section we develop a theory of antibalanced skew morphisms and classify all an-
tibalanced skew morphisms of cyclic groups.
A skew morphism ϕ of a finite group A will be called antibalanced if
ϕ−1(a) = ϕ(a−1)−1, for all a ∈ A.
Since 1 = ϕ(aa−1) = ϕ(a)ϕπ(a)(a−1), we have ϕ(a)−1 = ϕπ(a)(a−1). Thus, ϕ
is antibalanced if and only if ϕπ(a)(a−1) = ϕ−1(a−1), or equivalently, π(a) ≡ −1
(mod |Oa−1 |), for all a ∈ A. By Proposition 2.3, |Oa| = |Oa−1 |. It follows that ϕ
is antiblanced if and only if π(a) ≡ −1 (mod |Oa|) for all a ∈ A. Note that for any
a ∈ Kerϕ, |Oa| is 1 or 2.
Remark 3.1. It was proved in [16, Theorem 1] that a Cayley map CM(A,X, p) is regular
(on the arcs) if and only if there is a skew morphism ϕ of A such that the restriction of ϕ
to X is equal to p. Since X is a generating set of A and is closed under taking inverses, the
associated skew morphism ϕ has a generating orbit which is closed under taking inverses.
For brevity, such a skew morphism will be called a Cayley skew morphism.
Moreover, a regular Cayley map CM(A,X, p) was termed antibalanced if p−1(x) =
p(x−1)−1 for all x ∈ X [24]. It follows that a regular Cayley map is antibalanced if and
only if the associated Cayley skew morphism is antibalanced. However, neither generat-
ing orbit, nor inverse-closed orbit are assumed in the preceding definition of antibalanced
skew morphisms. Therefore, antibalanced skew morphisms may be regarded as a natural
generalization of the skew morphisms arising from antibalanced regular Cayley maps.
We give an example to clarify the concept.
Example 3.2. The cyclic group Z12 has exactly eight skew morphisms, four of which are
proper:
ϕ = (0)(2)(4)(6)(8)(10)(1, 3, 5, 7, 9, 11), πϕ = [1][1][1][1][1][1][5, 5, 5, 5, 5, 5];
ψ = (0)(2)(4)(6)(8)(10)(1, 11, 9, 7, 5, 3), πψ = [1][1][1][1][1][1][5, 5, 5, 5, 5, 5];
µ = (0)(2)(4)(6)(8)(10)(1, 5, 9)(3, 7, 11), πµ = [1][1][1][1][1][1][2, 2, 2][2, 2, 2];
γ = (0)(2)(4)(6)(8)(10)(1, 9, 5)(3, 11, 7), πγ = [1][1][1][1][1][1][2, 2, 2][2, 2, 2].
It is easily seen that all the above skew morphisms are antibalanced. Note that the first two
skew morphisms contain a generating orbit closed under taking inverses, but the last two
skew morphisms do not contain such an orbit. Therefore, ϕ and ψ are antibalanced Cayley
skew morphism, and µ and γ are antibalanced non-Cayley skew morphisms.
We summarise some properties of antibalanced skew morphisms as follows.
6 Art Discrete Appl. Math. 3 (2020) #P1.07
Lemma 3.3. Let ϕ be an antibalanced skew morphism of a finite group A, and let π be the
associated power function. Then the following hold:
(a) for any positive integer `, ϕ−`(a) = ϕ`(a−1)−1 for all a ∈ A;
(b) for any automorphism θ of A, the skew morphism µ = θ−1ϕθ is antibalanced;
(c) for any ϕ-invariant normal subgroup N of A, the induced skew morphism ϕ̄ of A/N
is antibalanced;
(d) for any c ∈ Kerϕ and a ∈ A, π(a) ≡ 1 (mod |Oc|).
Proof. (a) The case ` = 1 is the definition. Assume the result for `, i.e. ϕ−`(a) =
ϕ`(a−1)−1 for all a ∈ A. Then
ϕ−(`+1)(a) = ϕ−1(ϕ−`(a)) = ϕ−1(ϕ`(a−1)−1) = ϕ(ϕ`(a−1))−1 = ϕ`+1(a−1)−1,
and the result follows by induction.
(b) For any a ∈ A, we have
µ−1(a) = θ−1ϕ−1θ(a) = θ−1(ϕ(θ(a)−1))−1) =
(
θ−1(ϕ(θ(a−1))
)−1
= µ(a−1)−1,
so µ is antibalanced.
(c) Since ϕ−1(a) = ϕ(a−1)−1, we have ϕ̄−1(ā) = ϕ̄(ā−1)−1, and so ϕ̄ is antibal-
anced.
(d) For any c ∈ Kerϕ and any a ∈ A, we have
ϕ(c)a−1 = ϕ(c)[ϕ−1(ϕ(a))]−1 = ϕ(c)ϕ(ϕ(a)−1) = ϕ(cϕ(a)−1)
= ϕ−1(ϕ(a)c−1)−1 =
(
ϕ−1(ϕ(a))ϕ−πϕ
−1(ϕ(a))(c−1)
)−1
= (aϕ−π(a)(c−1))−1 = (aϕπ(a)(c)−1)−1 = ϕπ(a)(c)a−1,
so ϕπ(a)(c) = ϕ(c), and hence π(a) ≡ 1 (mod |Oc|).
Lemma 3.4. Let ϕ be an automorphism of a finite group A. Then ϕ is antibalanced if and
only if ϕ2 = 1, that is, ϕ is an involutory automorphism.
Proof. By definition, ϕ is antibalanced if and only if for all a ∈ A, ϕ−1(a) = ϕ(a−1)−1.
Since ϕ is an automorphism, ϕ(a−1)−1 = ϕ(a), and hence ϕ is antibalanced if and only if
for all a ∈ A, ϕ−1(a) = ϕ(a), that is, ϕ2(a) = a.
Corollary 3.5. Every antibalanced automorphism of the cyclic additive group Zn is of the
form ϕ(x) = sx, x ∈ Zn, where s2 ≡ 1 (mod n).
Let ϕ be a skew morphism of a finite groupA. Suppose that ϕ has an orbitX generating
A. The words of even length in the generators from X form a subgroup of A, which will
be called the even word subgroup of A with respect to X and denoted by A+X . Note that
the index of A+X in A is 1 or 2.
The following results generalize the properties of antibalanced Cayley skew morphisms
(or more precisely, antibalanced regular Cayley maps) obtained in [4]
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 7
Lemma 3.6. Let ϕ be a skew morphism of a finite group A containing an orbit X which
generates A, and let π be the associated power function. Then ϕ is antibalanced if and
only if π(x) ≡ −1 (mod |X|) for all x ∈ X and ϕ restricted to A+X is an involutory
automorphism. Furthermore, if ϕ is antibalanced, then ϕ is a smooth skew morphism of
skew type 1 or 2.
Proof. Since X is a generating orbit of ϕ, we have |ϕ| = |X| by Proposition 2.5, and the
value of π onA is completely determined by the value of π onX . Suppose that π(x) ≡ −1
(mod |X|) for all x ∈ X . Then by Proposition 2.1, for any x, y ∈ X ,
π(xy) =
π(x)∑
i=1
π(ϕi−1(y)) ≡ π(x)π(y) ≡ 1 (mod |X|).
SinceA = 〈X〉, every element ofA is expressible as a word of finite length in the elements
ofX . By induction, π(a) ≡ 1 (mod |X|) if a is an even word, and π(a) ≡ −1 (mod |X|)
if a is an odd word. Note that if a is an even word (resp. an odd word), then both ϕ(a) and
a−1 are also even words (resp. odd words). Thus, ϕ is a smooth skew morphism of skew
type 1 or 2 and ϕ restricted to A+X is an automorphism of A
+
X .
If ϕ is antibalanced, then it is evident that π(x) ≡ −1 (mod |X|) for all x ∈ X .
This implies that ϕ restricted to A+X is an automorphism of A
+
X , and hence, for any a ∈
A+X , ϕ(a
−1) = ϕ(a)−1. Since ϕ is antibalanced, we have ϕ(a−1)−1 = ϕ−1(a), and
hence, for any a ∈ A+X , ϕ−1(a) = ϕ(a). Therefore, ϕ restricted to A
+
X is an involutory
automorphism.
Conversely, assume that π(x) ≡ −1 (mod |X|) for all x ∈ X and ϕ restricted to A+X
is an involutory automorphism. For any even word a ∈ A+X , ϕ−1(a) = ϕ(a) = ϕ(a−1)−1.
For any odd word b,
1 = ϕ(b−1b) = ϕ(b−1)ϕπ(b
−1)(b) = ϕ(b−1)ϕ−1(b),
and so ϕ−1(b) = ϕ(b−1)−1. Therefore, ϕ is antibalanced.
Remark 3.7. Let ϕ be a skew morphism of a finite group A containing an orbit X which
generates A, and let π be the associated power function. Lemma 3.6 implies that
(a) if |A : A+X | = 1, then the skew morphism ϕ is antibalanced if and only if ϕ is an
involutory automorphism of A;
(b) if |A : A+X | = 2, then ϕ is antibalanced if and only if π(a) ≡ 1 (mod |ϕ|) for all
a ∈ A+X , π(a) ≡ −1 (mod |ϕ|) for all a ∈ A \ A
+
X and ϕ restricted to A
+
X is an
involutory automorphism.
The following lemma deals with antibalanced skew morphisms of abelian groups.
Lemma 3.8. Let ϕ be a skew morphism of a finite abelian group A containing an orbit X
which generates A, and let π be the associated power function. Then ϕ is antibalanced if
and only if π(x) ≡ −1 (mod |X|) for all x ∈ X .
Proof. By Lemma 3.6, it suffices to prove the sufficient part. Assume that π(x) ≡ −1
(mod |X|) for all x ∈ X . Then, π(a) ≡ 1 (mod |X|) if a is an even word, and π(a) ≡ −1
8 Art Discrete Appl. Math. 3 (2020) #P1.07
(mod |X|) if a is an odd word. Furthermore ϕ restricted toA+X is an automorphism ofA
+
X .
For any a ∈ A+X and for any odd word b,
ϕ(b)ϕ−1(a) = ϕ(ba) = ϕ(ab) = ϕ(a)ϕ(b) = ϕ(b)ϕ(a),
and hence ϕ2(a) = a. Thus, by Lemma 3.6, ϕ is antibalanced.
Now we are ready to determine antibalanced skew morphisms of cyclic groups.
Theorem 3.9. Let ϕ be an antibalanced skew morphism of the cyclic additive group Zn.
(a) If n is odd, then ϕ is an involutory automorphism of the form ϕ(x) = sx, x ∈ Zn,
where s2 ≡ 1 (mod n).
(b) If n is even, then ϕ and the associated power function π are of the form
ϕ(x) =
{
xs, x is even,
(x− 1)s+ 2r + 1, x is odd,
and π(x) =
{
1, x is even,
−1, x is odd,
(3.1)
where r, s are integers in Zn/2 such that
s2 ≡ 1 (mod n/2) and (r + 1)(s− 1) ≡ 0 (mod n/2). (3.2)
In this case, the order of ϕ is equal to n/ gcd(n, r(s+ 1)), and in particular ϕ is an
automorphism if and only if s ≡ 2r + 1 (mod n/2) and (2r + 1)2 ≡ 1 (mod n).
Proof. First suppose that ϕ is an antibalanced skew morphism of Zn with the associated
power function π. Note that the orbit X of ϕ containing 1 generates Zn. Let Z+n be the
even word subgroup of Zn with respect to X . Then |Zn : Z+n | = 1 or 2. By Lemma 3.6, ϕ
restricted to Z+n is an involutory automorphism.
If n is odd, then Z+n = Zn, so ϕ is an involutory automorphism of Zn, and the result
follows from Corollary 3.5.
Now assume that n is an even number. By Lemma 3.6, ϕ is a smooth skew morphism
of skew type 1 or 2, so 〈2〉 is a ϕ-invariant normal subgroup of Zn contained in Kerϕ,
and the induced skew morphism ϕ̄ of Zn/〈2〉 is the identity permutation. Thus, there are
integers r, s ∈ Zn/2 such that
ϕ(1) ≡ 2r + 1 (mod n) and ϕ(2) ≡ 2s (mod n),
where gcd(s, n/2) = 1. By Lemma 3.6,
π(x) = 1 if x is even, π(x) = −1 if x is odd.
It follows that
ϕ(x) =
{
xs, x is even,
ϕ(x− 1) + ϕ(1) = (x− 1)s+ 2r + 1, x is odd.
(3.3)
Since ϕ restricted to Z+n is an involutory automorphism, we have s2 ≡ 1 (mod n/2).
Furthermore, we have
2s = ϕ(2) = ϕ(1) + ϕ−1(1) = 2r + 1− 2rs+ 1 (mod n),
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 9
and hence (r + 1)(s− 1) ≡ 0 (mod n/2). Moreover, by induction we have
ϕj(1) ≡ 1 + 2r
j∑
i=1
si−1 (mod n).
Let k be the smallest positive integer such that ϕk(1) = 1. By Proposition 2.5, k = |ϕ|.
If k is odd, then s ≡ 1 (mod n/2), since the length of the orbit of ϕ containing 2 di-
vides k. Upon substitution we get k = n/ gcd(n, 2r). If k is even, then the congruence
2r
∑k
i=1 s
i−1 ≡ 0 (mod n) reduces to rk(s+ 1) ≡ 0 (mod n), so k = n/ gcd(n, r(s+
1)). Note that in either cases k = n/ gcd(n, r(s + 1)). In particular, if ϕ is an automor-
phism, then for any x ∈ Zn, ϕ(x) = xϕ(1) = x(2r + 1) and (2r + 1)2 ≡ 1 (mod n).
Conversely, we need to verify that ϕ given by (3.1) is an antibalanced skew morphism
of Zn, provided that the stated numerical conditions in (3.2) are fulfilled. It is easily seen
that ϕ(0) = 0 and ϕ is a bijection on Zn. Now for any x ∈ Zn and for any y ∈ Zn, if x is
even, then one can easily show that
ϕ(x) + ϕ(y) = ϕ(x+ y).
If x is odd and y is even, then
ϕ(x) + ϕ−1(y) = (x− 1)s+ 2r + 1 + ys = (x+ y − 1)s+ 2r + 1 = ϕ(x+ y).
Finally, if both x and y are odd, then ϕ(−2rs + (y − 1)s + 1) = y, and so ϕ−1(y) =
−2rs+ (y − 1)s+ 1. From the condition (r + 1)(s− 1) ≡ 0 (mod n/2) we deduce that
−2rs ≡ 2s− 2r − 2 (mod n) and hence ϕ−1(y) = (y + 1)s− 2r − 1. Consequently,
ϕ(x) + ϕ−1(y) = (x− 1)s+ 2r + 1 + (y + 1)s− 2r − 1 = (x+ y)s = ϕ(x+ y).
Therefore, ϕ is a skew morphism of Zn. By Lemma 3.8, it is antibalanced.
From the proof of Theorem 3.9 we obtain the following corollary.
Corollary 3.10. Let ϕ be an antibalanced skew morphism of Zn. If ϕ is of odd order, then
the restriction of ϕ to Kerϕ is the identity automorphism of Kerϕ.
Theorem 3.11. Let n = 2αpα11 p
α2
2 · · · p
α`
` be the prime power factorization of a positive
integer n. Then the number ν(n) of antibalanced skew morphisms of the cyclic additive
group Zn is determined by the following formula:
ν(n) =

2`, α = 0,∏̀
i=1
(pαii + 1), α = 1,
2
∏̀
i=1
(pαii + 1), α = 2,
6
∏̀
i=1
(pαii + 1), α = 3,
(4 + 2α−2 + 2α−1)
∏̀
i=1
(pαii + 1), α ≥ 4.
10 Art Discrete Appl. Math. 3 (2020) #P1.07
Proof. If α = 0, then n is odd. By Theorem 3.9(a), every antibalanced skew morphism of
Zn is an automorphism of the form ϕ(x) = xs, x ∈ Zn, where s2 ≡ 1 (mod n). It follows
that the number ν(n) is equal to the number of solutions of the quadratic congruence s2 ≡ 1
(mod n), which is equal to 2`.
Now assume α ≥ 1, so n is an even number. By Theorem 3.9(b), the number of
antibalanced skew morphisms of Zn is equal to the number of integer solutions (r, s) in
Zn/2 of the system {
s2 ≡ 1 (mod n/2),
(r + 1)(s− 1) ≡ 0 (mod n/2).
By the Chinese Remainder Theorem, (r, s) is a solution of the system if and only if it is a
solution of each of the following `+ 1 systems{
s2 ≡ 1 (mod 2α−1),
(r + 1)(s− 1) ≡ 0 (mod 2α−1)
(3.4)
and {
s2 ≡ 1 (mod pαii ),
(r + 1)(s− 1) ≡ 0 (mod pαii ),
i = 1, 2, . . . , `. (3.5)
We first determine the solutions of (3.5). By assumption, for each i, i = 1, 2, . . . , `,
pi is an odd prime. It follows from the congruence s2 ≡ 1 (mod pαii ) that either s ≡
1 (mod pαii ) or s ≡ −1 (mod p
αi
i ). If s ≡ 1 (mod p
αi
i ), then upon substitution the
congruence (r+ 1)(s− 1) ≡ 0 (mod pαii ) holds for every r ∈ Zpαii . On the other hand, if
s ≡ −1 (mod pαii ), then upon substitution the congruence (r+ 1)(s−1) ≡ 0 (mod p
αi
i )
reduces to r ≡ −1 (mod pαii ). Therefore, for each odd prime pi, the system (3.5) has
precisely (pαii + 1) solutions in Zpαii .
Now we turn to solutions of (3.4). If α = 1, then it only has the trivial solution
(r, s) = (1, 1). If α = 2, then (r, s) = (0, 1), (1, 1) in Z2. If α = 3, then (r, s) =
(0, 1), (1, 1), (2, 1), (3, 1), (1, 3), (3, 3) in Z4. If α ≥ 4, then by the congruence s2 ≡ 1
(mod 2α−1) we have s ≡ ±1, 2α−2±1 (mod 2α−1). Combining this with the congruence
(r+ 1)(s− 1) ≡ 0 (mod 2α−1) we obtain the following solutions (r, s) in Z2α−1 : (a) r ∈
Z2α−1 and s = 1; (b) r = 2α−1 − 1, 2α−2 − 1 and s = −1; (c) r = 2α−1 − 1, 2α−2 − 1
and s = 2α−2 − 1; (d) r ≡ 1 (mod 2) and s = 2α−2 + 1.
Finally, multiplying the numbers of solutions for the prime power cases we obtain the
number ν(n), as required.
4 Correspondence
A correspondence between complete regular dessins and pairs of certain skew morphisms
of cyclic groups has been established in [12, Theorem 5]. In this section we extend the
correspondence to reflexible complete regular dessins.
Definition 4.1. Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a pair of skew morphisms of the
cyclic additive groups Zn and Zm, associated with power functions π : Zn → Z|ϕ| and
π̃ : Zm → Z|ϕ̃|, respectively. The pair (ϕ, ϕ̃) will be called reciprocal if they satisfy the
following conditions:
(a) |ϕ| divides m and |ϕ̃| divides n,
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 11
(b) π(x) ≡ ϕ̃x(1) (mod |ϕ|) and π̃(y) ≡ ϕy(1) (mod |ϕ̃|) for all x ∈ Zn and y ∈ Zm.
Suppose that D is a complete regular dessin with underling graph Km,n. Take an
arbitrary pair of vertices u and v of valency m and n, respectively. Then the stabilizers Gu
and Gv of G = Aut(D) are cyclic of orders m and n, respectively. Assume Gu = 〈a〉 and
Gv = 〈b〉. Then by the regularity we have G = 〈a, b〉 and |G| = mn. Since the underlying
graph Km,n is simple, 〈a〉 ∩ 〈b〉 = 1. Consequently, from the product formula we deduce
that G = 〈a〉〈b〉. Thus each complete regular dessin determines a triple (G, a, b) such that
G = 〈a〉〈b〉 and 〈a〉 ∩ 〈b〉 = 1.
Now each of the cyclic factors 〈a〉 and 〈b〉 of G can be taken as the complement of the
other, so in the spirit of Proposition 2.10, there are a pair of skew morphisms ϕ and ϕ̃ of
the cyclic additive groups Zn and Zm such that
a−1bx = bϕ(x)a−π(x) and b−1ay = aϕ̃(y)b−π̃(y) (4.1)
where x ∈ Zn and y ∈ Zm. By induction we deduce that
a−kbx = bϕ
k(x)a−σ(x,k) and b−lay = aϕ̃
l(y)b−σ̃(y,l), (4.2)
where
σ(x, k) =
k∑
i=1
π(ϕi−1(x)) and σ̃(y, l) =
l∑
i=1
π̃(ϕ̃i−1(y)).
Inverting the above identities yields
b−xak = aσ(x,k)b−ϕ
k(x) and a−ybl = bσ̃(y,l)a−ϕ̃
l(y). (4.3)
Substituting for x = 1 and k = y we obtain b−1ay = aσ(1,y)b−ϕ
y(1). By comparing this
with the second identity in (4.1) we obtain
π̃(y) ≡ ϕy(1) (mod n) and ϕ̃(y) ≡ σ(1, y) (mod m).
Similarly, inserting y = 1 and l = x into the second identity in (4.3) we have a−1bx =
bσ̃(1,x)a−ϕ̃
x(1). A similar comparison with the first identity in (4.1) yields
π(x) ≡ ϕ̃x(1) (mod m) and ϕ(x) ≡ σ̃(1, x) (mod n).
By Proposition 2.10, |ϕ| = |〈a〉 : 〈a〉G| and |ϕ̃| = |〈b〉 : 〈b〉G|. Thus |ϕ| divides m and
|ϕ̃| divides n. In particular, the above four congruences are reduced to
π(x) ≡ ϕ̃x(1) (mod |ϕ|), π̃(y) ≡ ϕy(1) (mod |ϕ̃|)
and
ϕ(x) ≡ σ̃(1, x) (mod |ϕ̃|), ϕ̃(y) ≡ σ(1, y) (mod |ϕ|). (4.4)
It follows that every complete regular dessin with underlying graphKm,n determines a pair
of reciprocal skew morphisms (ϕ, ϕ̃) of the cyclic additive groups Zn and Zm. Conversely,
it is shown in [12, Proposition 4] that given a pair of reciprocal skew morphisms of the
cyclic groups Zn and Zm, a complete regular dessin with underlying graph Km,n may be
reconstructed in a canonical way. Therefore, we obtain a correspondence between complete
regular dessins and pairs of reciprocal skew morphisms of cyclic groups. See [12, Theorem
5] for details.
The following theorem extends this correspondence to reflexible complete regular dessins.
12 Art Discrete Appl. Math. 3 (2020) #P1.07
Theorem 4.2. The isomorphism classes of reflexible regular dessins with complete bipar-
tite underlying graphs Km,n are in one-to-one correspondence with pairs of reciprocal
antibalanced skew morphisms (ϕ, ϕ̃) of the cyclic groups Zn and Zm.
