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Contents
Realisation of groups as automorphism groups in permutational categories
Gareth A. Jones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Complex uniformly resolvable decompositions of Kv
Csilla Bujtás, Mario Gionfriddo, Elena Guardo, Lorenzo Milazzo,
Salvatore Milici, Zsolt Tuza . . . . . . . . . . . . . . . . . . . . . . . . . 23
General d-position sets
Sandi Klavžar, Douglas F. Rall, Ismael G. Yero . . . . . . . . . . . . . . . 33
On Hermitian varieties in PG(6, q2)
Angela Aguglia, Luca Giuzzi, Masaaki Homma . . . . . . . . . . . . . . . 45
Achromatic numbers of Kneser graphs
Gabriela Araujo-Pardo, Juan Carlos Díaz-Patino, Christian Rubio-Montiel . 57
Coarse distinguishability of graphs with symmetric growth
Jesús Antonio Álvarez López, Ramón Barral Lijó, Hiraku Nozawa . . . . . 71
On complete multipartite derangement graphs
Andriaherimanana Sarobidy Razafimahatratra . . . . . . . . . . . . . . . . 89
On 2-closures of rank 3 groups
Saveliy V. Skresanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Nonlinear maps preserving the elementary symmetric functions
Constantin Costara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A characterization of exceptional pseudocyclic association schemes by
multidimensional intersection numbers
Gang Chen, Jiawei He, Ilia Ponomarenko, Andrey Vasil’ev . . . . . . . . . 133
Volume 21, Number 1, Fall/Winter 2021, Pages 1–150
iii

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.01 / 1–22
https://doi.org/10.26493/1855-3974.1840.6