Proof. It is proved in [12, Theorem 5] that the isomorphism classes of regular dessinsD =
(G, a, b) with complete bipartite underlying graphsKm,n are in one-to-one correspondence
with the pairs (ϕ, ϕ̃) of reciprocal skew morphisms of the cyclic groups Zn and Zm. It
remains to show thatD is reflexible if and only if the corresponding pair of skew morphisms
(ϕ, ϕ̃) are both antibalanced.
First suppose that D = (G, a, b) is reflexible, then the identities in (4.1) determine a
pair of reciprocal skew morphisms (ϕ, ϕ̃) of the cyclic groups Zn and Zm. Since D is
reflexible, the assignment θ : a 7→ a−1, b 7→ b−1 extends to an automorphism of G. By the
identities in (4.2) derived from (4.1) we have
ab−x = bϕ
−1(−x)a−σ(−x,−1) and ba−y = aϕ̃
−1(−y)b−σ̃(−y,−1).
Applying the automorphism θ of G to the above identities we obtain
a−1bx = θ(ab−x) = θ(bϕ
−1(−x)a−σ(−x,−1)) = b−ϕ
−1(−x)aσ(−x,−1)
and
b−1ay = θ(ba−y) = θ(aϕ̃
−1(−y)b−σ̃(−y,−1)) = a−ϕ̃
−1(−y)bσ̃(−y,−1).
By comparing these with the identities in (4.1) we get ϕ(x) = −ϕ−1(−x) and ϕ̃(y) =
−ϕ̃−1(−y). Thus both ϕ and ϕ̃ are antibalanced.
Conversely, suppose that ϕ : Zn → Zn and ϕ̃ : Zm → Zm form a pair of antibalanced
reciprocal skew morphisms. Denote
Zn = {0, 1, . . . , (n− 1)} and Zm = {0′, 1′, . . . , (m− 1)′},
so that Zn and Zm are disjoint sets. Define two cyclic permutations ρ and ρ̃ on the sets Zn
and Zm by setting
ρ = (0, 1, . . . , (n− 1)) and ρ̃ = (0′, 1′, . . . , (m− 1)′).
We extend the permutations ϕ, ϕ̃, ρ and ρ̃ to permutations on Ω = Zn ∪ Zm in a natural
way, still denoted by ϕ, ϕ̃, ρ and ρ̃. Set a = ϕρ̃, b = ϕ̃ρ and G = 〈a, b〉. It is proved in [12,
Proposition 4] that |a| = m, |b| = n, 〈a〉 ∩ 〈b〉 = 1 and G = 〈a〉〈b〉, so D = (G, a, b) is a
complete regular dessin with underlying graph Km,n. Now define a bijection γ : Ω → Ω
on Ω to be γ(x) = −x and γ(y′) = −y′ for all x ∈ Zn and y′ ∈ Zm. Since both ϕ and ϕ̃
are antibalanced, we have
γa(x) = γϕρ̃(x) = γϕ(x) = −ϕ(x) = ϕ−1(−x)
= ϕ−1γ(x) = ϕ−1ρ̃−1(γ(x)) = a−1γ(x)
and
γa(y′) = γϕρ̃(y′) = γϕ((y + 1)′) = γ((y + 1)′) = −(y + 1)′
= (−y − 1)′ = ϕ−1ρ̃−1(−y′) = a−1γ(y′).
Thus γa = a−1γ. Similarly, γb = b−1γ. Hence, (G, a, b) ∼= (G, a−1, b−1), where
(G, a−1, b−1) denotes the mirror image ofD. Therefore, (G, a, b) is reflexible, as required.
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 13
We summarize two properties of reciprocal skew morphisms.
Lemma 4.3 ([12, 14]). Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a pair of reciprocal skew
morphisms of the cyclic additive groups Zn and Zm. Then
(a) ϕ(x) ≡
∑x
i=1 π̃(ϕ̃
i−1(1)) (mod |ϕ̃|) and ϕ̃(y) ≡
∑y
i=1 π(ϕ
i−1(1)) (mod |ϕ|),
(b) |Zm : Ker ϕ̃| divides |ϕ| and |Zn : Kerϕ| divides |ϕ̃|.
Lemma 4.4 ([14]). Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a pair of reciprocal skew
morphisms of the cyclic additive groups Zn and Zm. If one of the skew morphisms is an
automorphism, then the other is smooth. In particular, if one of the skew morphism is the
identity automorphism, then the other is an automorphism.
5 Classification
By Theorem 4.2, the classification of reflexible complete regular dessins is reduced to the
classification of reciprocal pairs of antibalanced skew morphisms of cyclic groups. The
aim of this section is to present a classification of the latter.
Proposition 5.1. Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a reciprocal pair of antibalanced
skew morphisms of the cyclic additive groups Zn and Zm, respectively. If both n and m
are odd, then (ϕ, ϕ̃) = (idn, idm) where idk denotes the identity automorphism of Zk,
k = n,m.
Proof. By Theorem 3.9(a), both ϕ and ϕ̃ are involutory automorphisms. The divisibility
condition on reciprocality implies that both ϕ and ϕ̃ are the identity automorphisms.
Theorem 5.2. Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a reciprocal pair of antibalanced
skew morphisms of the cyclic additive groups Zn and Zm, respectively. If n is odd and
m is even, then ϕ is an automorphism of the form ϕ(x) ≡ sx (mod n), and ϕ̃ is a skew
morphism of the form
ϕ̃(y) =
{
y, y is even,
y + 2u, y is odd
and π̃(y) =
{
1, y is even,
−1, y is odd
where s ∈ Zn and u ∈ Zm/2 are integers such that
gcd(n, s+ 1) gcd(m/2, u) ≡ 0 (mod m/2) and s2 ≡ 1 (mod n). (5.1)
Proof. By assumption, both ϕ and ϕ̃ are antibalanced. Since n is odd and m is even, by
Theorem 3.9, ϕ is an automorphism of the form ϕ(x) = sx, where s2 ≡ 1 (mod n) and
ϕ̃ is a skew morphism of the form
ϕ̃(y) =
{
ty, y is even,
t(y − 1) + 2u+ 1, y is odd,
for some t, u ∈ Zm/2 satisfying the following conditions:
t2 ≡ 1 (mod m/2) and (u+ 1)(t− 1) ≡ 0 (mod m/2).
14 Art Discrete Appl. Math. 3 (2020) #P1.07
Note that the order of ϕ is equal to the multiplicative order of s in Zn, which is a divisor of
2, and the order of ϕ̃ is the smallest positive integer ` such that
2u
∑̀
i=1
ti−1 ≡ 0 (mod m).
Now we employ the reciprocality to simplify these numerical conditions. By Defini-
tion 4.1(a), |ϕ̃| divides n. Since n is odd, |ϕ̃| is also odd, so by Corollary 3.10, t = 1, and
consequently, ϕ̃ reduces to the stated form and |ϕ̃| = m/ gcd(m, 2u). By Definition 4.1(b),
−1 ≡ π̃(1) ≡ ϕ(1) = s (mod m
gcd(m, 2u)
).
Thus, |ϕ̃| = m/ gcd(m, 2u) is a common divisor of (s + 1) and n. Since m is even,
gcd(m, 2u) = 2 gcd(m/2, u), and we obtain the first condition in (5.1), as required.
By exchanging the roles of ϕ and ϕ̃, and the associated parameters, we obtain all recip-
rocal pairs of antibalanced skew morphisms of Zn and Zm where n is even and m is odd.
The details are left to the reader.
Theorem 5.3. Let ϕ : Zn → Zn and ϕ̃ : Zm → Zm be a reciprocal pair of antibalanced
skew morphisms of the cyclic additive groups Zn and Zm, respectively. If both n and m
are even, then
ϕ(x) =
{
sx, x is even,
s(x− 1) + 2r + 1, x is odd,
π(x) =
{
1, x is even,
−1, x is odd
and
ϕ̃(y) =
{
ty, y is even,
t(y − 1) + 2u+ 1, y is odd,
π̃(y) =
{
1, y is even,
−1, y is odd
where r, s ∈ Zn/2 and u, t ∈ Zm/2 are integers such that
s2 ≡ 1 (mod n/2),
(r + 1)(s− 1) ≡ 0 (mod n/2),
gcd(m/2, u+ 1) gcd(n, r(s+ 1)) ≡ 0 (mod n/2)
(5.2)
and 
t2 ≡ 1 (mod m/2),
(u+ 1)(t− 1) ≡ 0 (mod m/2),
gcd(n/2, r + 1) gcd(m,u(t+ 1)) ≡ 0 (mod m/2).
(5.3)
Proof. By Theorem 3.9(b), the skew morphisms ϕ and ϕ̃ may be represented by the stated
form, where the parameters r, s ∈ Zn/2 and u, t ∈ Zm/2 are integers such that
s2 ≡ 1 (mod n/2), (r + 1)(s− 1) ≡ 0 (mod n/2)
and
t2 ≡ 1 (mod m/2), (u+ 1)(t− 1) ≡ 0 (mod m/2).
K. Hu and Y. S. Kwon: Reflexible complete regular dessins and antibalanced skew morphisms 15
In particular,
|ϕ| = n/ gcd(n, r(s+ 1)) and |ϕ̃| = m/ gcd(m,u(t+ 1)).
We now employ the reciprocality to simplify the numerical conditions. By Defini-
tion 4.1, we have |ϕ| = n/ gcd(n, r(s+ 1)) divides m, |ϕ̃| = m/ gcd(m,u(t+ 1)) divides
n,
−1 ≡ π(1) ≡ ϕ̃(1) ≡ 2u+ 1 (mod n/ gcd(n, r(s+ 1)))
and
−1 ≡ π̃(1) ≡ ϕ(1) ≡ 2r + 1 (mod m/ gcd(m,u(t+ 1))).
Thus, n/ gcd(n, r(s + 1)) divides gcd(m, 2(u + 1)) and m/ gcd(m,u(t + 1)) divides
gcd(n, 2(r + 1)), or equivalently,
gcd(n, r(s+ 1)) gcd(m/2, u+ 1) ≡ 0 (mod n/2)
and
gcd(m,u(t+ 1)) gcd(n/2, r + 1) ≡ 0 (mod m/2),
as required.
ORCID iDs
Kan Hu https://orcid.org/0000-0003-4775-7273
Young Soo Kwon https://orcid.org/0000-0002-1765-0806
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.08
https://doi.org/10.26493/2590-9770.1269.732
(Also available at http://adam-journal.eu)
Recipes for edge-transitive tetravalent graphs*
Primož Potočnik†
University of Ljubljana, Faculty of Mathematics and Physics,
Jadranska 19, SI-1000 Ljubljana, Slovenia
Stephen E. Wilson‡
Department of Mathematics and Statistics, Northern Arizona University,
Box 5717, Flagstaff, AZ 86011, USA
Received 2 September 2018, accepted 1 October 2019, published online 17 August 2020
Abstract
This paper presents all constructions known to the authors which result in tetravalent
graphs whose symmetry groups are large enough to be transitive on the edges of the graph.
Keywords: Graph, automorphism group, symmetry.
Math. Subj. Class. (2020): 05C15, 05C10
1 The census
This paper is to accompany the Census of Edge-Transitive Tetravalent Graphs, available at
http://jan.ucc.nau.edu/∼swilson/C4FullSite/index.html,
which is a collection of all known edge-transitive graphs of valence 4 up to 512 vertices.
The Census contains information for each graph. This information includes parame-
ters such as group order, diameter, girth etc., all known constructions, relations to other
graphs in the Census (coverings, constructions, etc.), and interesting substructures such as
colorings, cycle structures, and dissections.
*The authors wish to express their gratitude for the efforts of the anonymous referee of this paper. What you
have read here is greatly improved over the previous version because of the referee’s thorough, careful examination
and insightful suggestions.
†This author gratefully acknowledges the support of the US Department of State and the Fulbright Scholar
Program who sponsored his visit to Northern Arizona University in spring 2004.
‡Corresponding author.
E-mail addresses: primoz.potocnik@fmf.uni-lj.si (Primož Potočnik), stephen.wilson@nau.edu (Stephen
E. Wilson)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 3 (2020) #P1.08
We try to present most graphs as members of one or more parameterized families, and
one purpose of this paper is to gather together, here in one place, descriptions of each of
these families, to show how each is constructed, what the history of each is and how one
family is related to another. We also discuss in this paper the theory and techniques behind
computer searches leading to many entries in the Census.
We should point out that similar censi exist for edge transitive graphs of valence 3
[8, 10]. Unlike our census, these censi are complete in the sense that they contain all the
graphs up to a given order. The method used in these papers relies on the fact that in the
case of prime valence, the order of the automorphism group can be bounded by a linear
function of the order of the graph, making exhaustive computer searches possible – see [9]
for details.
Even though our census is not proved to be complete, it is complete in some segments.
In particular, the census contains all dart-transitive tetravalent graphs up to 512 vertices
(see [32, 36]) and all 12 -arc-transitive tetravalent graphs up to 512 vertices (see [33, 38]).
Therefore, if a graph is missing from our census, then it is semisymmetric (see Section 2).
2 Basic notions
A graph is an ordered pair Γ = (V, E), where V is an arbitrary set of things called vertices,
and E is a collection of subsets of V of size two; these are called edges. We let V(Γ) = V
and E(Γ) = E in this case. If e = {u, v} ∈ E , we say u is a neighbor of v, that u and v are
adjacent, and that u is incident with e and vice versa. A dart or directed edge is an ordered
pair (u, v) where {u, v} ∈ E . Let D(Γ) be the set of darts of Γ. The valence or degree of a
vertex v is the number of edges to which v belongs. A graph is regular provided that every
vertex has the same valence, and then we refer to that as the valence of the graph.
A digraph is an ordered pair ∆ = (V, E), where V is an arbitrary set of things called
vertices, and E is a collection of ordered pairs of distinct elements of V . We think of the
pair (u, v) as being an edge directed from u to v. An orientation is a digraph in which for
all u, v ∈ V , if (u, v) ∈ E then (v, u) /∈ E .
A symmetry, or automorphism, of a graph or a digraph Γ = (V, E) is a permutation of
V which preserves E . If v ∈ V(Γ) and σ is a symmetry of Γ, then we denote the image of
v under σ by vσ, and if ρ is also a symmetry of Γ, then the product σρ is a symmetry that
maps v to (vσ)ρ. Together with this product, the set of symmetries of Γ forms a group,
the symmetry group or automorphism group of Γ, denoted Aut(Γ). We are interested in
those graphs for which G = Aut(Γ) is big enough to be transitive on E . Such a graph is
called edge-transitive. Within the class of edge-transitive graphs of a given valence, there
are three varieties:
(1) A graph is symmetric or dart-transitive provided that G is transitive on D = D(Γ).
(2) A graph is 12 -arc-transitive provided that G is transitive on E and on V , but not
on D. A 12 -arc-transitive graph must have even valence [48]. The G-orbit of one
dart is then an orientation ∆ of Γ such that every vertex has k in-neighbours and k
out-neighbors, where 2k is the valence of Γ. The symmetry group Aut(∆) is then
transitive on vertices and on edges. We call such a ∆ a semitransitive orientation
and we say that a graph which has such an orientation is semitransitive [54].
(3) Finally, Γ is semisymmetric provided thatG is transitive on E but not on V (and hence
not onD). In this case, the graph must be bipartite, with each edge having one vertex
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 3
from each class. More generally, we say that a graph Γ is bi-transitive provided that
Γ is bipartite, and its group of color-preserving symmetries is transitive on edges
(and so on vertices of each color); a bi-transitive graph is thus either semisymmetric
or dart-transitive.
There is a fourth important kind of symmetricity that a tetravalent graph might have,
in which Aut(Γ) is transitive on vertices but has two orbits on edges. Graphs satisfying
this and certain other conditions are called LR structures. These are introduced and defined
in Section 18 but referred to in several places before that. These graphs, though not edge-
transitive themselves, can be used directly to construct semisymmetric graphs.
For σ ∈ Aut(Γ), if there is a vertex v ∈ V such that vσ is adjacent to v and vσ2 6= v we
call σ a shunt and then σ induces a directed cycle [v, vσ, vσ2, . . . , vσm], with vσm+1 = v,
called a consistent cycle. The remarkable theorem of Biggs and Conway [4, 30] says that if
Γ is dart-transitive and regular of degree d, then there are exactly d− 1 orbits of consistent
cycles. A later result [5] shows that a 12 -arc-transitive graph of degree 2e must have exactly
e orbits of consistent cycles. For tetravalent graphs, the dart-transitive ones have 3 orbits
of consistent cycles and the 12 -arc-transitive ones have two such orbits. An LR structure
might have 1 or 2 orbits of consistent cycles.
3 Computer generated lists of graphs
3.1 Semisymmetric graphs arising from amalgams of index (4, 4)
Let L and R be two finite groups intersecting in a common subgroup B and assume that no
non-trivial subgroup of B is normal in both L and R. Then the triple (L,B,R) is called
an amalgam. For example, if we let L = A4, the alternating group of degree 4, B ∼= C3,
viewed as a point-stabiliser in L, and R ∼= C12 containing B as a subgroup of index 4, then
(L,B,R) is an amalgam.
If G is a group that contains both L and R and is generated by them, then G is called a
completion of the amalgam. It is not too difficult to see that there exists a completion that
is universal in the sense that every other completion is isomorphic to a quotient thereof.
This universal completion is sometimes called the free product of L and R amalgamated
over B (usually denoted by L ∗B R) and can be constructed by merging together (disjoint)
presentations of L and R and adding relations that identify copies of the same element of
B in both L and R. For example, if the amalgam (L,B,R) is as above, then we can write
L = 〈x, y, b|x2, y2, [x, y], b3, xby, ybxy〉, R = 〈z|z12〉,
yielding
L ∗B R = 〈x, y, b, z|x2, y2, [x, y], b3, z12, xby, ybxy, z4 = b〉.
Completions of a given amalgam (L,B,R) up to a given order, say M , can be computed
using a LOWINDEXNORMALSUBGROUPS routine, developed by Firth and Holt [12] and
implemented in MAGMA [6].
Given a completion G of an amalgam (L,B,R), one can construct a bipartite graph,
called the graph of the completion, with white and black vertices being the cosets of L and
R in G, respectively, and two cosets Lg and Rh adjacent whenever they intersect. Note
that white (black) vertices are of valence [L : B] ([R : B], respectively). In particular, if B
is of index 4 in both L and R, then the graph is tetravalent. We shall say in this case that
the amalgam is of index (4, 4).
4 Art Discrete Appl. Math. 3 (2020) #P1.08
The group G acts by right multiplication faithfully as an edge-transitive group of au-
tomorphisms of the graph and so the graph of a completion of an amalgam always admits
an edge- but not vertex-transitive group of automorphisms, and so the graph is bi-transitive
(and thus either dart-transitive or semisymmetric). This now gives us a good strategy for
constructing tetravalent semisymmetric graphs of order at most M : Choose your favorite
amalgams (L,B,R) of index (4, 4), find their completions up to order 2M |B| and con-
struct the corresponding graphs.
We have done this for several amalgams of index (4, 4) and the resulting graphs ap-
pear in the census under the name SS. The graph SS[n, i] is the i-th graph in the list of
semisymmetric graphs of order n. These graphs are available in magma code at [33].
3.2 Dart-transitive graphs from amalgams
If Γ is a tetravalent dart-transitive graph, then its subdivision, obtained from Γ by inserting
a vertex of valence 2 on each edge, is edge-transitive but not vertex-transitive. This process
is reversible, by removing vertices of degree 2 in a bi-transitive graph of valence (4, 2) one
obtains a tetravalent dart-transitive graph.
In the spirit of the previous section, each such graph can be obtained from an amalgam
of index (4, 2). Amalgams of index (4, 2) were fully classified in [11, 34, 53], however,
unlike in the case of amalgams of index (3, 2), giving rise to cubic dart-transitive graphs, the
number of these amalgams is infinite. This fact, together with existence of relatively small
tetravalent dart-transitive graphs with very large automorphism groups (see Section 14)
made the straightforward approach used in [9, 8] in the case of cubic graphs impossible in
the case of valence 4. This obstacle was finally overcome in [37] and now a complete list
of dart-transitive tetravalent graphs of order up 640 is described in [36] and available in
magma code at [33]. We use AT[n, i] for these graphs to indicate the i-th graph of order n
in this magma file.
3.3 Semitransitive graphs from universal groups
Every semitransitive tetravalent graph arises from an infinite 4-valent tree T4 and a group
G acting on T4 semitransitively and having a finite stabiliser by quotienting out a normal
semiregular subgroup of G. All such groups G were determined in [27]. This result in
principle enables the same approach as used in the case of tetravalent dart-transitive graphs,
and indeed, by overcoming the issue of semitransitive graphs with large automorphism
groups (see [45]), a complete list of semitransitive tetravalent graphs (and in particular 12 -
arc-transitive graphs) of order up to 1000 was obtained in [38] and is available at [33].
We include the 12 -arc-transitive graphs from this list in our census with the designation
HT[n, i].
We now begin to show the notation and details of each construction used for graphs in
the Census.
4 Wreaths, unworthy graphs
A general Wreath graph, denoted W(n, k), has n bunches of k vertices each, arranged in a
circle; every vertex of bunch i is adjacent to every vertex in bunches i+ 1 and i− 1. More
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 5
precisely, its vertex set is Zn × Zk; edges are all pairs of the form {(i, r), (i+ 1, s)}. The
graph W(n, k) is regular of degree 2k. If n = 4, then W(n, k) is isomorphic to K2k,2k
and its symmetry group is the semidirect product of S2k × S2k with Z2. If n 6= 4, then
its symmetry group is the semidirect product of Snk with Dn; this group is often called the
wreath product of Dn over Sk.
Those of degree 4 are the graphs W(n, 2). Here, for simplicity, we can also notate
the vertex (i, 0) as Ai, and (i, 1) as Bi for i ∈ Zn, with edges then being of the forms
{Ai, Ai+1}, {Ai, Bi+1}, {Bi, Ai+1}, {Bi, Bi+1}. For example, Figure 1 shows W (7, 2).
Figure 1: W (7, 2)
A wreath graph W(n, 2) has dihedral symmetries ρ and µ, where ρ sends (i, j) to
(i + 1, j) and µ sends (i, j) to (−i, j). The important aspect of this graph is that for each
i ∈ Zn, there is a symmetry σi, called a “local” symmetry, which interchanges (i, 0) with
(i, 1) and leaves every other vertex fixed. Notice that ρ acts as a shunt for a cycle of length
n, and ρσ0 acts as a shunt for a cycle of length 2n. The third orbit of consistent cycles
are those of the form [Ai, Ai+1, Bi, Bi+1] The symmetry ρµσ0 is a shunt for the cycle
[A0, B0, A1, B1] in this orbit.
Since every σi for i 6= 0 is in the stabilizer of A0, we see that vertex stabilizers in these
graphs can be arbitrarily large. In fact, the order of the vertex-stabilizer grows exponentially
with respect to the order of the graph.
A graph Γ is unworthy provided that some two of its vertices have exactly the same
neighbors. The graph W (n, 2) is unworthy because for each i, the vertices Ai and Bi have
the same neighbors. The symmetry groups of vertex-transitive unworthy graphs tend to be
large due to the symmetries that fix all but two vertices sharing the same neighborhood.
The paper [39] shows that there are only two kinds of tetravalent edge-transitive graphs
which are unworthy. One is the dart-transitive W(n, 2) graphs. The other is the “sub-
divided double” of a dart-transitive graph; this is a semisymmetric graph given by this
construction:
Construction 4.1. Suppose that Λ is a tetravalent graph. We construct a bipartite graph
Γ = SDD(Λ) in the following way. The white vertices of Γ correspond to edges of Λ. The
black vertices correspond two-to-one to vertices of Λ; for each v ∈ V(Λ), there are two
vertices v0, v1 in V(Γ). An edge of Γ joins each e to each vi where v is a vertex of e in Λ.
6 Art Discrete Appl. Math. 3 (2020) #P1.08
The Folkman graph on 20 vertices [13] is constructible as SDD(K5). It is clear that
SDD(Λ) is tetravalent, and that if Λ is dart-transitive, then SDD(Λ) is edge-transitive.
The paper [39] shows that any unworthy tetravalent edge-transitive graph is isomorphic
to some W(n, 2) if it is dart-transitive, and to SDD(Λ) for some dart-transitive tetravalent
Λ if it is semisymmetric. There are no tetravalent unworthy 12 -arc-transitive graphs.
5 Circulants
In general, the circulant graph Cn(S) is the Cayley graph for Zn with generating set S.
Here S must be a subset of Zn which does not include 0, but does, for each x ∈ S, include
−x as well. Explicitly, vertices are 0, 1, 2, . . . , n − 1, considered as elements of Zn, and
two numbers i, j are adjacent if their difference is in S. Thus the edge set consists of all
pairs {i, i + s} for i ∈ Zn, and s ∈ S. If S = {±a1,±a2, . . . }, we usually abbreviate
the name of each graph Cn(S) as Cn(a1, a2, . . . ). The numbers ai are called jumps and
the set of edges of the form {j, j + ai} for a fixed i is called a jumpset. Figure 2 shows an
example, the graph C10(1, 3).
1
2
3
4
56
7
8
9
0
Figure 2: C10(1, 3)
For general n, if S is a subgroup of the group Z∗n of units mod n, then Cn(S) is dart-
transitive, though it may be so in many other circumstances as well. In the tetravalent case,
there are two possibilities for an edge-transitive circulant:
Theorem 5.1. If Γ is a tetravalent edge-transitive circulant graph with n vertices, then it
is dart-transitive and either:
(1) Γ is isomorphic to Cn(1, a) for some a such that a2 ≡ ±1 (mod n), or
(2) n is even, n = 2m, and Γ is isomorphic to C2m(1,m+ 1).
Proof. Note first that every edge-transitive circulant Cn(a, b) is dart-transitive, due to an
automorphism which maps a vertex i to the vertex a − i mod n, and thus inverts the edge
{0, a}.
In (1), the dihedral group Dn acts transitively on darts of each of the two jumpsets, and
because a2 ≡ ±1 (mod n), multiplication by a mod n induces a symmetry of the graph
which interchanges the two jumpsets.
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 7
On the other hand, in (2), C2m(1,m+1) is isomorphic to the unworthy graph W(m, 2),
with i and i+m playing the roles ofAi andBi. Thus, the sufficiency of (1) or (2) for edge-
transitivity is clear.
The necessity can be deduced either from a complete classification of dart-transitive
circulants of arbitrary valence proved in [19] and [23] or by a careful examination of short
cycles in dart-transitive circulants.
6 Toroidal graphs
The tessellation of the plane into squares is known by its Schläfli symbol, {4, 4}. Let T be
the group of translations of the plane that preserve the tessellation. Then T is isomorphic
to Z × Z and acts regularly on the vertices of the tessellation. If U is a subgroup of finite
index in T , let M = {4, 4}/U be the quotient space formed from {4, 4} by identifying
points if they are in the same orbit under U . The result is a finite map of type {4, 4} on the
torus, and the paper [17] shows that, even in a more general setting, every such map arises
in this way. We will call U the kernel ofM. A symmetry α of {4, 4} acts as a symmetry
ofM if and only if α normalizes U . Thus every suchM has its symmetry group Aut(M)
transitive on vertices, on horizontal edges, on vertical edges; further for each edge e of
M, there is a symmetry reversing the edge (and acting as a 180◦ rotation about its center).
Thus, Aut(M) is transitive on the edges ofM (and so must be dart-transitive) if and only
if U is normalized by some symmetry interchanging the horizontal and vertical parallel
classes. This must be a 90◦ rotation or a reflection about some axis at a 45◦ angle to the
axes.
We will use the symbols {4, 4}b,c, {4, 4}[b,c], {4, 4}<b,c>, intoduced below, to stand
for certain maps on the torus as well as their skeletons. As shown in [50], this can happen
in three different ways:
(1) {4, 4}b,c: For this graph and map, defined for b ≥ c ≥ 0, U is the group generated by
the translations (b, c) and (−c, b). These are the well-known rotary maps. Because
this U is normalized by 90◦ rotations, the map admits these rotations as symmetries.
{4, 4}b,c has D = b2 + c2 vertices, D faces and 2D edges. Figure 3 shows the case
when b = 3, c = 2.
1
1 2
2
3 4
4 5 6
5
87
7 8
9
9 10 11 12 13
13
1 2 3
4 5 6 7 8
9 10 11 12 13=
Figure 3: The map {4, 4}3,2
(2) {4, 4}<b,c>: This graph and map, defined for b − 1 > c ≥ 0, uses for U the group
generated by the translations (b, c) and (c, b). Because this U is normalized by re-
8 Art Discrete Appl. Math. 3 (2020) #P1.08
flections whose axes are at 45◦ to the axes, the map admits these reflections as sym-
metries. It has E = b2− c2 vertices, E faces and 2E edges. Figure 4 shows the map
{4, 4}<3,1>.
1
1
8
2
2
6
3
3
3
7
4 65
=
1
1 2 3
87
4 65
Figure 4: The map {4, 4}<3,1>
Notice that we exclude the case b = c + 1. The map {4, 4}<c+1,c> exists, but its
skeleton has parallel edges.
(3) {4, 4}[b,c]: For this graph and map, defined for b ≥ c ≥ 0, U is the group generated
by the translations (b, b) and (−c, c). Because this U is normalized by reflections
whose axes are at 45◦ to the axes, the map admits these reflections as symmetries. It
is defined only for b ≥ c > 1. It has F = 2bc vertices, F faces and 2F edges.
1
1 2 3 4 5
5 6
6 7 8 9
9 10 11 12
12
1 2 3 4 6 7
8 129 10 11
5
Figure 5: The map {4, 4}[3,2]
If c = 1, then the map {4, 4}[b,c] exists, but, again, its skeleton has multiple edges
and so is not a simple graph.
It is interesting to note that {4, 4}b+c,b−c is a double cover of {4, 4}b,c, while {4, 4}[b+c,b−c]
is a double cover of {4, 4}<b,c> and {4, 4}<b+c,b−c> is a double cover of {4, 4}[b,c]. In
each case, the covering is 2-fold because the kernel of the covering map has index 2 in the
kernel of the covered map.
Because {4, 4}b,0 is isomorphic to {4, 4}<b,0>, this map is reflexible. Because {4, 4}b,b
is isomorphic to {4, 4}[b,b], this map is also reflexible. All other {4, 4}b,c are chiral: i.e.,
rotary but not reflexible.
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 9
Now, every U of finite index in Z2 can be expressed in the form U = 〈(d, e), (f, g)〉,
where (d, e) and (f, g) are linearly independent. In particular, we claim that U can also
be expressed in the form U = 〈(r, 0), (s, t)〉, where t = GCD(e, g). To see that, let
e = e′t, g = g′t, and let m and n be Bezout multipliers, so that me′ + ng′ = 1. Now let
s = md+ nf and r = g′d− e′f . Then 〈(r, 0), (s, t)〉 has a fundamental region which is a
rectangle r squares wide, t squares high, with the left and right edges identified directly, and
the bottom edges identified with the top after a shift s squares to the right, as in Figure 6.
t
s
r
Figure 6: A standard form for maps of type {4, 4}
In the special case in which b and c are relatively prime, we have t = 1, and this forces
the graph to be circulant; in fact, the circulant graph Cr(1, s). Here, s2 ≡ −1 for {4, 4}b,c,
and s2 ≡ 1 for both {4, 4}<b,c> and {4, 4}[b,c]. On the other hand, every tetravalent
circulant graph is toroidal or has an embedding on the Klein bottle. More precisely, if
a2 ≡ −1(mod n), then Cn(1, a) ∼= {4, 4}b,c for some b, c. If a2 ≡ 1(mod n), then
Cn(1, a) ∼= {4, 4}<b,c> or {4, 4}[b,c] for some b, c. Of the graphs C2m(1,m + 1) ∼=
W(m, 2), if m is even then it is {4, 4}[m2 ,2]. On the other hand, if m is odd, it has an
embedding on the Klein bottle, though that embedding is not edge-transitive [55].
7 Depleted wreaths
The general Depleted Wreath graph DW(n, k) is formed from W(n, k) by removing the
edges of k disjoint n-cycles, each of these cycles containing one vertex from each of the k
bunches. More precisely, its vertex set is Zn × Zk. Its edge set is the set of all pairs of the
form {(i, r), (i+1, s)} for i ∈ Zn and r, s ∈ Zk, r 6= s. Its vertices are of degree 2(k−1).
It is tetravalent when k = 3. Figure 7 shows part of DW(n, 3).
It is not hard to see that if n > 4, then the group of symmetries of this graph acts
imprimmitively in two different ways. One system of blocks is the collection of sets of the
form Ri = {(i, r)|r ∈ Z3}, defined for each i ∈ Zn. Another system is the collection of
sets of the form Qr = {(i, r)|i ∈ Zn}, defined for each r ∈ Z3. Then for n > 4, the group
of symmetries of DW(n, 3) is isomorphic to S3 ×Dn.
From this, we can see that the symmetry group has an element of order 3n exactly when
3 does not divide n. More precisely, if n ≡ 1 (mod 3) then DW(n, 3) ∼= C3n(1, n+ 1) and
if n ≡ 2 (mod 3) then DW(n, 3) ∼= C3n(1, n− 1); in the remaining case, n ≡ 0 (mod 3),
DW(n, 3) is not a circulant.
A primary result in [39] is that, with one exception on 14 vertices, any edge-transitive
tetravalent graph in which each edge belongs to at least two 4-cycles is toroidal. Because
every edge of DW(n, 3) belongs to two 4-cycles of the form (i−1, x)−(i, y)−(i+1, x)−
10 Art Discrete Appl. Math. 3 (2020) #P1.08
Figure 7: Part of DW(n, 3)
(i, z)−(i−1, x), where {x, y, z} = {0, 1, 2}, the Depleted Wreaths are also toroidal. If n is
even, then DW(n, 3) ∼= {4, 4}[n2 ,3], while if n is odd, then DW(n, 3)
∼= {4, 4}<n+32 ,n−32 >.
8 Spidergraphs
The Power Spidergraph PS(k, n; r) is defined for k ≥ 3, n ≥ 5, and r such that rk ≡ ±1
(mod n), but r 6≡ ±1 (mod n). Its vertex set is Zk × Zn, and vertex (i, j) is connected
by edges to vertices (i + 1, j ± ri). It may happen that this graph is not connected; if
so, we re-assign the name to the connected component containing (0, 0). Directing each
edge from (i, j) to (i + 1, j ± ri) gives a semitransitive orientation, and so the resulting
graph is always semitransitive. (In this presentation, we wish to not include toroidal graphs
as spidergraphs. This is what moves us to require r 6≡ ±1 (mod n) and the consequent
n ≥ 5.)
Closely related is the Mutant Power Spidergraph MPS(k, n; r). It is defined for k ≥ 3,
n even and n ≥ 8, and r such that rk ≡ ±1 (mod n), but r 6≡ ±1 (mod n). Its vertex set is
Zk ×Zn. For 0 ≤ i < k− 1, vertex (i, j) is connected by edges to vertices (i+ 1, j ± ri);
vertex (k−1, j) is connected to (0, j±rk−1+n/2). Marušič [25] and Šparl [51] have shown
that every tetravalent tightly-attached graph is isomorphic to some PS or MPS graph, and
that the graph is 12 -arc-transitive in all but a few cases: if r
2 ≡ ±1 (mod n), then the graph
is dart-transitive. The very special graph Σ = PS(3, 7; 2) is dart-transitive. If m is an
integer not divisible by 7 and r is the unique solution mod n = 7m to r ≡ 5 (mod 7),
r ≡ 1 (mod m), then PS(6, n; r) (which is a covering of Σ) is dart-transitive.
The paper [51] defines and notates these graphs in ways which differ from this paper,
and the difference is worthy of note. If m,n, r, t are integers satisfying (1) m,n are even
and at least 4, (2) rm ≡ 1 (mod n) and (3) s = 1+r+r2 + . . . rm−1 +2t is equivalent to 0
(mod n), then [51] defines the graph Xe(m,n; r, t) (“e” stands for “even”) to have vertices
[i, j] with i ∈ Zm and j ∈ Zn and edges from [i, j] to [i + 1, j] and [i + 1, j + ri] when
0 ≤ i < m− 1, while [m− 1, j] is connected to [0, j + t] and [0, j + rm−1 + t].
The argument in [54] shows that if (1), (2) and (3) hold, then rm ≡ 1 (mod 2n). Then
it is not hard to see that if the integer s is equivalent to 0 mod 2n, then Xe(m,n; r, t)
is isomorphic to PS(m, 2n; r); if s is equivalent to n mod 2n, then it is isomorphic to
MPS(m, 2n; r).
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 11
9 Attebery graphs
The following quite general construction is due to Casey Attebery [3]. We will first define
the digraph Att[A, T, k; a, b] and then define the graph Att(A, T, k; a, b) to be the under-
lying graph of the digraph. The parameters are: an abelian group A, an automorphism T of
A, an integer k at least 3, and two elements a and b of A. Let c = b− a and define ai, bi, ci
to be aT i, bT i, cT i respectively for i = 0, 1, 2, . . . , k. We require that:
(1) {ak, bk} = {a, b},
(2) A is generated by c0, c1, c2, . . . , ck−1 and Σk−1i=0 ai; and
(3) a+ b is in the kernel of the endomorphism T ∗ = Σk−1i=0 T
i.
Then the vertex set of the digraph Att[A, T, k; a, b] and of the graph Att(A, T, k; a, b) is
defined to be A × Zk. In the digraph, edges lead from each (x, i) to (x + ai, i + 1) and
(x+ bi, i+ 1). Then the digraph is a semitransitive orientation of the graph [3]. The graph
is thus semitransitive, and it is often, but not always, 12 -arc-transitive.
Not all Attebery graphs are implemented in the Census. There are four special cases
which are:
1. If A = Zn and T is multiplication by r, the Attebery graph is just PS(k, n; r), and
thus the Attebery graphs are generalizations of the spidergraphs.
2. The graph called C±1(p; st, s) in [14]. This is an Attebery graph with A = Zsp, k =
st, T : (a1, a2, . . . , as)→ (a2, a3, . . . , as, a1),−b = a = (1, 0, 0, . . . , 0).
3. The graph called C±e(p; 2st, s) in [14] with e2 ≡ −1 (mod p). This is an Attebery
graph with A = Zsp, k = 2st, T : (a1, a2, . . . , as) 7→ (ea2, ea3, . . . , eas, ea1),−b =
a = (1, 0, 0, . . . , 0).
4. Define AMC(k, n,M) to be Att(A, T, k; a, b) where A is Zn × Zn, M is a 2 × 2
matrix over Zn satisfying Mk = ±I , T is multiplication by M , and a = (1, 0), b =
(−1, 0).
The second and third of these are generalized to the graph CPM(n, s, t, r), defined in
section 15 of this paper.
It is intriguing that even though the matrix
M =
[
1 −4
4 1
]
does not satisfy the condition M4 = ±I , the graph AMC(4, 12,M) is nevertheless edge-
transitive, and in fact semisymmetric. Even more striking is that so far we have no other
construction for this graph.
10 The separated box product
Suppose that ∆1 and ∆2 are digraphs in which every vertex has in- and out-valence 2. We
allow ∆1 and ∆2 to be non-simple.
We form the separated box product ∆1#∆2 (first defined in [41]) as the underlying
graph of the orientation whose vertex set is V(∆1) × V(∆2) × Z2, and whose edge set
contains two types of edges: “horizontal” edges join (a, x, 0) → (b, x, 1), and “vertical”
edges (b, x, 1)→ (b, y, 0), where a→ b in ∆1 and x→ y in ∆2.
12 Art Discrete Appl. Math. 3 (2020) #P1.08
An orientation is reversible provided that it is isomorphic to its reversal. As shown in
Theorem 5.1 of [41], there are several useful cases of this construction:
• When ∆1 = ∆2 and ∆1 is reversible, then ∆1#∆2 is dart-transitive.
• When ∆1 = ∆2 and ∆1 is not reversible, ∆1#∆2 has a semitransitive orientation
and so might be dart-transitive or 12 -transitive.
• When ∆1 is not isomorphic to ∆2 or its reverse but both are reversible, then ∆1#∆2
has an LR structure.
• When ∆1 is not isomorphic to ∆2 but is isomorphic to the reverse of ∆2, then
∆1#∆2 is at least bi-transitive and is semisymmetric in all known cases.
In the Census, we use for ∆1 and ∆2 directed graphs from the census of 2-valent dart-
transitive digraphs [38, 33] with notation ATD[n, i] for the ith digraph of order n from that
census. We also allow the “sausage digraph” DCycn: an n-cycle with each edge replaced
by two directed edges, one in each direction. See [41] for more details.
11 Rose windows
The Rose Window graph Rn(a, r) has 2n vertices: Ai, Bi for i ∈ Zn. The graph has four
kinds of edges:
Rim: Ai −Ai+1
In-Spoke: Ai −Bi
Out-spoke: Bi −Ai+a
Hub: Bi −Bi+r
For example, Figure 8 shows R12(2, 5).
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Figure 8: R12(2, 5)
We will soon mention these graphs as examples of bicirculant graphs, and more gener-
ally later as examples of polycirculant graphs.
The paper [20] shows that every edge-transitive Rn(a, r) is dart-transitive and is iso-
morphic to one of these:
a. Rn(2, 1). (This graph is isomorphic to W(n, 2).)
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 13
b. R2m(m+ 2,m+ 1).
c. R12k(3k ± 2, 3k ∓ 1).
d. R2m(2b, r), where b2 ≡ ±1 (mod m), r is odd and r ≡ 1 (mod m).
12 Bicirculants
A graph Γ is bicirculant provided that it has a symmetry ρ which acts on V as two cycles of
the same length. A rose window graph is bicirculant, as the symmetry sending Ai to Ai+1
and Bi to Bi+1 is the required ρ.
The paper [21] classifies all edge-transitive tetravalent bicirculant graphs. Besides the
Rose Window graphs there is one other class of graphs, called BC4 in that paper, and called
BC here and in the Census. The graph BCn(a, b, c, d) has 2n vertices: Ai, Bi for i ∈ Zn.
The edges are all pairs of the form {Ai, Bi+e} for i ∈ Zn, e ∈ {a, b, c, d}. It is easy to
see that any such graph is isomorphic to one of the form BCn(0, a, b, c) where a divides n.
The edge-transitive graphs in this class consist of three sporadic examples and three infinite
families. The sporadics are:
BC7(0, 1, 2, 4), BC13(0, 1, 3, 9), BC14(0, 1, 4, 6)
Of the three infinite families of graphs BCn(0, a, b, c), there are two in which we can
choose a = 1, and a third, less easy to describe, in which none of the parameters is rela-
tively prime to n:
(I) BCn(0, 1,m+ 1,m2 +m+ 1), where (m+ 1)(m2 + 1) = 0 (mod n)
(II) BCn(0, 1, d, 1− d), where 2d(1− d) = 0 (mod n)
(III) BCkrst(0, r, rs′s+ st, rt′t+ st+ rst), where
(1) r, s, t are all integers greater than 1,
(2) s, t are odd,
(3) r, s, t are relatively prime in pairs,
(4) k ∈ {1, 2},
(5) if k = 2, then r is even,
(6) s′ is an inverse of s mod krt, and t′ is an inverse of t mod krs.
13 Semiregular symmetries and their diagrams
A symmetry σ is semiregular provided that it acts on the |V| = kn vertices as k cycles
of length n. We can visually represent such a graph and symmetry with a diagram. This
is a graph-like object, with labels. Each “node” represents one orbit under σ. If σ =
(u0, u1, . . . , un−1)(v0, v1, . . . , vn−1) . . . (w0, w1, . . . , wn−1) and there is an edge from u0
to va, then there is an edge from each ui to va+i (indices computed modulo n). This
matching between u′is and v
′
is is represented in the diagram by a directed edge from node
u to node v with label a (or one from v to u with label −a). If a = 0, then the label a and
the direction of the edge in the diagram can be omitted.
If there is an edge from u0 to ub (and thus one from each ui to ui+b), we represent this
by a loop at u with label b. In the special case in which n is even and b = n2 , there are only
14 Art Discrete Appl. Math. 3 (2020) #P1.08
n
2 edges in the orbit and we represent them with a semi-edge at u. This convention makes
the valence of u in the diagram the same as the valences of all of the ui’s in the graph.
For example, the graph in Figure 9 has the symmetry
ϕ = (u0u1u2u3u4u5)(v0v1v2v3v4v5)(w0w1w2w3w4w5).
u0
u1
u2
u3
u4
u5
v0
v1
v2
v3
v4
v5
w0
w1
w2
w3
w4
w5
Figure 9: A trivalent graph having a semiregular symmetry
1
u v w2
Mod 6
Figure 10: The diagram of the semiregular symmetry ϕ
The corresponding diagram is shown in Figure 10. The label “Mod 6” in this diagram
informs the reader that all numbers which label edges in the diagram are to be considered
as elements of Z6.
It should be pointed out that what we were describing above is simply the notion of a
quotient of a graph (as defined, for example, in [24]) by the cyclic group generated by the
semiregular element σ, and the diagram which we obtain is the voltage graph describing
the graph Γ.
Further, the graph can be reconstructed from the diagram. This is simply the ordinary
voltage graph construction with voltage group Zn and voltages of the darts as shown with
the integers drawn at the darts in the diagram (if neither the dart nor its inverse dart have
any integers drawn next to them, then the corresponding voltage is assumed to be 0).
In constructions, we often refer to the generic form of the diagram, in which the modu-
lus is unspecified and parameter names are used as labels for some edges, as in Figure 11.
As an example, consider the bicirculant graphs Rn(a, r) and BCn(0, a, b, c), which can
be represented by the diagrams shown in Figure 12.
In the following, we use diagrams to define many families of graphs.
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 15
a
u v wb
Figure 11: A generic diagram
1 r
a
A B
a
A Bb
c
Figure 12: Diagrams of graphs
13.1 Propellors
A B Ca
b
c d
Figure 13: Diagram for the graph Prn(a, b, c, d)
A Propellor Graph is a graph with the diagram shown in Figure 13. This means that
the graph has 3n vertices:Ai, Bi, Ci for i ∈ Zn. There are 6 kinds of edges:
Tip: Ai −Ai+a
Ci − Ci+d
Flat: Ai −Bi
Bi − Ci
Blade: Bi −Ai+b
Bi − Ci+c
Propellor graphs have been investigated by Matthew Sterns. He conjectures in [46] that
the edge-transitive propellor graphs are isomorphic to
I Prn(1, 2d, 2, d), where n is even and d2 ≡ ±1 (mod n)
II Prn(1, b, b+ 4, 2b+ 3), where n is divisible by 4, 8b+ 16 ≡ 0 (mod n), and b ≡ 1
(mod 4).
16 Art Discrete Appl. Math. 3 (2020) #P1.08
III These five sporadic examples:
Pr5(1, 1, 2, 2),Pr10(1, 1, 2, 2), Pr10(1, 4, 3, 2), Pr10(1, 1, 3, 3), Pr10(2, 3, 1, 4).
Notice first that in every case except the last of the sporadic cases, the A-tip edges form
a single cycle. The first of the infinite families consists of the 2-weaving graphs: those
graphs in which some symmetry of the graphs sends this cycle to a cycle of the form AB
AB AB. . . The second class consists of 4-weaving graphs: those in which some symmetry
sends the A-tip cycle to a cycle of the form ABCB ABCB AB. . . There is also a class of
5-weaving graphs in which some image of the tip cycle is of the form AABCB AABCB
AAB. . ., but the requirements for this class force n to divide 10, resulting in the first four
of the sporadic cases. This leaves the last sporadic graph as something of a mystery.
A more recent paper [47] proves that if a or d is relatively prime to n, then the conjec-
ture holds.
13.2 Metacirculants
In [29], Marušič and Šparl considered tetravalent graphs which are properly called weak
metacirculant, though we will simply refer to them as metacirculant in this paper. A graph
is metacirculant provided that it has a symmetry ρ which acts on the mn vertices as m
cycles of length n and another symmetry σ which normalizes 〈ρ〉 and permutes the m ρ-
orbits in a cycle of length m. That paper considers four classes of 12 -transitive tetravalent
metacirculant graphs. The four classes together include all such graphs.
The Type I graphs are the Power Spidergraphs PS(m,n; r) and MPS(m,n; r). Papers
[25, 29, 51, 54] completely determine which of these are 12 -transitive and which are dart-
transitive. The Type II graphs are called Y there and will be called MSY here and in the
census. These also have been classified, in unpublished work [2]. See Section 13.2.1 below.
The graphs of Type III we will call MC3 in this census. They have been studied with
a few results. See Section 13.2.3. While the general Type IV graphs are very unruly,
there are some results known; for instance, the paper [1] classifies Type IV 12 -arc-transitive
metacirculants of girth 4. A subclass of Type IV graphs, called Z in [29] and MSZ in the
Census, have received some concentrated study. See Section 13.2.2 for a description of the
graph.
13.2.1 MSY
The graph MSY(m,n; r, t) has the diagram shown in Figure 14.
More precisely, its vertex set is Zm × Zn, with two kinds of edges:
(1) (i, j)− (i, j + ri) for all i and j, and
(2) (i, j)− (i+ 1, j) for 0 ≤ i < m− 1 and (m− 1, j)− (0, j + t) for all j.
The graph is seen to be metacirculant (with ρ sending (i, j) to (i, j + 1) and σ sending
(i, j) to (i + 1, rj) for i 6= m − 1 and sending (m − 1, j) to (0, rj + t)) if and only if
rm = 1 and rt = t. Here, all equalities are equivalences mod n. The paper [52] proves
that MSY(m,n; r, t) is metacirculant and 12 -transitive if and only if it is isomorphic to one
in which:
(1) n = dm for some integer d at least 3,
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 17
3
210
-1
5
4
1
r-1
r
r2
r3
r4
r5
t
Figure 14: Diagram for the graph MSY(m,n; r, t)
(2) rm = 1,
(3) r2 6= ±1,
(4) m(r − 1) = t(r − 1) = (r − 1)2 = 0,
(5) 〈m〉 = 〈t〉 in Zn,
(6) there is a unique c in Zd satifying cm = t and ct = m,
(7) there is a unique k in Zd satifying kt = −km = r − 1, and
(8) either m 6= 4 or t 6= 2 + 2r.
The paper [2] shows that MSY(m,n; r, t) is metacirculant and edge-transitive if and
only if it is isomorphic to one in which rm = 1, rt = t and one of three things happens:
(1) m = GCD(t, n) (then t = sm, n = n′m and GCD(n′, s) = 1);
(2) r ≡ 1 (mod m) and so r = km+ 1 for some k;
(3) st = m;
(4) kt = −km.
or
(1) m = GCD(t, n) (then t = sm, n = n′m and GCD(n′, s) = 1);
(2) r ≡ 1 (mod m) and so r = km+ 1 for some k;
(3) st = −m;
(4) kt = km.
or [m,n, r, t] is one of these four sporadic examples:
[5, 11, 5, 0], [5, 22, 5, 11], [5, 33, 16, 0], [5, 66, 31, 33].
13.2.2 MSZ
The graph MSZ(m,n; k, r) has a diagram isomorphic to the circulant graph Cm(1, k). It
has vertex set Zm × Zn. The vertex (i, j) is adjacent to (i+ 1, j) and to (i+ k, j + ri).
18 Art Discrete Appl. Math. 3 (2020) #P1.08
13.2.3 MC3
The diagram for this family has an even number of nodes arranged in a circle. With c = 1
or 2, each node is joined by an edge to nodes that are c steps away in the circle. Each node
is joined to the node opposite by two edges.
We call the graph Γ = MC3(m,n, a, b, r, t, c); here, m must be even, c is 1 or 2, and
a, b, r, t are numbers mod n such that rt = t, rm = 1, and {a+t, b+t} = {−arm2 ,−brm2 }.
The vertex set is Zm × Zn. One kind of edge connects each (i, j) to (i + c, j) for i =
0, 1, 2, . . . ,m− c− 1, and (i, j) to (i + c, j + t) for i ∈ {m− c,m− 1}. A second kind
of edge joins each (i, j) to (i+ m2 , j + ar
i) and (i+ m2 , j + br
i).
Because it has been shown that each such graph which is 12 -arc-transitive is also a
PS,MPS,MSY or MSZ, little attention has been given to it. However, many MC3’s are
dart-transitive and many are LR structures (see section 18, where we will refer to the two
kinds of edges as green and red). Each of the following families is such an example:
1. m is divisible by 4, n is divisible by 2, r2 = ±1, a = 1, b = −1, t = 0,
2. m is not divisible by 4, n is divisible by 4, r2 = 1, a = 1, b = −1, t = n2 ,
3. n is divisible by 4, r2 = 1, a = 1, b = n/2− 1, t = n/2.
These were found and proved to be LR strucures (see section 18) by Ben Lantz [22],
and there are LR examples not covered by these families. Further, many of the MC3 graphs
are dart-transitive. There are many open questions about this family.
13.3 Other diagrams
A number of other diagrams have been found to give what appears to be an infinite number
of examples of edge-transitive graphs. The first of these is the Long Propellor, LoPrn(a, b, c, d, e),
shown in Figure 15.
ba
A B
ec
d
C D
Figure 15: Diagram for the graph LoPrn(a, b, c, d, e)
Next is the Woolly Hat, WHn(a, b, c, d), shown in Figure 16. This diagram appears to
give no edge-transitive covers, but it does yield a family of LR structures (see Section 18),
as yet unclassified.
The Kitten Eye, KEn(a, b, c, d, e), shown in Figure 17, has dart-transitive covers.
The Curtain, Curtainn(a, b, c, d, e), shown in Figure 18, has both dart-trasitive and LR
covers.
14 Praeger-Xu constructions
The graph called C(2, n, k) in [44] is also described in [14] in two different ways. In this
Census, we will name the graph PX(n, k). In order to describe the graph, we need some
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 19
A
B C
a
c
d
b
Figure 16: Diagram for the graph WHn(a, b, c, d, e)
A
B
C
a
c
db
D e
Figure 17: Diagram for the graph KEn(a, b, c, d, e)
notation about bit strings. A bit string of length k is the concatenation of k symbols, each
of them a ’0’ or a ’1’. For example x = 0011011110 is a bit string of length 10. If x is
a bit string of length k, then xi is its i-th entry, and xi is the string identical to x in every
place except the i-th. Also 1x is the string of length k + 1 formed from x by placing a ’1’
in front; similar definitions hold for the (k + 1)-strings 0x, x0, x1. Finally, the string x̄ is
the reversal of x.
The vertices of the graph PX(n, k) are ordered pairs of the form (j, x), where j ∈ Zn
and x is a bit string of length k. Edges are all pairs of the form {(i, 0x), (i+1, x0)}, {(i, 0x), (i+
1, x1)}, {(i, 1x), (i+1, x0)}, {(i, 1x), (i+1, x1)}, where x is any bit string of length k−1.
We first wish to consider some symmetries of this graph. First we note ρ and µ given
by (j, x)ρ = (j + 1, x) and (j, x)µ = (−j, x̄) These are clearly symmetries of the graph
and act on it as Dr.
For b ∈ Zn, we define the symmetry τb to be the permutation which interchanges
(b − i, x) with (b − i, xi) for i = 1, 2, 3, . . . , k and leaves all other vertices fixed. If
n > k, then the symmetries τ0, τ1, . . . , τn−1 commute with each other and thus generate
an elementary abelian group of order 2n, while the symmetries ρ, µ and τ0 generate a
semidirect product Zn2 oDn of order n2n+1. Unless n = 4, this is also the full symmetry
group of the graph (see [44]).
The Praeger-Xu graphs generalize two families of graphs: PX(n, 1) = W(n, 2) and
PX(n, 2) = R2n(n+ 2, n+ 1).
15 Gardiner-Praeger constructions
The paper [14] constructs two families of tetravalent graphs whose groups contain large
normal subgroups such that the factor graph is a cycle. The first is C±1(p, st, s), and the
second is C±e(p, 2st, s). In the Census, we use a slight generalization of both, which we
notate CPM(n, s, t, r), where n is any integer at least 3, s is an integer at least 2, t is a
20 Art Discrete Appl. Math. 3 (2020) #P1.08
b
a
A B
ec
d
C D
Figure 18: Diagram for the graph Curtainn(a, b, c, d, e)
positive integer, and r is a unit mod n. We first form the digraph CPM[n, s, t, r]. Its vertex
set is Zsn ×Zst. Directed edges are of the form ((x, i), (x± riej , i+ 1)), where j is i mod
s, and ej is the j-th standard basis vector for Zsn. If rst is ±1 mod n, then CPM[n, s, t, r]
is a semitransitive orientation for its underlying graph CPM(n, s, t, r). When the graph is
not connected, we re-assign the name CPM(n, s, t, r) to the component containing (0, 0).
If n is odd, then the graph has stns vertices. If n is even then it has st(n2 )
s vertices if t is
even and twice that many if t is odd.
Some special cases are known: For r = 1,CPM(n, s, t, 1) ∼= C±1(n, st, s). If r2 = 1,
then CPM(n, s, 2t, r) is C±r(p, 2st, s). When s = 1,CPM(n, 1, t, r) ∼= PS(n, t; r). If
s = 1 and t = 4, then the graph is a Wreath graph. When s = 2 and t is 1 or 2, then the
graph is toroidal. Other special cases are conjectured:
The convention in the following conjectures is that q is a number whose square is one
mod m and p is the parity function:
p(t) =
{
1 if t is odd
2 if t is even (15.1)
With that said, we believe that:
1. If t is not divisible by 3, then CPM(3, 2, t, 1) ∼= PS(6,m; q) where m = 3t.
2. If t is not divisible by 5, then CPM(5, 2, t, 1) ∼= CPM(5, 2, t, 2) ∼= PS(10,m; q)
where m = 5tp(t).
3. If t is not divisible by 3, then CPM(6, 2, t, 1) ∼= PS(6,m; q) where m = 12tp(t) .
4. If t is not divisible by 4, then CPM(8, 2, t, 1) ∼= CPM(8, 2, t, 3) ∼= MPS(8,m; q)
where m = 16tp(t) .
5. For all s, CPM(4, s, t, 1) ∼= PX( 2stp(t) , s).
16 Graphs Γ± of Spiga, Verret and Potočnik
It was proved in [37] that a tetravalent graph Γ whose automorphism group G is dart-
transitive is either 2-arc-transitive (and then |Gv| ≤ 2436), a PX-graph (see Section 14),
one of eighteen exceptional graphs, or it satisfies the inequality
|V (Γ)| ≥ 2|Gv| log2(|Gv|/2). (∗)
This result served as the basis for the construction of a complete list of dart-transitive
tetravalent graphs (see [36] for details).
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 21
Moreover, in [35] it was proved that a graph attaining the bound (∗) has t2t+2 vertices
for some t ≥ 2 and is isomorphic to one of the graphs (t, ), for  ∈ {0, 1}, defined below
as coset graphs of certain groups G0t or G
1
t . Recall that the coset graph Cos(G,H, a) on
a group G relative to a subgroup H ≤ G and an element a ∈ G is defined as the graph
with vertex set the set of right cosets G/H = {Hg | g ∈ G} and with edge set the set
{{Hg,Hag} | g ∈ G}.
For  ∈ {0, 1} and t ≥ 2, let Gt be the group defined as follows:
Gt = 〈x0, . . . , x2t−1, z, a, b | x2i = z2 = b2 = za2t = (ab)2 = 1,
[xi, z] = 1 for 0 ≤ i ≤ 2t− 1,
[xi, xj ] = 1 for |i− j| 6= t,
[xi, xt+i] = z for 0 ≤ i ≤ t− 1,
xai = xi+1 for 0 ≤ i ≤ 2t− 1,
xbi = xt−1−i for 0 ≤ i ≤ 2t− 1〉.
In either group, we let H = 〈x0, . . . , xt−1, b〉, and define graphs
PPM(t, ) = Cos(Gt, H, a),
denoted Γ+t (for  = 0) and Γ
−
t (for  = 1) in [35]. We should point out that a graph
Γ = PPM(t, ) is a 2-cover of the Praeger-Xu graph PX(2t, t). Furthermore, the girth
of PPM(t, ) is generally 8, the only exceptions being that PPM(2, 0) has girth 4, and
PPM(3, 0) has girth 6. Finally, PPM(2, 0) ∼= PX(4, 3), while in all other cases the graph
Γ is not isomorphic to a PX graph.
The 2016 edition of the Census seems, mysteriously, to have missed implementing this
construction. The next edition will include these graphs. Meanwhile, we know that exactly
6 of them fall in the range of sizes addressed in the Census. The resulting entries are:
PPM(2,0) = C4[32,2] = {4, 4}4,4.
PPM(2,1) = C4[32,5] = MSY(4, 8, 5, 4).
PPM(3,0) = C4[96,27] = KE24(1, 22, 8, 3, 7).
PPM(3,1) = C4[96,24] = KE24(1, 13, 4, 21, 5).
PPM(4,0) = C4[256,72] = UG(ATD[256, 128]).
PPM(4,1) = C4[256,78] = UG(ATD[256, 146]).
17 From cubic graphs
In this section, we describe five constructions, each of which constructs a tetravalent graph
from a smaller cubic (i.e., trivalent) graph in such a way that the larger graph inherits many
symmetries from the smaller graph. Throughout this section, assume that Λ is a cubic
graph, and that it is dart-transitive. Our source of these graphs is Marston Conder’s census
of symmetric cubic graphs of up to 10,000 vertices [8].
17.1 Line graphs
The line graph of Λ is a graph Γ = L(Λ) whose vertices are, or correspond to, the edges
of Λ. Two vertices of Γ are joined by an edge exactly when the corresponding edges of Λ
share a vertex. Every symmetry of Λ acts on Γ as a symmetry, though Γ may have other
symmetries as well. Clearly, if Λ is edge-transitive, then Γ is vertex-transitive. If Λ is dart-
transitive, then Γ is edge-transitive, and if Λ is 2-arc-transitive, then Γ is dart-transitive.
22 Art Discrete Appl. Math. 3 (2020) #P1.08
17.2 Dart graphs
The Dart Graph of Λ is a graph Γ = DG(Λ) whose vertices are, or correspond to, the darts
of Λ. Edges join a dart (a, b) to the dart (b, c) whenever a and c are distinct neighbors of b.
Clearly, DG(Λ) is a two-fold cover of L(Λ).
17.3 Hill capping
For every vertex A of Λ, we consider the symbols (A, 0), (A, 1), though we will ususally
write them as A0, A1. Vertices of Γ = HC(Λ) are all unordered pairs {Ai, Bj} of symbols
where {A,B} is an edge of Λ. Edges join each vertex {Ai, Bj} to {Bj , C1−i} where A
and C are distinct neighbors of B.
If Λ is bipartite and 2-arc-transitive then Γ is dart-transitive. If Λ is bipartite and not
2-arc-transitive then Γ is semisymmetric. If Λ is not bipartite and is 2-arc-transitive then Γ
is 12 -arc-transitive.
HC(Λ) is clearly a fourfold cover of L(Λ); it is sometimes but not always a twofold
cover of DG(Λ). The Hill Capping is described more fully in [15].
17.4 3-arc graph
The three-arc graph of Λ, called A3(Λ) in the literature [18] and called TAG(Λ)in the
Census, is a graph whose vertices are the darts of Λ, with (a, b) adjacent to (c, d) exactly
when [b, a, c, d] is a 3-arc in Λ. Thus, a and c are adjacent, b 6= c and a 6= d. This graph is
dart-transitive if Λ is 3-arc-transitive.
18 Cycle decompositions
A cycle decomposition of a tetravalent graph Λ is a partition C of its edges into cycles.
Every edge belongs to exactly one cycle in C and each vertex belongs to exactly two cycles
of C. Aut(C) is the group of all symmetries of Λ which preserve C. One possibility for such
a symmetry is a swapper. If v is a vertex on the cycle C, a C−swapper at v is a symmetry
which reverses C while fixing v and every vertex on the other cycle through v.
If C is a cycle decomposition of Λ, the partial line graph of C, written P(C) and notated
PL(C) in the Census, is a graph Γ whose vertices are (or correspond to) the edges of Λ,
and whose edges are all {e, f} where e and f are edges which share a vertex but belong to
different cycles of C.
Because Aut(C) acts on Γ as a group of its symmetries, the partial line graph is use-
ful for constructing graphs having a large symmetry group. Almost all tetravalent dart-
transitive graphs have cycle decompositions whose symmetry group is transitive on darts.
These are called “cycle structures” in [31].
If Λ is 12 -arc-transitive, then it has a cycle decomposition A into ’alternating cycles’
[25]. If the stabilizer of a vertex has order at least 4, then P(A) has a 12 -arc-transtive action
and may actually be 12 -arc-transitive.
Many graphs in the census are constructed from smaller ones using the partial line
graph. Important here are the Praeger-Xu graphs. Each PX(n, k) has a partition C of
its edges into 4-cycles of the form:(i, 0x), (i + 1, x0), (i, 1x), (i + 1, x1). Then P(C) is
isomorphic to PX(n, k + 1). A special case of this is that family (b) of Rose Window
graphs is P applied to family (a), the wreath graphs.
The toroidal graphs have a cycle decomposition in which each cycle consists entirely
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 23
of vertical edges or entirely of horizontal edges. The partial line graph of this cycle decom-
position is another toroidal graph. For the rotary case, P({4, 4}b,c) = {4, 4}b+c,b−c. The
other two families of toroidal maps need not have edge-transitive partial line graphs.
It may happen that a cycle decomposition C is a ’suitable LR structure’ [42]; this means
that C has a partition intoR and G (the ’red’ and the ’green’ cycles) such that every vertex
belongs to one cycle from each set, that the subgroup of Aut(C) which sends R to itself
is transitive on the vertices of Λ, that Aut(C) has all possible swappers, that no element
of Aut(C) interchanges R and G and, finally, that no 4-cycle alternates between R and
G. With all of that said, if C is a suitable LR structure, then P(C) is a semisymmetric
tetrtavalent graph in which each edge belongs to a 4-cycle. Further, every such graph is
constructed in this way. [42]
19 Some LR structures
The Census uses the partial line graph construction on a number of families of LR struc-
tures, shown here.
19.1 Barrels
The barrels are the most common of the suitable LR structures. The standard barrel is
Br(k, n; r), where k is an even integer at least 4, n is an integer at least 5 and r is a number
mod n such that r2 = ±1 (mod n) but r 6= ±1 (mod n). The vertex set is Zk × Zn. Green
edges join each (i, j) to (i, j + ri). Red edges join each (i, j) to (i+ 1, j).
The mutant barrel is MBr(k, n; r), where k is an even integer at least 2, n is an even
integer≥ 8 and r is a number mod n such that r2 = ±1 (mod n) but r 6= ±1 (mod n). The
vertex set is Zk ×Zn. Green edges join each (i, j) to (i, j+ ri). Red edges join each (i, j)
to {
(i+ 1, j) if i 6= k − 1
(0, j + n2 ) if i = k − 1
.
19.2 Cycle structures
If Λ is a tetravalent graph admiting a cycle structure C, we can form an LR struture from it
in two steps:
1. Replace each vertex v with two vertices, each incident with the two edges of one
of the two cycles in C containing v; think of these as green edges. Join the two vertices
corresponding to v with two parallel red edges. Call this C ′. We can cover C ′ in different
ways.
2a. Double cover C ′. We assign weights or voltages to red edges so that each pair has
one 0 and one 1. Voltages for green edges are assigned in one of two ways: (0) every green
edge gets voltage 0 or (1) one edge in each green cycle gets voltage 1, and the rest get
0. The double covers corresponding to these two assignments are called CS(Λ, C, 0) and
CS(Λ, C, 1), respectively, and they are, in most cases, suitable LR structures, as shown in
[43].
2b. Cover C ′ k-fold for some k at least 3. We assign weights or voltages to red edges
so that each pair has a 1 in each direction. All green edges are assigned voltage 0. The
k-cover corresponding to this assignment is called CSI(Λ, C, k), and [43] shows that each
24 Art Discrete Appl. Math. 3 (2020) #P1.08
such is a suitable LR structure.
19.3 Bicirculants
Consider a bicoloring of the edges of the bicirculant BCn(0, a, b, c) with green edges link-
ing Ai to Bi and Bi+a and red edges linking Ai to Bi+b and Bi+c. We call this coloring
BCn({0, a}, {b, c}). The paper [40] shows several cases in which BCn({0, a}, {b, c}) is a
suitable LR structure:
1. a = 1 − r, b = 1, c = s, where r, s ∈ Z∗n \ {−1, 1}, r2 = s2 = 1, r 6∈
{−s, s}, and (r − 1)(s− 1) = 0.
2. n = 2m, a = m, b = 1, c ∈ Z2m \{1,−1,m+ 1,m−1} such that c2 ∈ {1,m+ 1}.
3. n = 4k, a = 2k, b = 1, c = k + 1, for k ≥ 3
Moreover, it is conjectured in that paper that every suitable BCn({0, a}, {b, c}) is isomor-
phic to at least one of these three.
19.4 MSY’s and MSZ’s
The graph MSY(m,n; r, t) has an LR structure, with edges of the first kind being red and
those of the second kind being green, if and only if 2t = 0 and r2 = ±1. Many examples
of MSZ graphs being suitable LR structures are known, but no general classification has
been attempted.
19.5 Stack of pancakes
The structure is called SoP(4m, 4n). Let r = 2n+ 1. The vertex set is Z4m × Z4n × Z2.
Red edges join (i, j, k) to (i, j±rk, k); for a fixed i and j, green edges join the two vertices
(2i, j, 0) and (2i, j, 1) to the two vertices (2i+ 1, j, 0) and (2i+ 1, j, 1) if j is even, to the
two vertices (2i− 1, j, 0) and (2i− 1, j, 1) if j is odd.
The paper [43] shows that this is a suitable LR structure for all m and n, and that
the symmetry group of it and of its partial line graph, can have arbitrarily large vertex-
stabilizers.
19.6 Rows and columns
The LR structure RC(n, k) has as vertices all ordered pairs (i, (r, j)) and ((i, r), j), where
i and j are in Zn, and r is in Zk, where k and n are integers at least 3. Green edges join
(i, (r, j)) to (i± 1, (r, j)) and ((i, r), j) to ((i, r), j ± 1), while red edges join (i, (r, j)) to
((i, r ± 1), j) and so ((i, r), j) to (i, (r ± 1, j)).
This structure is referred to in both [40] and [43].
19.7 Cayley constructions
Suppose a group A is generated by two sets, R and G, of size two, neither containing the
identity, and each containing the inverse of each of its elements. Then we let Cay(A;R,G)
be the structure whose vertex set is A, whose red edges join each a to sa for s ∈ R and
whose green edges join each a to sa for s ∈ G. The paper [40] shows that if A admits two
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 25
automorphisms, one fixing each element of R but interchanging the two element of G and
the other vice versa, then Cay(A;R,G) is an LR structure. The condition RG 6= GR is
equivalent to the structure not having alternating 4-cycles.
Many examples occur in the case where A is the dihedral group Dn. One family of this
type is the first group of bicirculants BCn({0, 1− r}, {1, s}) shown in subsection 19.3.
The paper [40] shows several other algebraically defined structures. First, there are
examples for the group Dn where the swappers do not arise from group automorphisms.
Second, there is a Cayley construction for the structure RC(n, k) of the previous subsec-
tion.
Further, the body of the paper shows six ’linear’ constructions in which A is an exten-
sion of some Zkn:
Construction 19.1. For n and k both at least 3, let A be a semidirect product of Zkn with
the group generated by the permutation σ = (123 . . . k − 1k) acting on the coordinates.
Let e1 be the standard basis element (100 . . . 0), let R = {e1,−e1} and G = {σ, σ−1}.
We define the LR structure AffLR(n, k) to be Cay(A;R,G).
Construction 19.2. Let ProjLR(k, n) be AffLR(k, n) factored out by the cyclic group
generated by (1, 1, . . . , 1).
Construction 19.3. Let ProjLR◦(2k, n) be AffLR(2k, n) factored out by the group gen-
erated by all di = ei − ei+k, where ei is the standard basis element having a 1 in position
i and zeroes elsewhere.
Construction 19.4. Let A be a semidirect product of Z2k2 with the group generated by
the permutation γ = (1, 2, 3, . . . , k)(k + 1, k + 2, . . . , 2k). acting on the coordinates.
Let R = {e1, ek+1} and G = {γ, γ−1}. We define the LR structure AffLR2(k) to be
Cay(A;R,G).
Let d1 be the 2k-tuple in which the first k entries are 1 and the last k entries are 0; let d2
be the 2k-tuple in which the first k entries are 0 and the last k entries are 1; let d = d1 + d2.
Construction 19.5. Let ProjLR′(k) be AffLR2(k) factored out by the group generated by
d1 and d2.
Construction 19.6. Let ProjLR”(k) be AffLR2(k) factored out by the group generated
by d.
The paper [40] proves that all six of these constructions lead to suitable LR structures
(except for a few cases).
20 Base graph-connection graph constructions
This section deals with a family of constructions called base graph - connection graph
(BGCG) constructions. In these, we construct an edge-transitive bipartite graph Γ from a
set of copies of the subdivision (see section 3.2) of a base graph B, connected according
to a connection graph C and some other information.
For a tetravalent graphB, letB∗ be its subdivision, i.e., the graph resulting by replacing
each edge of B with a path of length 2. We refer to its 4-valent vertices as black and its
valence-2 vertices as white. We will often refer to an edge of B and the white vertex on the
path that replaces it in B∗ as being equivalent or corresponding.
26 Art Discrete Appl. Math. 3 (2020) #P1.08
In the constructions we often define a partition P of the edges of a graph (equivalently,
of the white vertices of its subdivision) into sets of size 2, and refer to such a P as a pairing.
We can regard the partition as a coloring of the edges of the graph. We can also think of
the pairing as an involutory permutation by writing y = xP when {x, y} ∈ P .
Given simple graphs B and C, let X be a disjoint union of copies of B, one copy Br
for each vertex r of C. Refer to this X as BC . A pairing P of X is compatible with C
provided that it satisfies these two conditions:
1. if {x, y} ∈ P with x in Br and y in Bs, r 6= s, then {r, s} is an edge of C, and
2. every edge of C is so represented.
These ingredients are combined in the following construction:
Construction 20.1. Given a tetravalent base graph B, a connection graph C, and a pair-
ing P on X = BC which is compatible with C, form a graph Γ by identifying each pair of
white vertices of X∗ which corresponds to an element of P . We refer to Γ as a BGCG of
B and C with respect to P , and write Γ = BGCG(B,C,P).
Theorem 20.2 ([49]). If B is tetravalent, C is a graph, and P is a pairing of BC which is
compatible with C, then Γ = BGCG(B,C,P) is a bipartite tetravalent graph.
Definition 20.3. If X is a (connected or disconnected) tetravalent graph and P is a pairing
onX , we will call P a dart-transitive pairing provided that the subgroup of Aut(X) which
preserves P is transitive on the darts of X .
Theorem 20.4 ([49]). If B and C are each dart-transitive and if P is a dart-transitive
pairing on X = BC which is compatible with C, and if Γ = BGCG(B,C,P) is defined,
then Γ is a bipartite edge-transitive graph.
Theorem 20.4 is the basis for several BGCG constructions, each depending on the na-
ture of C.
Theorem 20.5 (C = K1). Suppose that Q is a dart-transitive pairing on B. Then
BGCG(B,K1,Q) is an edge-transitive tetravalent graph.
Theorem 20.6 (C = K2). Suppose that Q is a dart-transitive pairing on B, and consider
the vertex set ofK2 to be {1, 2}. If e is an edge ofB, let (e, 1) and (e, 2) be the correspond-
ing edges in X = BK2 . Define P on X by (e, 1)P = (eQ, 2). Then BGCG(B,K2,P) is
an edge-transitive tetravalent graph.
No truly general technique yet exists in the case when C is the n-cycle Cn. We give
here two constructions from [56] and remark that there are many more for C = Cn.
Construction 20.7. Suppose that
1. Q is a dart-transitive pairing of B invariant under a dart-transitive group G,
2. there is a partition {R,G} of the edges ofB into two ‘colors’ (R= ‘red’, G= ‘green’)
which is also invariant under G,
3. each pair in Q meets bothR and G,
4. the vertex set of Cn is Zn.
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 27
Then we can form P1 and P2 on X = BCn as follows: for each pair {e, f} in Q, with
e ∈ R, and for each i ∈ Zn, we let (e, i)P1 = (f, i + 1). If n is even, then for i even, we
let (e, i)P2 = (e, i+ 1), and for i odd, we let (f, i)P2 = (f, i+ 1).
Theorem 20.8 (C = Cn). With P1 and P2 as above, BGCG(B,Cn,P1) and BGCG(B,
Cn,P2) are edge-transitive tetravalent graphs.
In the Census, we have no efficient way to describe the diferent pairs, (Q, {R,G}), and
so we simply number them. If the pair is numbered i, the Census reports BGCG(B,Cn,P1)
as “BGCG(B,Cn, i)” and BGCG(B,Cn,P2) as “BGCG(B,Cn, i′)”.
21 From regular maps
A map is an embedding of a graph or multigraph on a compact connected surface such that
each component of the complement of the graph (these are called faces) is topologically a
disk. A symmetry of a map is a symmetry of the graph which extends to a homeomorphism
of the surface. A map M is rotary provided that for some face and some vertex of that
face, there is a symmetry R which acts as rotation one step about the face and a symmetry
S which acts as rotation one step about the vertex. A mapM is reflexible provided that it
is rotary and has a symmetry X acting as a reflection fixing that face and vertex. IfM is
rotary but not reflexible, we call it chiral. See [15] for more details.
If M is rotary, its symmetry group, Aut(M), is transitive on faces and on vertices.
Thus all faces have the same number p of sides and all vertices have the same degree q. We
then say thatM has type {p, q}.
21.1 Underlying graphs
The underlying graph of a rotary mapM is called UG(M) and is always dart-transitive.
If q = 4, it belongs in this census.
21.2 Medial graphs
The vertices of the medial graph, MG(M), are the edges of M. Two are joined by
an edge if they are consecutive in some face (and so in some vertex). If M is rotary,
MG(M) is always edge-transitive. If M is reflexible or if it is self-dual in such a way
that some orientation-preserving homeomorphism of the surface interchangesM with its
dual, D(M), then MG(M) is dart-transitive. If not, it is quite often, but not quite always,
1
2 -transitive. No one seems to know a good criterion for this distinction.
21.3 Dart graphs
The vertices of the dart graph, DG(M), are the darts ofM. Two are joined by an edge if
they are head-to-tail consecutive in some face. The graph DG(M) is a twofold cover of
MG(M) and is often the medial graph of some larger rotary map. It can be dart-transitive
or 12 -transitive; again, no good criterion is known.
21.4 HC of maps
The Hill Capping of a rotary map M is defined in a way completely analogous to the
capping of a cubic graph Λ: we join {Ai, Bj} to {Bj , C1−i} where A,B and C are ver-
tices which are consecutive around some face. The graph HC(M) is a 4-fold covering
28 Art Discrete Appl. Math. 3 (2020) #P1.08
of MG(M) and can be dart-transitive or semisymmetric or 12 -transitive or even not edge-
transitive.
21.5 XI of maps
Suppose thatM is a rotary map of type {p, q} for some even q = 2n. Then each corner of
the map (formed by two consecutive edges in one face) is opposite at that vertex to another
corner; we will call such a pair of corners an ’X’. As an example, consider Figure 19, which
shows one vertex, of degree 6, in a map. The X’s are pairs a, b, c of opposite corners.
1 2
6 3
5 4
a b c
abc
Figure 19: X’s and I’s in a map
We form the bipartite graph XI(M) in this way: The black vertices are the X’s, the
white vertices are the edges of M, and edges of XI(M) are all pairs {x, e} where x is
an X and e is one of the four edges of x. Continuing our example, the black vertex a is
adjacent to white vertices 1, 3, 4, 6, while b is adjacent to 1, 2, 4, 5 and c to 2, 3, 5, 6.
It is interesting to see the construction introduced here as a special case of two previous
constructions. First it is made from MG(M) using a BGCG construction in which each
edge of MG(M) is paired with the one opposite it at the vertex of M containing the
corresponding corner.
Secondly, it is P of an LR structure called a locally dihedral cycle structure as outlined
at the end of [43].
It is clear that ifM is reflexible, then XI(M) is edge-transitive; it is surprising, though,
that sometimes (criteria still unknown) XI of a chiral map can also be edge-transitive.
22 Sporadic graphs
There are a few graphs in the Census which are given familiar names rather than a para-
metric form. These are: K5 = C5(1, 2), the Octahedron = K2,2,2. Also there is the graph
Odd(4); its vertices are subsets of {1, 2, 3, 4, 5, 6, 7} of size 3. Two are joned by an edge
when the sets are disjoint. Finally, there is the graph denoted Gray(4) due to a construc-
tion by Bouwer [7] which generalizes the Gray graph to make a semisymmetric graph of
valence n on 2nn vertices, in this case, 512. In fact we wanted to extend the Census to 512
vertices in order to include this graph.
Out of more than 7000 graphs in this Census, 400 of them have as their listed names one
of the tags AT[n, i],HT[n, i] or SS[n, i] from the computer-generated censi. These graphs,
then, must have no other known constructions. Each of these might be truly sporadic or,
perhaps, might belong to some interesting family not yet recognized.
P. Potočnik and S. Wilson: Recipes for edge-transitive tetravalent graphs 29
Each of these is a research project in its own right, a single example waiting to be
meaningfully generalized.
23 Open questions
1. In compiling the Census, whenever we wanted to include a parameterized family
(Cn(1, a) (section 5), BCn(a, b, c, d) (section 12), Prn(a, b, c, d) (section 13.1), etc.
), it was very helpful to have some established theorems which either completely
classified which values of the parameters gave edge-transitive graphs or restricted
those parameters in some way. In families for which no such theorems were known,
we were forced into brute-force searches, trying all possible values of the parameters.
In many cases this caused our computers to run out of time or space before finishing
the search. There are many families for which no such results or only partial results
exist. So, our first and most pressing question is :
For which values of parameters are the following graphs edge-transitive (or LR):
AMC(k, n,M) (section 9),
MSZ(m,n; k, r) (section 13.2.2),
MC3(m,n, a, b, r, t, c) (section 13.2.3),
LoPrn(a, b, c, d, e) (section 13.3),
WHn(a, b, c, d) (section 13.3),
KEn(a, b, c, d, e) (section 13.3),
Curtainn(a, b, c, d, e) (section 13.3),
CPM(n, s, t, r) (section 15).
2. The general class IV metacirculants (of which the graphs MSZ are merely a part)
has begun to be explored. It is parameterized in [1], where the authors point out that
not all values of the parameters which make the graph metacirculent make it 12 -arc-
transitive. Then the same questions as above are relevant: When are these graphs
isomorphic? dart-transitive? 12 -arc-transitive? LR structures?
3. Given a and n with a2 ≡ ±1 (mod n), which toroidal graph is isomorphic to
Cn(1, a)? See sections 5 and 6. The toroidal graphs are very common and almost
every family includes some as special cases. In researching a family, we often point
out that certain values of the parameters give toroidal graphs and so will not be stud-
ied with this family. However, it is often difficult to say exactly which toroidal graph
is given by the indicated parameters. This is simply the first of many such questions.
4. Under what conditions on their parameters can two spidergraphs be isomorphic?
This is a question mentioned in [54]. Many examples of isomorphism theorems are
given there as well as examples which show that not all isomorphisms have been
found.
5. The Attebery construction presents many challenges. First, for the general construc-
tion, under what conditions are the graphs Att(A, T, k; a, b) and Att(A, T ′, k; a′, b′)
isomorphic? It is clear that if P is any automorphism of A, then Att(A, T, k; a, b) ∼=
Att(A,P−1TP, k; aP, bP ), but it is almost certain that there are other isomorphisms.
Even in the special AMC construction, we do not know when AMC(k, n,M) can
be isomorphic to AMC(k′, n′,M ′). This question must be answered before we can
make much progress on other aspects of the Attebery graphs.
30 Art Discrete Appl. Math. 3 (2020) #P1.08
6. Some graphs with the same diagrams as the metacirculants PS,MPS,MSY,MSZ,MC3
are not themselves metacirculants but are nevertheless edge-transitive. For example
consider the graph KE12(1, 3, 8, 5, 1) It is isomorphic to the graph whose vertices
are Z4 × Z12, with each (1, i) adjacent to (2, i) and (2, i + 1), each (2, i) to (3, i)
and (3, i + 4), each (3, i) to (4, i + 4) and (4, i + 5), and each (4, i) to (1, i) and
(1, i + 10). This diagram is the same ‘sausage graph’ that characterizes the PS and
MPS graphs and yet the graph is not isomorphic to any PS or MPS graph. This
happens rarely enough that the exceptional cases might be classifiable.
7. The Praeger-Xu graphs, including W(n, 2) and R2m(m + 2,m + 1), present many
problems computationally. Some aspects, such as semitransitive orientations, cycle
structures and regular maps have been addressed in [16]. Can we establish theo-
rems determining their dart-transitive colorings, their BGCG dissections, and other
properties?
8. When do the constructions DG, HC, TAG, applied to some cubic graph Λ or some
rotary mapM, simply result in the line graph or medial graph of some larger graph
or map?
9. The BGCG constructions we have used here are only the beginning of this topic. We
have given some constructions for cases where the connection graph is K1,K2, or
Ck, but for other connection graphs, we have no general techniques at all.
10. How can XI of a chiral map be edge-transitive?
11. The 3-arc graph of a cubic graph (see section 17.4) is the partial line graph of some
cycle decomposition; what decomposition? . . . of what graph?
12. In section 19, we use cycle structures to construct LR structures. What is an efficient
way to find all isomorphism classes of cycle structures for a given dart-transitive
tetravalent graph?
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ISSN 2590-9770
The Art of Discrete and Applied Mathematics 3 (2020) #P1.09
https://doi.org/10.26493/2590-9770.1381.332
(Also available at http://adam-journal.eu)
Observations and answers to questions about
edge-transitive maps∗
Marston D.E. Conder† , Isabel Holm
Department of Mathematics, University of Auckland
Private Bag 92019, Auckland 1142, New Zealand
Thomas W. Tucker‡
Department of Mathematics, Colgate University, Hamilton, NY, USA
Received 21 February 2019, accepted 7 January 2020, published online 8 September 2020
Abstract
A map is a 2-cell embedding of a connected graph or multigraph on a closed surface,
and a map is called edge-transitive if its automorphism group has a single orbit on edges.
There are 14 classes of edge-transitive maps, determined by the effect of the automorphism
group. In this paper we make some observations about these classes, and answer three open
questions from a 2001 paper by Širán, Tucker and Watkins, by showing that (a) in each of
the classes 1, 2P , 2P ex, 3, 4P and 5P , there exists a self-dual edge-transitive map, (b)
there exists an edge-transitive map with simple underlying graph on an orientable surface
of genus g for every integer g ≥ 0, and (c) there exists an orientable surface that carries an
edge-transitive map of each of the 14 classes, and indeed that these three things still hold
when we insist that both the map and its dual have simple underlying graph. We also give
the maximum number of automorphisms of an edge-transitive map on an orientable surface
of given genus g > 1, and consider some special cases in which the automorphism group
(or its subgroup of orientation-preserving automorphisms) is prescribed. For example, we
show that a certain soluble group of order 576 is the smallest group that occurs as the
automorphism group of some edge-transitive map in each of the 14 classes.
Keywords: Graph embedding, regular map, automorphism, edge-transitive, simple underlying graph,
duality.
Math. Subj. Class. (2020): 57M15, 05C10, 05E18, 57M60
∗The authors acknowledge helpful use of the MAGMA system [1] in constructing examples, searching for
patterns in the resulting output, testing and confirming conjectures, and analysing specific examples.
†Research supported in part by the N. Z. Marsden Fund, Grant UOA 1626
‡The author has been supported by Simons Foundation Award 317689.
E-mail address: m.conder@auckland.ac.nz (Marston D.E. Conder), ihol325@aucklanduni.ac.nz (Isabel
Holm), ttucker@colgate.edu (Thomas W. Tucker)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 3 (2020) #P1.09
1 Introduction
A regular map is a symmetric embedding of a connected graph or multigraph on a closed
surface: the automorphism group of the embedding has a single orbit on ‘flags’ (which are
like incident vertex-edge-face triples), or more loosely on ‘arcs’ (which are incident vertex-
edge pairs). The theory of such discrete objects has a long and interesting history, dating
back to the work of Brahana, Burnside, Dyck and others, and recent work has produced
infinite families of examples [10] as well as complete lists of those on hyperbolic surfaces
of small genera [3, 5, 6].
In contrast, relatively little is known about the more general case of edge-transitive
maps, for which the automorphism group has a single orbit on edges. By work of Graver
and Watkins [11], it is known that these can be divided into 14 classes according to certain
properties (determined by automorphisms preserving a given edge or one of the vertices or
faces incident with it). That work was taken further in [18] by Širán, Tucker and Watkins,
who showed there exist finite maps in each class. They also posed a number of questions,
some of which were answered by Alen Orbanić in his 2006 PhD thesis. Further questions
were also posed by Orbanić et al in [17]. Some of the remaining questions and related
ones were discussed at a BIRS workshop (on Symmetries of surfaces, maps and dessins) at
Banff in September 2017, and in this paper we present the answers to many of those.
In particular, we answer Questions 3, 4 and 6 from [18], by showing the following:
(a) In each of the classes 1, 2P , 2P ex, 3, 4P and 5P , there exists a self-dual edge-
transitive map, indeed one for which the map is non-degenerate, in that both the map
and its dual have simple underlying graph;
(b) There exists a non-degenerate edge-transitive map on an orientable surface of genus
g for every integer g ≥ 0, and
(c) There exists an orientable surface that carries a edge-transitive map of each of the 14
classes, and indeed a non-degenerate one of each class.
We also give the maximum number of automorphisms of an edge-transitive map on an
orientable surface of given genus g > 1, and consider some special cases in which the au-
tomorphism group (or its subgroup of orientation-preserving automorphisms) is prescribed.
For example, we show that a certain soluble group of order 576 is the smallest group that
occurs as the automorphism group of some edge-transitive map in each of the 14 classes.
Before doing that in Sections 3 to 8, we provide some further information about the 14
classes, including ‘universal’ groups that determine the effect of the automorphism group
of every map in the class, and then we conclude the papers with some remarks and further
questions.
Many of the findings we describe in this paper resulted from computations involving the
universal groups and their quotients, using the MAGMA system, or were guided by them. In
cases where the outcomes depended almost entirely on computations, we summarise them
in tables in an Appendix at the end.
2 Details and properties of the 14 classes
In this section we present some background information on each class that can be helpful in
constructing or analysing examples of edge-transitive maps. Much of this information can
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 3
also be found in references [11] or [18] or [16] or [17], but unfortunately the table in [11]
that was copied as Table I in [18] contains a number of shortcomings that make it difficult
to follow. First, the columns headed Gv and Gf refer to the stabiliser of a particular vertex
v or face f , but for some classes the stabilisers of the vertex u and/or face g should be given
as well. And in fact the column headed Gv looks more appropriate for Gu, but even then,
its entry for class 4∗ should be 〈σ 4u , θug〉, not 〈σ 4u , θuf 〉. Also in Table II of [18], some
relations are missing, namely τ2 = 1 for type 4, and λ2 = 1 for type 4∗, and φ2 = 1 for
type 4P . The information below is more accurate and comprehensive.
Figure 1: Reflecting generators for the universal group of class 1
In each case we let e = {u, v} be a given edge, and let f and g be the faces incident with
e, as illustrated in Figure 1.
Associated with each class is a universal group U , which has the property that if M
is any ET map of the given class, then its automorphism group Aut(M) is a quotient
of U . This group U is generated by particular elements that can be described in terms
of their effect on the vertices, edges and faces labelled in Figure 1. Here we compose
automorphisms from left to right.
Conversely, if A is any quotient of U in which the images of the generators and certain
other elements have the appropriate orders (to avoid collapse), then there exists an ET map
M on which A acts edge-transitively as a group of automorphisms, in the appropriate way.
This map can be constructed using cosets of the stabilisers of the vertices u and v, the
edge e and the faces f and g coming from Figure 1, in a similar way to the well-known
construction for regular maps from groups (see [3], for example).
Class 1 consists of the fully regular maps. In this case, the universal group is
U1 = 〈 a, b, c | a2 = b2 = c2 = (ac)2 = 1 〉
where a, b and c are automorphisms that act like local reflections, such that the stabilisers
of the vertex u, edge e and face f are the subgroups generated by {b, c}, {a, c} and {a, b}
respectively. (These correspond to the elements λe, θuf and τe in the notation of [18].)
The universal group U for any one of the 14 classes can be embedded into the group
U1 above, as a subgroup of index dividing 4, with transversal a subgroup of 〈a, c〉. We
give this embedding, which is unique except for classes 4, 4∗ and 4P , where there are
two possibilities (that can be interchanged under conjugation by a, c and either a or c,
respectively).
4 Art Discrete Appl. Math. 3 (2020) #P1.09
In each case, we also give generators for the orientation-preserving subgroup (which
is the intersection of U with the subgroup 〈ab, bc〉 of U1) in the orientable case, and state
whether or not the automorphism group of a map in the given class acts transitively on
vertices, on faces, and/or on Petrie polygons of the map. Note that when the action (on
vertices, faces, or Petrie polygons) is not transitive, there are two orbits (since each edge is
incident with at most two vertices, and most two faces, and at most two Petrie polygons).
Then we describe the stabilisers of the vertices u and v, the edge e, and the faces f and g,
in terms of both generators for U and generators for U1. We also indicate the effect of an
orientation-reversing element (usually but not always b) by conjugation on the generators
for the orientation-preserving subgroup, when such an element exists, and similarly, the
effect of a map duality on the chosen set of generators for U , in the orientable case.
Finally, for each element g ∈ {a, c, ac} in the stated transversal for U in U1 (but lying
outside U ), we describe the effect of the automorphism ψg of U induced by conjugation
by g, taking each generator h of U to g−1hg. These ψg are what we may call ‘barred’
automorphisms, in the sense that if A is the automorphism group of an ET map in the
given class, then A has no automorphism that has the same effect on the images in A of the
generators of U as ψg has on the generators themselves.
Class 1
Universal group: U = 〈x, y, z | x2 = y2 = z2 = (xz)2 = 1 〉;
Embedding of generators in U1 : (x, y, z) 7→ (a, b, c), with transversal {1};
Orientation-preserving subgroup U+ is generated by r = xy = ab and s = xz = ac
(and r−1s = yz = bc), subject to the single relation s2 = 1;
Automorphism group of map is transitive on vertices, faces and Petrie polygons;
Stabiliser of vertex u is generated by {y, z} = {b, c};
Stabiliser of vertex v is generated by {xyx, z} = {aba, c};
Stabiliser of edge e is generated by {x, z} = {a, c};
Stabiliser of face f is generated by {x, y} = {a, b};
Stabiliser of face g is generated by {x, zyz} = {a, cbc};
A reflection (by a) takes (r, s) = (ab, ac) 7→ (ba, ca) = (r−1, s−1);
An orientable duality takes (a, b, c) 7→ (c, b, a), preserving class 1;
There are no barred automorphisms.
Class 2
Universal group: U = 〈x, y, z | x2 = y2 = z2 = 1 〉;
Embedding of generators in U1 : (x, y, z) 7→ (b, c, aba), with transversal {1, a};
Orientation-preserving subgroup U+ is generated by
r=xy = bc and s=zy = abac = abca, and is free of rank 2;
Automorphism group of map is transitive on faces and Petrie polygons but not on vertices;
Stabiliser of vertex u is generated by {x, y} = {b, c};
Stabiliser of vertex v is generated by {y, z} = {c, aba};
Stabiliser of edge e is generated by {y} = {c};
Stabiliser of face f is generated by {x, z} = {b, aba};
Stabiliser of face g is generated by {yxy, yzy} = {cbc, acbca} = {cbc, cabac};
A reflection (by c) takes (r, s) = (bc, abac) 7→ (cb, caba) = (r−1, s−1);
An orientable map duality takes (b, c, aba) 7→ (b, a, cbc), interchanging classes 2 and 2∗;
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 5
Extending automorphism ψa takes (x, y, z) = (b, c, aba) 7→ (aba, c, b) = (z, y, x).
Class 2∗
Universal group: U = 〈x, y, z | x2 = y2 = z2 = 1 〉;
Embedding of generators in U1 : (x, y, z) 7→ (a, b, cbc), with transversal {1, c};
Orientation-preserving subgroup U+ is generated by r = xy = ab and s = yz = (bc)2,
and is free of rank 2;
Automorphism group of map is transitive on vertices and Petrie polygons but not on faces;
Stabiliser of vertex u is generated by {y, z} = {b, cbc};
Stabiliser of vertex v is generated by {xyx, xzx} = {aba, acbca} = {aba, cabac};
Stabiliser of edge e is generated by {x} = {a};
Stabiliser of face f is generated by {x, y} = {a, b};
Stabiliser of face g is generated by {x, z} = {a, cbc};
A reflection (by b) takes (r, s) = (ab, (bc)2) 7→ (ba, (cb)2) = (r−1, s−1);
An orientable map duality takes (a, b, cbc) 7→ (c, b, aba), interchanging classes 2∗ and 2;
Extending automorphism ψc takes (x, y, z) = (a, b, cbc) 7→ (a, cbc, b) = (x, z, y).
Class 2P
Universal group: U = 〈x, y, z | x2 = y2 = z2 = 1 〉;
Embedding of generators in U1 : (x, y, z) 7→ (ac, b, cbc), with transversal {1, a} or {1, c};
Orientation-preserving subgroup U+ is generated by r = x = ac, s = yz = (bc)2 and
t = yxz = babc, subject to the two relations r2 = (st−1)2 = 1;
Automorphism group of map is transitive on vertices and faces but not on Petrie polygons;
Stabiliser of vertex u is generated by {y, z} = {b, cbc};
Stabiliser of vertex v is generated by {xzx, xyx} = {aba, acbca} = {aba, cabac};
Stabiliser of edge e is generated by {x} = {ac};
Stabiliser of face f is generated by {y, xzx} = {b, aba};
Stabiliser of face g is generated by {z, xyx} = {cbc, acbca} = {cbc, cabac};
A reflection (by b) takes
(r, s, t) = (ac, (bc)2, babc) 7→ (bacb, (cb)2, abcb) = (ts−1, s−1, rs−1);
An orientable map duality takes (ac, b, cbc) 7→ (ac, b, aba), preserving class 2P ;
Extending automorphism ψc takes (x, y, z) = (ac, b, cbc) 7→ (ca, cbc, b) = (x, z, y).
Class 2ex
Universal group: U = 〈x, y | x2 = 1 〉;
Embedding of generators in U1 : (x, y) 7→ (c, ab), with transversal {1, a} or {1, b};
Orientation-preserving subgroup U+ is generated by r = y = ab and s = xyx = cabc,
and is free of rank 2;
Automorphism group of map is transitive on vertices, faces and Petrie polygons;
Stabiliser of vertex u is generated by {x, y−1xy} = {c, bcb};
Stabiliser of vertex v is generated by {x, yxy−1} = {c, abcba} = {c, (aba)c(aba)};
Stabiliser of edge e is generated by {x} = {c};
Stabiliser of face f is generated by {y} = {ab};
Stabiliser of face g is generated by {xyx} = {cabc} = {acbc};
A reflection (by c) takes (r, s) = (ab, cabc) 7→ (cabc, ab) = (s, r);
6 Art Discrete Appl. Math. 3 (2020) #P1.09
An orientable map duality takes (c, ab) 7→ (a, cb), interchanging classes 2ex and 2∗ex;
Extending automorphism ψa takes (x, y) = (c, ab) 7→ (c, ba) = (x−1, y−1).
Class 2∗ex
Universal group: U = 〈x, y | x2 = 1 〉;
Embedding of generators in U1 : (x, y) 7→ (a, bc), with transversal {1, b} or {1, c};
Orientation-preserving subgroup U+ is generated by r = y = bc and s = xyx = abca,
and is free of rank 2;
Automorphism group of map is transitive on vertices, faces and Petrie polygons;
Stabiliser of vertex u is generated by {y} = {bc};
Stabiliser of vertex v is generated by {xyx} = {abca} = {abac};
Stabiliser of edge e is generated by {x} = {a};
Stabiliser of face f is generated by {x, yxy−1} = {a, bab};
Stabiliser of face g is generated by {x, y−1xy} = {a, cbabc};
A reflection (by a) takes (r, s) = (bc, abca) 7→ (abca, bc) = (s, r);
An orientable map duality takes (a, bc) 7→ (c, ba), interchanging classes 2∗ex and 2ex;
Extending automorphism ψc takes (x, y) = (a, bc) 7→ (a, cb) = (x−1, y−1).
Class 2P ex
Universal group: U = 〈x, y | x2 = 1 〉;
Embedding of generators in U1 : (x, y) 7→ (ac, ab), with transversal {1, a};
Orientation-preserving subgroup U+ is equal to U ;
Automorphism group of map is transitive on vertices, faces and Petrie polygons;
Stabiliser of vertex u is generated by {y−1x} = {bc};
Stabiliser of vertex v is generated by {yx} = {abac} = {abca};
Stabiliser of edge e is generated by {x} = {ac};
Stabiliser of face f is generated by {y} = {ab};
Stabiliser of face g is generated by {xyx} = {cbac} = {cbca};
There is no orientation-reversing automorphism in U ;
An orientable map duality takes (ac, ab) 7→ (ac, bc), preserving class 2P ex;
Extending automorphism ψa takes (x, y) = (ac, ab) 7→ (ca, ba) = (x−1, y−1).
Class 3
Universal group: U = 〈x, y, z, w | x2 = y2 = z2 = w2 = 1 〉;
Embedding of generators in U1 : (x, y, z, w) 7→ (b, cbc, acbca, aba),
with transversal {1, a, c, ac};
This group U is also a subgroup of index 2 in the universal groups for classes 2, 2∗ and 2P
(but not in the universal groups for classes 2ex, 2∗ex or 2P ex);
Orientation-preserving subgroup U+ is generated by r = wx = (ab)2 and s = xy = (bc)2
and t = zx = (acb)2 = (cab)2, and is free of rank 3;
Automorphism group of map is not transitive on vertices, faces or Petrie polygons;
Stabiliser of vertex u is generated by {x, y} = {b, cbc};
Stabiliser of vertex v is generated by {w, z} = {aba, acbca} = {aba, cabac};
Stabiliser of edge e is trivial;
Stabiliser of face f is generated by {x,w} = {b, aba};
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 7
Stabiliser of face g is generated by {y, z} = {cbc, acbca};
A reflection (by b) takes
(r, s, t) = ((ab)2, (bc)2, (acb)2) 7→ ((ba)2, (cb)2, (bac)2) = (r−1, s−1, t−1);
An orientable duality takes (b, cbc, acbca, aba) 7→ (b, aba, acbca, cbc) or (acbca, cbc, b, aba),
preserving class 3;
Extending automorphisms ψa, ψc and ψac respectively take (x, y, z, w) = (b, bc, bac, ba)
to (ba, bac, bc, b), (bc, b, ba, bac) and (bac, ba, b, bc),
that is, to (w, z, y, x), (y, x, w, z), and (z, w, x, y).
Class 4
Universal group: U = 〈x, y, z | x2 = y2 = 1 〉
Embedding of generators in U1 : (x, y, z) 7→ (b, cbc, acba), with transversal {1, a, c, ac};
This group U is also a subgroup of index 2 in the universal group for class 2
(but not in the universal groups for classes 2∗, 2P , 2ex, 2∗ex or 2P ex);
Orientation-preserving subgroup U+ is generated by
r = yx = (cb)2 and s = z = acba and t = yzx = cbabab, and is free of rank 3;
Automorphism group of map is transitive on faces and Petrie polygons but not on vertices;
Stabiliser of vertex u is generated by {x, y} = {b, cbc};
Stabiliser of vertex v is generated by {z} = {acba} = {caba};
Stabiliser of edge e is trivial;
Stabiliser of face f is generated by {x, z−1yz} = {b, abababa};
Stabiliser of face g is generated by
{y, zxz−1} = {cbc, acbababca} = {cbc, a(cbc)a(cbc)a(cbc)a};
A reflection (by b) takes
(r, s, t) = ((cb)2, acba, cbabab) 7→ ((bc)2, bacbab, bcbaba) = (r−1, r−1t, r−1s);
An orientable duality takes (b, cbc, acba) 7→ (b, aba, cabc), interchanging classes 4 and 4∗;
Extending automorphismψc takes (x, y, z) = (b, cbc, acba) 7→ (cbc, b, abac) = (y, x, z−1).
Class 4∗
Universal group: U = 〈x, y, z | x2 = y2 = 1 〉;
Embedding of generators in U1 : (x, y, z) 7→ (b, aba, cabc), with transversal {1, a, c, ac};
This group U is also a subgroup of index 2 in the universal group for class 2∗
(but not in the universal groups for classes 2, 2P , 2ex, 2∗ex or 2P ex);
Orientation-preserving subgroup U+ is generated by
r = yx = (ab)2 and s = z = cabc and t = yzx = abcbcb, and is free of rank 3;
Automorphism group of map is transitive on vertices and Petrie polygons but not on faces;
Stabiliser of vertex u is generated by {x, z−1yz} = {b, cbcbcbc};
Stabiliser of vertex v is generated by
{y, zxz−1} = {aba, cabcbcbac} = {aba, c(aba)c(aba)c(aba)c};
Stabiliser of edge e is trivial;
Stabiliser of face f is generated by {x, y} = {b, aba};
Stabiliser of face g is generated by {z} = {cabc} = {acbc};
A reflection (by b) takes
(r, s, t) = ((ab)2, cabc, abcbcb) 7→ ((ba)2, bacbcb, babcbc) = (r−1, r−1t, r−1s);
An orientable duality takes (b, aba, cabc) 7→ (b, cbc, acba), interchanging classes 4∗ and 4;
8 Art Discrete Appl. Math. 3 (2020) #P1.09
Extending automorphismψa takes (x, y, z) = (b, aba, cabc) 7→ (aba, b, cbca) = (y, x, z−1).
Class 4P
Universal group: U = 〈x, y, z | x2 = y2 = 1 〉;
Embedding of generators in U1 : (x, y, z) 7→ (b, acbca, abc), with transversal {1, a, c, ac};
This group U is also a subgroup of index 2 in the universal group for class 2P
(but not in the universal groups for classes 2, 2∗, 2ex, 2∗ex or 2P ex);
Orientation-preserving subgroup U+ is generated by
r = zx = abcb and s = xz = babc and t = zy = ababac, and is free of rank 3;
Automorphism group of map is transitive on vertices and faces but not on Petrie polygons;
Stabiliser of vertex u is generated by {x, z−1yz} = {b, cbcbcbc};
Stabiliser of vertex v is generated by {y, zxz−1} = {acbca, abcbcba} = {cabac,
(aba)c(aba)c(aba)};
Stabiliser of edge e is trivial;
Stabiliser of face f is generated by {x, zyz−1} = {b, abababa};
Stabiliser of face g is generated by {y, z−1xz} = {acbca, cbababc};
A reflection (by b) takes
(r, s, t) = (abcb, babc, ababac) 7→ (babc, abcb, bababacb) = (s, r, st−1r);
An orientable duality takes (b, acbca, abc) 7→ (b, acbca, cba) or (acbca, b, abc),
preserving class 4P ;
Extending automorphism ψac takes
(x, y, z) = (b, acbca, abc) 7→ (acbca, b, cba) = (y, x, z−1).
Class 5
Universal group: U = 〈x, y | − 〉 (free of rank 2);
Embedding of generators in U1 : (x, y) 7→ (bc, abca), with transversal {1, a, c, ac};
This group U is also a subgroup of index 2 in the universal groups for classes 2, 2∗ex
and 2P ex (but not in the universal groups for classes 2∗, 2P or 2ex);
Orientation-preserving subgroup U+ is equal to U ;
Automorphism group of map is transitive on faces and Petrie polygons but not on vertices;
Stabiliser of vertex u is generated by {x} = {bc};
Stabiliser of vertex v is generated by {y} = {abca} = {abac};
Stabiliser of edge e is trivial;
Stabiliser of face f is generated by {yx−1} = {(ab)2};
Stabiliser of face g is generated by {y−1x} = {c(ab)2c} = {a(cbc)a(cbc)};
There is no orientation-reversing automorphism in U ;
An orientable duality takes (bc, abca) 7→ (ba, cbac), interchanging classes 5 and 5∗;
Extending automorphisms ψa, ψc and ψac respectively take (x, y) = (bc, abca)
to (abca, bc) = (y, x), (cb, caba) = (x−1, y−1) and (acba, cb) = (y−1, x−1).
Class 5∗
Universal group: U = 〈x, y | − 〉 (free of rank 2);
Embedding of generators in U1 : (x, y) 7→ (ab, cabc), with transversal {1, a, c, ac};
This group U is also a subgroup of index 2 in the universal groups for classes 2∗, 2ex
and 2P ex (but not in the universal groups for classes 2, 2P or 2∗ex);
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 9
Orientation-preserving subgroup U+ is equal to U ;
Automorphism group of map is transitive on vertices and Petrie polygons but not on faces;
Stabiliser of vertex u is generated by {x−1y} = {(bc)2};
Stabiliser of vertex v is generated by {xy−1} = {a(bc)2a} = {(aba)c(aba)c};
Stabiliser of edge e is trivial;
Stabiliser of face f is generated by {x} = {ab};
Stabiliser of face g is generated by {y} = {cabc} = {acbc};
There is no orientation-reversing automorphism in U ;
An orientable duality takes (ab, cabc) 7→ (cb, acba), interchanging classes 5∗ and 5;
Extending automorphisms ψa, ψc and ψac respectively take (x, y) = (ab, cabc)
to (ba, cbca) = (x−1, y−1), (cabc, ab) = (y, x) and (cbac, ba) = (y−1, x−1).
Class 5P
Universal group: U = 〈x, y | − 〉 (free of rank 2);
Embedding of generators in U1 : (x, y) 7→ (abc, acb), with transversal {1, a, c, ac};
This group U is also a subgroup of index 2 in the universal groups for classes 2P , 2ex
and 2∗ex (but not in the universal groups for classes 2, 2∗ or 2P ex);
Orientation-preserving subgroup U+ is generated by
r = xy = (ab)2 and s = y−1x = (bc)2 and t = xy−1 = a(bc)2a, and is free of rank 3;
Automorphism group of map is transitive on vertices and faces but not on Petrie polygons;
Stabiliser of vertex u is generated by {y−1x} = {(bc)2};
Stabiliser of vertex v is generated by {xy−1} = {a(bc)2a} = {(aba)c(aba)c};
Stabiliser of edge e is trivial;
Stabiliser of face f is generated by {xy} = {(ab)2};
Stabiliser of face g is generated by {yx} = {c(ab)2c} = {a(cbc)a(cbc)};
Conjugation by the orientation-reversing automorphism x = abc takes
(r, s, t) = ((ab)2, (bc)2, a(bc)2a) 7→ (cababc, cbabcbcabc, bcbc) = (t−1rs, s−1r−1trs, s);
An orientable duality takes (abc, acb) 7→ (cba, acb) or (abc, bca), preserving class 5P ;
Extending automorphisms ψa, ψc and ψac respectively take (x, y) = (abc, acb)
to (bca, cba) = (y−1, x−1), (cab, abc) = (y, x) and (cba, bac) = (x−1, y−1).
It can also be helpful to see how the 14 universal subgroups can be embedded not just
in U1 but also in each other. These inclusions are illustrated in Figure 2.
Indeed here we may note that the 14 classes correspond precisely to the 14 conjugacy
classes of subgroups of U1 that are complementary to some subgroup of the edge-stabiliser
〈a, c〉, and that this gives a purely algebraic way of finding them, much more easily than in
the approach taken in [11]. It is also easy to determine these classes using the combinatorial
approach of ‘symmetry type graphs’ in [17].
3 Maximum orders of automorphism groups for ET maps of genus
greater than 1
Theorem 3.1. Let A be a group of automorphisms of an edge-transitive map on some
orientable surface of genus g > 1, or some non-orientable surface of genus p > 2. Then
|A| ≤ |A|maxo or |A| ≤ |A|maxnono, respectively, where |A|maxo and |A|maxnono are given
in Table 1 for each of the 14 classes of edge-transitive maps. Moreover, these bounds are
10 Art Discrete Appl. Math. 3 (2020) #P1.09
1
2∗2 2P 2ex 2∗ex 2P ex
3 4 4∗ 4P 5 5∗ 5P
ψcψa ψc ψa ψc ψa
Figure 2: Inclusions among the universal groups of the 14 classes
sharp for certain values of g and p in each class.
Proof. First, we might as well take A as the full automorphism group of the map in each
case, and then the bounds can be proved easily using the orders of stabilisers of vertices,
edges and faces, and the Euler-Poincaré formula.
For example, in class 1 we know that if k = o(xy) = o(ab) and m = o(yz) = o(bc),
then |V | = |A :Au| = |A|/k and |E| = |A :Ae| = |A|/4 and |F | = |A :Af | = |A|/m,
so the Euler characteristic is χ = |V | − |E| + |F | = |A|(1/k − 1/4 + 1/m). Then the
maximum possible negative value of this is −|A|/84, achievable when {k,m} = {3, 7},
and giving |A| ≤ −84χ = 168(g − 1) or 84(p − 2) in the orientable and non-orientable
cases, respectively. Similarly, for class 4, we have |V | = |A :Au|+ |A :Av| = |A|/o(z) +
|A|/(2o(xy)) and |E| = |A| and |F | = |A : Af | = |A|/o(xzyz−1), and this gives the
maximum possible negative value of χ = |V | − |E| + |F | as −|A|/24, achievable when
(o(z), o(xy), o(xzyz−1)) = (3, 4, 1) or (3, 1, 4). The other cases are similar, and are left
as an exercise for the reader.
Sharpness can be proved by exhibiting examples for which the bounds are attained.
If M is a fully regular map of type {3, 7} or {7, 3}, such as Klein’s map of genus 3,
then Aut(M) is generated by elements a, b, c satisfying a2 = b2 = c2 = (ab)3 = (bc)7 =
(ac)2 = 1, and |Aut(M)| = 168(g−1) or 84(p−2), and this proves sharpness for class 1.
But then also the triple (x, y, z) = (a, b, c) satisfies the defining relations for the universal
group of class 2 maps, but there exists no automorphism of the group that takes (x, y, z) to
(z, y, x), since o(xy) = 3 while o(zy) = o(yz) = 7. Hence this triple gives an ET map in
class 2, with χ = (|A|/6 + |A|/14) − |A|/2 + |A|/4 = −|A|/84, and therefore the same
genus and the same number of automorphisms as M . Also its dual is in class 2∗ and has
the same genus and same automorphism group as well.
Similarly, if M is an orientably-regular but chiral map of type {3, 7} or {7, 3}, then
it has class 2P ex and 84(g − 1) automorphisms. Also any irreflexible generating pair
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 11
Classes |A|maxo |A|maxnono Sufficient conditions for achieving maximum
1 168(g − 1) 84(p− 2) (o(xy), o(yz)) = (3, 7) or (7, 3)
2, 2∗ 168(g − 1) 84(p− 2) {o(xy), o(yz), o(xz)} = {2, 3, 7}
2P 24(g − 1) 12(p− 2) (o(yz), o(xyxz)) = (2, 3) or (3, 2)
2ex, 2∗ex 48(g − 1) 24(p− 2) (o(y), o(xy−1xy)) = (3, 4)
2P ex 84(g − 1) N/A {o(y), o(xy)} = {3, 7}
3 24(g − 1) 12(p− 2) {o(xy), o(zw), o(yz), o(xw)} = {2, 2, 2, 3}
4, 4∗ 48(g − 1) 24(p− 2) (o(z), o(xy), o(xzyz−1))
= (3, 4, 1) or (3, 1, 4)
4P 8(g − 1) 4(p− 2) (o(xz−1yz), o(xzyz−1)) = (1, 2) or (2, 1)
5, 5∗ 84(g − 1) N/A {o(x), o(y), o(xy−1)} = {2, 3, 7}
5P 12(g − 1) 6(p− 2) {o(xy), o(xy−1)} = {2, 3}
Table 1: Bounds on the number of automorphisms
(x, y) for Aut(M) such that x3 = y7 = (x−1y)2 = 1 satisfies the conditions for a
canonical generating pair for the automorphism group of a map of class 5 or 5∗, with
χ = (|A|/3 + |A|/7) − |A| + |A|/2 = −|A|/42, and hence the same genus and same
number of automorphisms as M .
Small examples of maps in classes 2P , 3 and 4P attaining the upper bound on the num-
ber automorphisms can be found in Sections 4.5 of Alen Orbanić’s thesis [16]. In particular,
his lists include orientable maps with (g, |A|) = (6, 60), (2, 24) and (15, 112), respectively,
and non-orientable maps with (p, |A|) = (30, 168), (3, 12) and (16, 56), respectively.
Additional computations (using MAGMA [1]) give examples attaining the bounds for
the remaining five classes. In class 2ex, there exists an orientable map of genus g = 26 with
|A| = 1200 = 48(g−1), and a non-orientable map of genus p = 1346 with |A| = 32256 =
24(p− 2), and their duals give corresponding examples in class 2∗ex. Also the same thing
happens in classes 4 and 4∗. Finally, in class 5P there exists an orientable map of genus
g = 14 with |A| = 156 = 12(g − 1), as given also in [16, §4.5], and a non-orientable map
of genus p = 226 with |A| = 1344 = 6(p− 2). This completes the proof. 
Many other examples than those mentioned in the last two paragraphs of the above
proof can be found. Further details are available form the first author upon request.
4 Self-dual non-degenerate orientable ET maps
In this brief section, we answer Question 6 of [18], about self-duality for ET maps in classes
1, 2P , 2P ex, 3, 4P and 5P . In fact we can do even more, by proving the following:
Theorem 4.1. In each of the classes 1, 2P , 2P ex, 3, 4P and 5P , there exist self-dual edge-
transitive orientable maps such that the map and its dual have simple underlying graph.
Proof. Computations using MAGMA produce the non-degenerate self-dual maps
summarised in Table 3, with defining relations for their automorphism groups given in
Table 4, in the Appendix. 
12 Art Discrete Appl. Math. 3 (2020) #P1.09
5 Edge-transitive maps with simple underlying graphs
Question 3 in [18] asked the following: Does every closed orientable surface support some
non-degenerate, edge-transitive map? (It is not difficult to construct degenerate regular
maps (of class 1) on orientable surfaces of all possible genera.)
Examples of small genera are already widely known, such as the five Platonic maps on
the sphere (genus 0) and the regular maps {4, 4}2q on the torus (see [10]).
We can complete an answer to the above question positively by proving the following:
Theorem 5.1. For every integer g ≥ 2, there exists an edge-transitive map of class 2 on the
orientable surface of genus g, such that both the map and its dual have simple underlying
graph (indeed with underlying graph being a complete bipartite graph in each case).
We have two proofs of this theorem. We give one of them, and then give a brief de-
scription of the other, based on an alternative construction for the family of maps involved.
Proof. We start by taking the following group, which is a quotient of the universal group
for ET maps of class 2 (obtained by adding two extra relations):
G = 〈x, y, z | x2 = y2 = z2 = [x, z] = (xyzy)2 = 1 〉.
In this group G, the subgroup N generated by a = (xy)2 = [x, y] and b = (yz)2 = [y, z]
is normal, with
ax = a−1, ay = a−1 and az = zxyxyz = zx(yxyz)−1 = zxzyxy = (xy)2 = a,
and bx = xyzyzx = (xyzy)−1zx = yzyxzx = (yz)2 = b, by = b−1 and bz = b−1,
and the quotient G/N is elementary abelian of order 8. Moreover, by Reidemeister-
Schreier theory (explained in [12, §12 & §13] and implemented as the Rewrite command
in MAGMA [1]), the subgroup N is free abelian of rank 2.
Now for any even positive integers k andm, letN (k,m) be the subgroup ofN generated
by ak/2 = (xy)k and bm/2 = (yz)m. Then N (k,m) is normal in G, with conjugation of its
generators by x, t = y and z following the same pattern as given for a and b above, and the
quotient Q(k,m) = G/N (k,m) is isomorphic to an extension of N/N (k,m) ∼= Ck/2×Cm/2
by G/N ∼= C2 × C2 × C2. In particular, Q(k,m) = G/N (k,m) has order 2km.
We now use these quotients Q(k,m) of G to construct bipartite ET maps of class 2 with
the required properties, one for each choice of the pair (k,m) with k 6= m. For notational
convenience, from now on we will let x, y, z, a and b denote the images in Q(k,m) of the
elements given above. Equivalently, we simply assume that the elements a = (xy)2 and
b = (yz)2 have orders k/2 and m/2 respectively. Also we now define Q = Q(k,m), and
let N be the normal subgroup generated by a and b in Q, isomorphic to Ck/2×Cm/2, with
quotient Q/N ∼= C2 × C2 × C2.
For the underlying graph of the map for a given pair (k,m), we take the vertices of one
part as the m right cosets of the dihedral subgroup V1 of order 2k generated by x and y,
and the vertices of the other part as the k right cosets of the dihedral subgroup V2 of order
2m generated by y and z, and define adjacency by non-trivial intersection.
For example, the vertex V1 = 〈x, y〉 is adjacent to the k cosets of V2 = 〈y, z〉 of
the form V2(xy)i for i ∈ Zk, with V2(xy)ix = V2(yx)−ix = V2(xy)−i−1 for all such
i. Note that the coset intersections are given by V1 ∩ V2(xy)i = {(xy)i, y(xy)i} =
{(xy)i, (xy)−i−1x} for i ∈ Zk. In particular, the valency of the vertex V1 is k, and hence
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 13
V1 is adjacent to every vertex of the second part. Similarly, the vertex V2 = 〈y, z〉 has
neighbours V1(zy)j for 0 ≤ j < m, and hence it is adjacent to every one of the m vertices
of the first part.
For the faces of the map, we use the right cosets of the subgroupH of order 4 generated
by x and z, and define incidence again by non-empty intersection. (We preserve the symbol
F for the set of faces.) For example, since H = {1, x, z, xz}, the face H itself is incident
with the vertices V1 and V1z in the first part, and the vertices V2 and V2x in the second part.
Here we note that z 6∈ V1 = 〈x, y〉, for otherwise Q = 〈x, y, z〉 = 〈x, y〉 which has order
2k < 2km, and similarly x 6∈ V2 = 〈y, z〉, because the latter has order 2m < 2km. In
particular, the face H has four different vertices, and hence also four different edges.
Also the group Q acts by right multiplication on this map, with two orbits of sizes m
and k on vertices, namely the two parts of the graph, and a single orbit on faces, and a
single orbit on edges. Indeed the edges can be identified with right cosets of the subgroup
K of order 2 generated by y, with K(xy)i = {(xy)i, y(xy)i} = {(xy)i, (xy)−i−1x} for
all i ∈ Zk. It follows that every vertex of the first part has valency k, and every vertex of
the second part has valency m, and hence the graph is isomorphic to the complete bipartite
graph Km,k. Similarly, the map has 2km/4 = km/2 faces, all of which have length 4
(with four distinct vertices), and so the dual of the map is simple too.
Next, if k 6= m then the parts of the underlying graph have different sizes, so the map
cannot lie in class 1, and hence it has class 2. (This also follows from the fact that Q has
no automorphism taking (x, y, z) to (z, y, x), since the orders of xy and zy are k and m.)
Finally, the map is orientable, since the subgroup generated by xy and zy (= (yz)−1)
has index 2 inQ (with imageC2×C2 inQ/N ∼= C2×C2×C2), and its Euler characteristic
is χ = |V | − |E| + |F | = (m + k) −mk + (mk/2) = m + k −mk/2, so its genus g is
(2 − χ)/2 = (4 − 2m − 2k +mk)/4 = (k − 2)(m − 2)/4. Taking m = 4 gives genus
g = (k − 2)/2 = k/2− 1, which can be any integer greater than 1 when k/2 > 2.
This completes the construction and proof. 
An alternative way of constructing these maps is to add six extra relations to the uni-
versal group for ET maps of class 3, to give the group with presentation
〈x, y, z, w | x2 = y2 = z2 = w2 = [x, z] =
= [y, z] = [x,w] = [y, w] = (xy)k = (zw)m = 1 〉,
which is isomorphic to the direct product of 〈x, y〉 ∼= Dk and 〈z, w〉 ∼= Dm.
The above map with underlying graphKm,k can be constructed from this group, but the
group admits an automorphism of order 2 that takes (x, y, z, w) to (y, x, w, z), and another
taking (x, y, z, w) to (z, w, x, y) when k = m, and hence the map has class 1 or class 2,
depending on whether or not k = m. The proof is straightforward.
Next, a natural question related to Question 3 in [18] (but not posed in [18]) is the
following: Does there exist a simple graph X that is the underlying graph of a map in each
of the 14 classes of edge-transitive maps?
Note that any such graph must be arc-transitive and hence regular (because it underlies
an ET map of class 1), with valency divisible by 4 (because it underlies an ET map of class
5), and also bipartite (because it underlies an ET map of class 2). Accordingly, there are
natural candidates to check. In an early search we found that the complete bipartite graph
K8,8 underlies ET maps of 11 of the 14 classes, namely all of them except classes 2∗ex,
2P ex and 5. A little further work led us to a positive answer to the question:
14 Art Discrete Appl. Math. 3 (2020) #P1.09
Theorem 5.2. In each of the 14 classes of edge-transitive maps, there exists an orientable
map with underlying graph isomorphic to the complete bipartite graph K16,16.
Proof. Computations using MAGMA produce the maps with underlying graphK16,16 sum-
marised in Table 5, with defining relations for their automorphism groups given in Table 6,
in the Appendix. 
6 Orientable ET maps of genus 14
Question 4 in [18, Section 6] is the following: What is the largest number of automorphism-
group types for edge-transitive maps contained by one surface? Is there some surface that
supports all 14 types? We can now answer both parts of this.
Theorem 6.1. The orientable surface of genus 14 carries edge-transitive maps of all 14
classes.
Proof. Computations using MAGMA produce the maps summarised in Table 7, with defin-
ing relations for their automorphism groups given in Table 8, in the Appendix. 
In particular, the answer to the second part of Question 4 in [18, Section 6] is “Yes”,
and the answer to the first part is 14. Also we can prove that 14 is the smallest genus for
which this happens, by MAGMA computations and Theorem 3.1 (to bound the order of
the groups required for consideration in a search for examples). Again, further details are
available from the first author upon request. On the other hand, there are other surfaces of
higher genus that carry ET maps of all 14 classes, as explained in the next section.
7 Non-degenerate orientable ET maps of genus 17
A natural extension of Question 4 in [18, Section 6] is the following: Is there some surface
that supports at least one non-degenerate ET map of each of the 14 classes? The answer to
this question is also “Yes”:
Theorem 7.1. The orientable surface of genus 17 carries edge-transitive maps of all 14
classes, with the property that the map and its dual have simple underlying graph.
Proof. Computations using MAGMA produce the non-degenerate maps summarised in Ta-
ble 9, with defining relations for their automorphism groups given in Table 10, in the Ap-
pendix. 
8 Edge-transitive maps with prescribed automorphism group(s)
Another very natural question is this: Which finite groups occur as the automorphism group
of at least one ET map in each of the 14 classes? This same question can be asked with a
restriction to simple or non-degenerate maps.
Clearly a necessary condition is that the group can be generated by two elements. On
the other hand, for orientable ET maps with more than two edges, it cannot be cyclic,
because in [18] it was shown that if an orientable ET map M has at least three edges, then
Aut(M) is a non-abelian group, except in the case where M has class 4P , and Aut(M)
is isomorphic to the direct product Cn × C2 where n ≡ 2 mod 4. In that exceptional
case the stabiliser of every vertex and every face is the subgroup of order 4 generated by
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 15
the images of the two involutory generators x and y of the universal group for class 4P ,
and the underlying graph of the map has double edges. Similarly it can be shown easily
that if a non-orientable ET map M has at least three edges and Aut(M) is abelian, then
either M has class 3 (with two vertices and eight edges) and Aut(M) ∼= C2 × C2 × C2,
or M has class 4 or 4∗ and Aut(M) ∼= Cn × C2 for some even n, or M has class 4P and
Aut(M) ∼= Cn × C2 for some n divisible by 4. Again in these cases the underlying graph
of the map has multiple edges.
We can also extend these observations to abelian groups that occur as the orientation-
preserving subgroup of Aut(M) for some orientable ET map M . It is not difficult to show
that there are no such maps in classes 2P , 2ex, 2∗ex, 2P ex, 4, 4∗, 5, 5∗ and 5P (mainly
because the abelian groups involved admit outer automorphisms that put the map into a
higher class), but there exist infinite families of examples in the other five classes, namely
1, 2, 2∗, 3 and 4P .
In class 1 these maps have just one or two vertices, and just one or two faces, with
multiple edges. In class 4P the maps have (|V |, |E|, |F |) = (k, 2kn, k) for arbitrary in-
tegers k ≥ 1 and n ≥ 2 or 3, but again none of them is simple. In class 2, some of the
maps are simple but all have non-simple dual, while in class 2∗, all of the maps are non-
simple (but some have simple dual). The only class containing non-degenerate orientable
ET maps is class 3, and in these case there are four infinite families of examples with
(|V |, |E|, |F |) = (4k+4, 16k, 4k+4) and Aut(M) ∼= C2k ×C2×C2 for all k ≥ 2, with
two infinite families containing self-dual examples, and the other two consisting of duals
of each other.
Details are available from the first author upon request.
A major contribution in the non-abelian case for the above question was made by
Gareth Jones in [13], where he considered this for non-abelian simple groups, the sym-
metric groups Sn, soluble groups, and nilpotent groups. In particular, he showed in [13,
Theorem 1.2] that a given non-abelian simple group is the automorphism group of some
ET map of a given class unless it appears in a list of known exceptions, copied in Table 2
below.
Class Non-abelian simple groups that do not occur
1 L3(q), U3(q), L4(2e), U4(3), U5(2), A6, A7, M11, M22, M23, McL
2, 2∗, 2P U3(3)
2ex, 2∗ex, 2P ex L2(q), L3(q), U3(q), A7
3 –
4, 4∗, 4P –
5, 5∗, 5P L2(q)
Table 2: Non-occurrences of non-abelian simple groups for a given class of ET map
Note that in 11 of the 14 classes (all except 2P ex, 5 and 5∗), the maps are non-
orientable, for the obvious reason that a non-abelian simple group has no subgroup of
index 2.
It follows easily from Table 2 that the smallest simple group that occurs for all 14
classes is the Suzuki group Sz(8), of order 29120. In fact, Sz(8) is the automorphism
16 Art Discrete Appl. Math. 3 (2020) #P1.09
group of some non-degenerate ET map in every one of the 14 classes; details are available
from the first author on request. The same holds for the next smallest examples, which are
M12, J1 and A9, and in particular, A5, A6, A7 and A8 do not occur in this way.
Indeed by [13, Theorem 1.1], the alternating groupAn of degree n is the automorphism
group of some ET map in any given class, except in the following cases: n ∈ {3, 4, 6, 7, 8}
for class 1; n ∈ {1, 2, 3, 4} for classes 2, 2∗, 2P and 3; n ∈ {1, 2, 3, 4, 5, 6, 7} for classes
2ex, 2∗ex and 2P ex; n ∈ {1, 2, 3} for classes 4, 4∗ and 4P ; and n ∈ {1, 2, 3, 4, 5, 6} for
classes 5, 5∗ and 5P . In particular,A9 is the smallest alternating group that occurs for all 14
classes. Moreover, non-degenerate ET maps occur in all classes for A9 (and A10); details
are available from the first author on request.
Similarly, by [13, Theorem 1.1], the symmetric group Sn of degree n is the automor-
phism group of some ET map in any given class, except in the following cases: n = 1 for
classes 2, 2∗, 2P , 3, 4, 4∗ and 4P ; and n ∈ {1, 2, 3, 4, 5} for classes 2ex, 2∗ex, 2P ex, 5,
5∗ and 5P . In particular, S6 is the smallest symmetric group that occurs for all 14 classes.
Moreover, non-degenerate ET maps occur in all classes for S6 (and S7, S8, S9 and S10);
details are available from the first author on request.
Incidentally, the symmetric group S6 is the smallest insoluble group that occurs as the
automorphism group of some ET map in every one of the 14 classes, and again, the same
holds under the assumption that the map is simple or non-degenerate.
Finally, we have the following:
Theorem 8.1. The smallest finite group G that occurs as the automorphism group of some
ET map in every one of the 14 classes is the 8654th group of order 576 (in the database of
all groups of order up to 2000).
Proof. Computations using MAGMA produce maps of each class for this group, indeed
non-degenerate maps of all 14 classes, and these are summarised in Table 11, with defining
relations for their automorphism groups given in Table 12, in the Appendix. The same
computations show that no group of smaller or equal order (necessarily divisible by 4) has
the required property. 
Note that just one of the maps given in Table 11 is non-orientable, namely the one in
class 2P . There are also orientable examples in class 2P with simple underlying graph, but
none with simple dual. Again, further details about these maps are available from the first
author on request.
9 Some final remarks and questions
Answers always raise more questions. We ask a few of them for this topic below.
For each of the 14 classes of ET maps, define the non-degenerate genus spectrum for
that class to be the set of genera of orientable surfaces that carry a non-degenerate map
of that class. By our Theorem 5.1 (and knowledge of ET maps of genus 0 and 1), the
non-degenerate genus spectrum for class 2 (and therefore also 2∗) is the set of all non-
negative integers. The corresponding question for classes 1 (regular maps) and/or 2P ex
(chiral maps), however, is a challenging open question (see [7, 8]).
Indeed, questions about genus spectra questions are notoriously difficult. Another ex-
ample comes from group actions. Given a finite group G, its symmetric genus σ(G) is
defined as the smallest non-negative integer g for which G has a faithful action on some
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 17
compact Riemann surface of genus g. Much but not all of the spectrum for σ is known;
see [9]. In contrast, the strong symmetric genus σo(G) is the smallest g such that G has a
faithful action on such a surface of genus g, preserving orientation, and the spectrum for
σo contains all non-negative integers, by a theorem of May and Zimmerman in [15], which
incidentally uses the groups Ck × Dm in a similar way to the way we used the groups
Dk ×Dm in proving Theorem 5.1. The following question(s) could constitute a long-term
project:
Question 9.1. What are the genus spectra for the 14 classes of ET maps, under the restric-
tion to non-degenerate maps, or to simple maps, or to ET maps in general?
The symmetric groups Sn play a prominent role in [18], producing ET maps in each
class, but with restrictions on the congruence class n modulo 12. Those restrictions are
removed in [13], where it is also shown that Sn realises all classes for every n ≥ 6, and An
realises all classes for every n ≥ 9.
Similarly, the first author of this paper showed in [4] that Sn and An realise the upper
bounds |A|maxo and |A|maxnono given in Table 1 for class 1 (and hence also for classes 2
and 2∗), for all but finitely many n, and many years later the same in [2, Theorem 6.3] for
An for class 2P ex (and hence also classes 5 and 5∗), again for all but finitely many n.
Question 9.2. For which other classes do An and/or Sn realise the upper bounds |A|maxo
and |A|maxnono in Table 1, possibly with restrictions on n?
Edge-transitive maps on non-orientable surfaces have been largely ignored in the lit-
erature. Even in this paper, most of our theorems, examples and tables concern maps on
orientable surfaces, and yet there should be non-orientable versions:
Question 9.3. What are the analogues of Theorems 4.1, 5.1, 5.2, 6.1 and 7.1 for non-
orientable ET maps (in the 11 classes other than 2P ex, 5 and 5∗)? How might the informa-
tion given in the tables in the Appendix differ for non-orientable maps?
Finally, for a map on a non-orientable surface S with automorphism group G, a stan-
dard technique is to pass to the orientable double cover of S, where G × C2 acts, with
G preserving orientation and C2 orientation-reversing and fixed-point free (in effect, an
antipodal symmetry). If the given map (on S) is edge-transitive, then so is the lifted map
under the group G × C2. On the other hand, the lifted map may actually be ‘unstable’, in
that it has a symmetry group larger than G × C2, which means that the lifted map may be
in a different class, with larger edge-stabiliser. Many examples of this phenomenon have
been provided recently by Gareth Jones [14].
Question 9.4. How common is instability for edge-transitive maps on non-orientable sur-
faces?
ORCID iDs
Marston D.E. Conder https://orcid.org/0000-0002-0256-6978
Isabel Holm https://orcid.org/0000-0002-6806-7514
Thomas W. Tucker https://orcid.org/0000-0002-7868-6925
18 Art Discrete Appl. Math. 3 (2020) #P1.09
References
[1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, vol-
ume 24, pp. 235–265, 1997, doi:10.1006/jsco.1996.0125, computational algebra and number
theory (London, 1993).
[2] E. Bujalance, M. D. E. Conder and A. F. Costa, Pseudo-real Riemann surfaces and chiral regular
maps, Trans. Amer. Math. Soc. 362 (2010), 3365–3376, doi:10.1090/S0002-9947-10-05102-0.
[3] M. Conder and P. Dobcsányi, Determination of all regular maps of small genus, J. Combin.
Theory Ser. B 81 (2001), 224–242, doi:10.1006/jctb.2000.2008.
[4] M. D. E. Conder, Generators for alternating and symmetric groups, J. London Math. Soc. (2)
22 (1980), 75–86, doi:10.1112/jlms/s2-22.1.75.
[5] M. D. E. Conder, Regular maps and hypermaps of Euler characteristic −1 to −200, J. Combin.
Theory Ser. B 99 (2009), 455–459, doi:10.1016/j.jctb.2008.09.003.
[6] M. D. E. Conder, Lists of regular and chiral maps on surfaces of Euler characteristic −1 to
−600, 2012, https://www.math.auckland.ac.nz/˜conder.
[7] M. D. E. Conder and J. Ma, Regular maps with simple underlying graphs, J. Combin. Theory
Ser. B 110 (2015), 1–18, doi:10.1016/j.jctb.2014.07.001.
[8] M. D. E. Conder, J. Širáň and T. W. Tucker, The genera, reflexibility and simplicity of regular
maps, J. Eur. Math. Soc. (JEMS) 12 (2010), 343–364, doi:10.4171/JEMS/200.
[9] M. D. E. Conder and T. W. Tucker, The symmetric genus spectrum of finite groups, Ars Math.
Contemp. 4 (2011), 271–289, doi:10.26493/1855-3974.127.eb9.
[10] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, volume 14
of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related
Areas], Springer-Verlag, Berlin-New York, 4th edition, 1980.
[11] J. E. Graver and M. E. Watkins, Locally finite, planar, edge-transitive graphs, Mem. Amer.
Math. Soc. 126 (1997), vi+75, doi:10.1090/memo/0601.
[12] D. L. Johnson, Topics in the theory of group presentations, volume 42 of London Mathematical
Society Lecture Note Series, Cambridge University Press, Cambridge-New York, 1980.
[13] G. A. Jones, Automorphism groups of edge-transitive maps, Acta Math. Univ. Comenian. (N.S.)
88 (2019), 841–847.
[14] G. A. Jones, Unstable maps, Acta Math. Univ. Comenian. (N.S.) 88 (2019), 341–350, doi:
10.1007/s10623-019-00685-y.
[15] C. L. May and J. Zimmerman, There is a group of every strong symmetric genus, Bull. London
Math. Soc. 35 (2003), 433–439, doi:10.1112/S0024609303001954.
[16] A. Orbanić, Edge-transitive maps, Ph.D. thesis, University of Ljubljana, 2006.
[17] A. Orbanić, D. Pellicer, T. Pisanski and T. W. Tucker, Edge-transitive maps of low genus, Ars
Math. Contemp. 4 (2011), 385–402, doi:10.26493/1855-3974.249.3a6.
[18] J. Širáň, T. W. Tucker and M. E. Watkins, Realizing finite edge-transitive orientable maps, J.
Graph Theory 37 (2001), 1–34, doi:10.1002/jgt.1000.abs.
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 19
Appendix
Below are the tables mentioned in the proofs of Theorems 5.1, 5.2, 6.1, 7.1 and 4.1. Note
that there are two tables for each theorem: the first gives details about a map from each of
the relevant classes, and the second gives defining relations for the automorphism groups
of those maps (in terms of the canonical generators for the relevant universal group).
The following notation is used in the odd-numbered tables. First ‘SD’ and ‘NSD’ indicate
that the map is self-dual or non-self-dual, respectively. (We do not indicate ‘NSD’ in the
obvious cases, where |V | 6= |F |.) Next, Cn, Dn, An and Sn denote the cyclic group of
order n, the dihedral group of degree n (and order 2n), and the alternating and symmetric
groups of degree n, while Group(n, k) denotes the k th group in the database of all groups
of order up to 2000 (except 1024), available in the MAGMA system [1]. Finally, V4 denotes
the direct product C2 × C2 (the Sylow 2-subgroup of A4), while K oH denotes a semi-
direct product with kernel K and complement H , and Cn ok Cm denotes the semi-direct
product 〈 a, b | am = bn = 1, a−1ba = bk 〉.
Class |V | |E| |F | Genus |A| Au, Av Af , Ag Comments
1 4 6 4 0 24 D3 D3 Aut(M) ∼= S4
2P 8 16 8 1 32 V4 V4 Aut(M) ∼=
Group(32, 43)
2P ex 5 10 5 1 20 C4 C4 Aut(M) ∼= C5o2C4
3 12 32 12 5 32 D4, V4 D4, V4 Aut(M) ∼= D4 × V4
4P 12 48 12 13 48 V4 V4 Aut(M) ∼= A4 × V4
5P 14 42 14 8 42 C3 C3 Aut(M) ∼= C7o2C6
Table 3: Non-degenerate self-dual maps with the smallest number of edges in six of the 14
classes
Class Defining relations for A = Aut(M)
1 x2 = y2 = z2 = (xz)2 = (xy)3 = (yz)3 = 1
2P x2 = y2 = z2 = (yz)2 = (xz)4 = (xyxz)2 = (xy)3zxzy = 1
2P ex x2 = y4 = xyxy2xy−1 = 1
3 x2 = y2 = z2 = w2 = (xz)2 = (yz)2 = (yw)2 = (zw)2 =
(xy)4 = xywxwy = 1
4P x2 = y2 = (xy)2 = (xz)3 = [x, z]2 = [xy, z] = 1
5P x6 = xy2x2y−1 = xy−2x2y = 1
Table 4: Defining relations for the automorphism groups of the maps in Table 3
20 Art Discrete Appl. Math. 3 (2020) #P1.09
C
lass
|F
|
G
enus
|A
u
t(M
)|
A
u ,
A
v
A
f ,A
g
C
om
m
ents
1
16
105
1024
D
1
6
D
3
2
A
u
t(M
)
notin
database
2
128
49
512
D
1
6 ,D
1
6
V
4
A
u
t(M
) ∼=
G
ro
u
p
(5
1
2,3
0
4
7
1
)
2
∗
80
201
512
D
8
D
1
6 ,D
4
A
u
t(M
) ∼=
G
ro
u
p
(5
1
2,3
2
9
1
7
)
2
P
32
225
512
D
8
D
8
SD
,A
u
t(M
) ∼=
G
ro
u
p
(5
1
2,3
2
9
1
7)
2ex
16
105
512
D
8
C
3
2
A
u
t(M
) ∼=
G
ro
u
p
(5
1
2,1
0
5
6
)
2
∗ex
16
105
512
C
1
6
D
1
6
A
u
t(M
) ∼=
G
ro
u
p
(5
1
2,9
5
5
)
2
P
ex
16
105
512
C
1
6
C
3
2
A
u
t(M
) ∼=
G
ro
u
p
(5
1
2,9
5
5
)
3
32
97
256
D
8 ,D
8
D
8 ,
D
8
SD
,A
u
t(M
) ∼=
G
ro
u
p
(2
5
6,7
2
2
)
4
32
97
256
D
8 ,C
1
6
D
4
A
u
t(M
) ∼=
G
ro
u
p
(2
5
6,5
6
)
4
∗
48
89
256
D
4
D
4 ,
C
1
6
A
u
t(M
) ∼=
G
ro
u
p
(2
5
6,9
5
)
4
P
32
97
256
D
4
D
4
SD
,A
u
t(M
) ∼=
G
ro
u
p
(2
5
6,9
5
)
5
16
105
256
C
1
6 ,C
1
6
C
1
6
A
u
t(M
) ∼=
G
ro
u
p
(2
5
6,4
1
)
5
∗
96
65
256
C
8
C
4 ,
C
8
A
u
t(M
) ∼=
G
ro
u
p
(2
5
6,1
1
7
)
5
P
32
97
256
C
8
C
8
SD
,A
u
t(M
) ∼=
G
ro
u
p
(2
5
6,1
1
7
)
Table
5:E
T
m
aps
ofall14
types
w
ith
underlying
graph
K
1
6
,1
6
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 21
C
la
ss
D
efi
ni
ng
re
la
tio
ns
fo
rA
=
A
u
t(
M
)
1
x
2
=
y
2
=
z
2
=
[x
,z
]
=
(x
y
x
z
y
z
y
)2
=
(x
y
x
y
x
y
x
y
z
)2
=
(x
y
)1
3
z
y
x
y
z
y
=
(y
z
)1
6
=
1
2
x
2
=
y
2
=
z
2
=
(x
z
)2
=
(x
y
x
y
z
)2
=
(x
y
)1
3
z
y
x
y
z
y
=
(y
z
)1
6
=
1
2
∗
x
2
=
y
2
=
z
2
=
(x
z
)4
=
(y
z
)8
=
x
y
x
y
x
z
y
x
y
z
=
x
y
x
z
x
z
y
z
x
z
=
1
2P
Sa
m
e
de
fin
in
g
re
la
tio
ns
as
fo
rc
la
ss
2∗
ab
ov
e
2e
x
x
2
=
(x
y
4
)2
=
(x
y
−
2
)2
(x
y
2
)2
=
(x
y
x
y
x
y
−
2
)2
=
x
y
x
y
x
y
−
1
x
y
x
y
−
1
x
y
x
y
−
1
x
y
3
=
1
2
∗ e
x
x
2
=
x
y
−
1
x
y
x
y
−
3
x
y
3
=
(x
y
2
)2
(x
y
−
2
)2
=
y
1
6
=
1
2P
ex
x
2
=
x
y
4
x
y
1
2
=
x
y
x
y
−
2
x
y
3
x
y
−
2
=
x
y
x
y
−
4
x
y
−
1
x
y
4
=
x
y
−
1
x
y
(x
y
−
1
)3
(x
y
)3
=
1
3
x
2
=
y
2
=
z
2
=
w
2
=
(x
z
)2
=
(y
w
)4
=
(y
z
)8
=
(x
w
y
)2
=
(x
y
)2
(x
w
)2
=
(x
y
)2
z
y
w
z
=
1
4
x
2
=
y
2
=
x
y
x
z
x
z
−
1
=
x
y
x
z
−
1
x
z
=
x
y
x
y
z
y
z
y
z
−
1
y
z
−
1
y
=
z
1
6
=
1
4
∗
x
2
=
y
2
=
(x
y
)4
=
x
y
z
x
y
z
−
1
=
x
z
(y
z
)3
=
x
z
−
1
x
z
−
2
x
z
x
z
2
=
(x
y
)2
z
8
=
1
4P
Sa
m
e
de
fin
in
g
re
la
tio
ns
as
fo
rc
la
ss
4∗
ab
ov
e
5
x
y
x
3
y
−
1
=
x
−
1
y
−
1
x
5
y
=
y
1
6
=
1
5
∗
x
4
=
y
8
=
[x
2
,y
2
]
=
x
y
x
y
−
1
x
−
1
y
x
y
−
1
=
x
−
1
y
−
3
(x
y
)3
=
1
5P
Sa
m
e
de
fin
in
g
re
la
tio
ns
as
fo
rc
la
ss
5∗
ab
ov
e
Ta
bl
e
6:
D
efi
ni
ng
re
la
tio
ns
fo
rt
he
au
to
m
or
ph
is
m
gr
ou
ps
of
th
e
m
ap
s
in
Ta
bl
e
5
22 Art Discrete Appl. Math. 3 (2020) #P1.09
C
lass
|V
|
|E
|
|F
|
|A
u
t(M
)|
A
u ,
A
v
A
f ,A
g
C
om
m
ents
1
2
35
7
140
D
3
5
D
1
0
A
u
t(M
) ∼=
D
5 ×
D
7
2
2
29
1
58
D
2
9 ,D
2
9
D
2
9
A
u
t(M
) ∼=
D
2
9
2
∗
1
29
2
58
D
2
9
D
2
9 ,D
2
9
M
ap
dualto
the
one
above
2
P
2
30
2
60
D
1
5
D
1
5
SD
,
A
u
t(M
) ∼=
D
1
5 ×
C
2
2
ex
4
40
10
80
D
1
0
C
8
A
u
t(M
) ∼=
C
5
o
(C
8 o
5
C
2 )
2
∗ex
10
40
4
80
C
8
D
1
0
M
ap
dualto
the
one
above
2
P
ex
26
78
26
156
C
6
C
6
N
SD
,
A
u
t(M
) ∼=
(C
1
3 o
1
0
C
6 )×
C
2
3
2
30
2
30
D
1
5 ,D
1
5
D
1
5 ,D
1
5
SD
,
A
u
t(M
) ∼=
D
1
5
4
12
40
2
40
D
1
0 ,
C
4
D
1
0
A
u
t(M
) ∼=
D
5 ×
C
4
4
∗
2
40
12
40
D
1
0
D
1
0 ,
C
4
M
ap
dualto
the
one
above
4
P
13
52
13
52
V
4
V
4
SD
,
A
u
t(M
) ∼=
C
2
6 ×
C
2
5
9
40
5
40
C
8 ,
C
1
0
C
8
A
u
t(M
) ∼=
C
5
o
2
C
8
5
∗
5
40
9
40
C
8
C
8 ,
C
1
0
M
ap
dualto
the
one
above
5
P
26
78
26
78
C
3
C
3
SD
,
A
u
t(M
) ∼=
C
1
3
o
3
C
6
Table
7:E
T
m
aps
ofall14
types
on
the
orientable
surface
ofgenus
1
4
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 23
Class Defining relations for A = Aut(M)
1 x2 = y2 = z2 = (xz)2 = (xyxyz)2 = (xyz)2(yz)5 = 1
2 x2 = y2 = z2 = xzyz = (xy)14xz = 1
2∗ x2 = y2 = z2 = yzxz = (yx)14yz = 1
2P x2 = y2 = z2 = (xyz)2 = (xz)2(yz)3 = (xy)6 = 1
2ex x2 = y8 = [x, y4] = (xy)4y4 = xyxy2xy2xy−1 = 1
2∗ex Same defining relations as for class 2ex above
2P ex x2 = y6 = (xyxy2)2 = (xy)3(xy−1)3 = xy−2xy2xy3xy3 = 1
3 x2 = y2 = z2 = w2 = xyxzwy = xyxwxw = (xz)3 = 1
4 x2 = y2 = z4 = [x, z] = [xz, y] = (xy)3(zy)2 = 1
4∗ Same defining relations as for class 4 above
4P x2 = y2 = (xy)2 = [x, z] = [y, z] = yz13 = 1
5 x2yx−2y = xy3x−1y−1 = xy−1x3y2 = 1
5∗ Same defining relations as for class 5 above
5P x6 = (xy)3 = (xy−1)3 = xy−1x−1y3 = 1
Table 8: Defining relations for the automorphism groups of the maps in Table 7
24 Art Discrete Appl. Math. 3 (2020) #P1.09
C
lass
|V
|
|E
|
|F
|
|A
u
t(M
)|
A
u ,A
v
A
f ,A
g
C
om
m
ents
1
32
96
32
384
D
6
D
6
SD
,
A
u
t(M
) ∼=
G
ro
u
p
(3
8
4,5
6
0
2
)
2
40
96
24
192
D
4 ,
D
6
D
4
A
u
t(M
) ∼=
G
ro
u
p
(1
9
2,5
9
1
)
2
∗
24
96
40
192
D
4
D
4 ,D
6
M
ap
dualto
the
one
above
2
P
16
64
16
128
D
4
D
4
SD
,
A
u
t(M
) ∼=
G
ro
u
p
(1
2
8,3
3
2
)
2
ex
32
128
64
256
D
4
C
4
A
u
t(M
) ∼=
G
ro
u
p
(2
5
6,5
1
1
)
2
∗ex
64
128
32
256
C
4
D
4
M
ap
dualto
the
one
above
2
P
ex
32
96
32
192
C
6
C
6
SD
,
A
u
t(M
) ∼=
G
ro
u
p
(1
9
2,1
0
0
8
)
3
16
64
16
64
V
4 ,
V
4
V
4 ,V
4
A
u
t(M
) ∼=
G
ro
u
p
(6
4,7
3
)
4
40
96
24
96
V
4 ,C
6
V
4
A
u
t(M
) ∼=
G
ro
u
p
(9
6,1
1
8
)
4
∗
24
96
40
96
V
4
V
4 ,C
6
M
ap
dualto
the
one
above
4
P
16
64
16
64
V
4
V
4
SD
,
A
u
t(M
) ∼=
G
ro
u
p
(6
4,4
2
)
5
48
96
16
96
C
3 ,
C
6
C
6
A
u
t(M
) ∼=
G
ro
u
p
(9
6,7
1
)
5
∗
16
96
48
96
C
6
C
3 ,C
6
M
ap
dualto
the
one
above
5
P
16
64
16
64
C
4
C
4
SD
,
A
u
t(M
) ∼=
G
ro
u
p
(6
4,4
6
)
Table
9:N
on-degenerate
E
T
m
aps
ofall14
types
on
the
orientable
surface
ofgenus
1
7
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 25
C
la
ss
D
efi
ni
ng
re
la
tio
ns
fo
rA
=
A
u
t(
M
)
1
x
2
=
y
2
=
z
2
=
(x
z
)2
=
(x
y
)6
=
(x
y
z
y
)3
=
(y
z
)6
=
[x
y
x
y
x
,z
y
z
y
z
]
=
1
2
x
2
=
y
2
=
z
2
=
(x
y
)4
=
(x
z
)4
=
(y
z
)6
=
(x
y
z
)2
(x
z
y
)2
=
x
(y
z
)2
x
(z
y
)2
=
1
2∗
x
2
=
y
2
=
z
2
=
(x
y
)4
=
(y
z
)4
=
(x
z
)6
=
(y
x
z
)2
(y
z
x
)2
=
y
(x
z
)2
y
(z
x
)2
=
1
2
P
x
2
=
y
2
=
z
2
=
(x
z
)4
=
(y
z
)4
=
(x
y
)3
z
x
z
y
=
x
y
x
z
y
z
(x
y
z
)2
=
1
2e
x
x
2
=
y
4
=
(x
y
)2
(x
y
2
)2
(x
y
−
1
)2
=
1
2∗
ex
Sa
m
e
de
fin
in
g
re
la
tio
ns
as
fo
rc
la
ss
2e
x
ab
ov
e
2
P
ex
x
2
=
y
6
=
(x
y
3
)4
=
x
y
x
y
2
x
y
−
2
x
y
−
1
=
x
y
x
y
−
1
x
y
−
1
x
y
2
x
y
x
y
−
2
=
1
3
x
2
=
y
2
=
z
2
=
w
2
=
(y
w
)2
=
(x
y
)4
=
(y
z
)4
=
x
y
z
x
z
w
=
y
z
y
w
z
w
=
1
4
x
2
=
y
2
=
z
6
=
(x
y
)2
=
[x
,z
2
]
=
(y
z
2
)2
=
(x
z
y
z
)2
=
(y
z
)2
(y
z
−
1
)2
=
x
z
x
z
−
1
x
z
−
1
y
x
z
−
1
y
=
1
4∗
Sa
m
e
de
fin
in
g
re
la
tio
ns
as
fo
rc
la
ss
4
ab
ov
e
4
P
x
2
=
y
2
=
x
y
z
−
1
x
y
z
=
x
y
z
−
1
y
z
−
1
y
=
x
y
x
z
3
x
z
−
1
=
1
5
x
3
=
y
6
=
(x
y
)4
=
x
y
−
1
x
−
1
y
x
−
1
y
2
=
x
y
2
x
−
1
y
−
1
x
y
3
=
1
5∗
Sa
m
e
de
fin
in
g
re
la
tio
ns
as
fo
rc
la
ss
5
ab
ov
e
5
P
x
2
y
−
1
x
2
y
=
x
−
1
y
−
1
x
y
3
=
1
Ta
bl
e
10
:D
efi
ni
ng
re
la
tio
ns
fo
rt
he
au
to
m
or
ph
is
m
gr
ou
ps
of
th
e
m
ap
s
in
Ta
bl
e
9
26 Art Discrete Appl. Math. 3 (2020) #P1.09
Class |V | |E| |F | Genus Au, Av Af , Ag Comments
1 48 144 48 25 D6 D6 NSD, Orientable
2 120 288 48 61 D4, D6 D6 Orientable
2∗ 48 288 120 61 D6 D4, D6 Map dual to the one above
2P 48 288 48 97 D6 D6 Non-orientable
2ex 48 288 96 73 D6 C6 Orientable
2∗ex 96 288 48 73 C6 D6 Map dual to the one above
2P ex 96 288 96 49 C6 C6 SD, Orientable
3 240 576 120 109 V4, D3 D6, D4 Orientable
4 240 576 144 97 D6, C3 V4 Orientable
4∗ 144 576 240 97 V4 D6, C3 Map dual to the one above
4P 72 576 144 181 D4 V4 Orientable
5 240 576 96 121 C4, C6 C6 Orientable
5∗ 96 576 240 121 C6 C4, C6 Map dual to the one above
5P 96 576 96 193 C6 C6 SD, Orientable
Table 11: Non-degenerate ET maps of all 14 types with Aut(M) ∼= Group(576, 8654)
M. D. E. Conder et al.: Observations and answers to questions about edge-transitive maps 27
C
la
ss
D
efi
ni
ng
re
la
tio
ns
fo
rA
=
A
u
t(
M
)
1
x
2
=
y
2
=
z
2
=
(x
z
)2
=
(x
y
)6
=
(y
z
)6
=
(x
y
x
y
z
y
)3
=
(x
y
z
)6
=
(x
y
z
y
)2
x
z
y
z
y
x
y
z
y
z
=
1
2
x
2
=
y
2
=
z
2
=
(x
y
)4
=
(x
z
)6
=
(y
z
)6
=
(x
y
x
z
)3
=
x
y
z
y
x
y
z
y
x
z
x
z
=
1
2
∗
x
2
=
y
2
=
z
2
=
(x
y
)4
=
(x
z
)6
=
(y
z
)6
=
(x
y
x
z
)3
=
x
y
z
y
x
y
z
y
x
z
x
z
=
1
2P
x
2
=
y
2
=
z
2
=
(z
x
)3
=
(x
y
)4
=
(y
z
)6
=
(x
z
x
y
z
y
)3
=
x
y
x
z
y
x
z
y
z
x
y
z
=
1
2e
x
x
2
=
y
6
=
(x
y
)6
=
(x
y
2
)4
=
(x
y
)2
(x
y
−
1
)2
(x
y
3
)2
=
1
2
∗ e
x
Sa
m
e
de
fin
in
g
re
la
tio
ns
as
fo
rc
la
ss
2e
x
ab
ov
e
2P
ex
Sa
m
e
de
fin
in
g
re
la
tio
ns
as
fo
rc
la
ss
2e
x
ab
ov
e
3
x
2
=
y
2
=
z
2
=
w
2
=
(x
y
)2
=
(x
z
)2
=
(w
z
)3
=
(y
z
)4
=
(x
z
w
y
w
)2
=
(x
w
x
w
z
)2
=
x
y
z
w
x
z
y
w
y
z
y
w
=
x
w
y
z
w
z
y
z
w
z
y
w
=
1
4
x
2
=
y
2
=
z
3
=
(y
z
)4
=
(x
z
−
1
y
z
)2
=
(x
z
−
1
)2
(x
z
)2
=
x
y
x
z
−
1
x
y
x
z
y
z
−
1
y
=
1
4
∗
Sa
m
e
de
fin
in
g
re
la
tio
ns
as
fo
rc
la
ss
4
ab
ov
e
4P
x
2
=
y
2
=
z
6
=
(x
y
)2
=
(y
z
)4
=
[y
,z
3
]
=
(x
z
y
z
−
1
)2
=
x
y
z
y
z
−
2
x
z
x
z
−
1
=
(x
z
3
)2
x
z
−
3
=
1
5
x
4
=
y
6
=
x
−
1
y
x
−
2
y
x
y
2
=
[x
2
,y
]2
=
(x
y
)4
x
−
1
y
−
1
x
y
−
1
=
1
5
∗
Sa
m
e
de
fin
in
g
re
la
tio
ns
as
fo
rc
la
ss
5
ab
ov
e
5P
Sa
m
e
de
fin
in
g
re
la
tio
ns
as
fo
rc
la
ss
5
ab
ov
e
Ta
bl
e
12
:D
efi
ni
ng
re
la
tio
ns
fo
rt
he
au
to
m
or
ph
is
m
gr
ou
ps
of
th
e
m
ap
s
in
Ta
bl
e
11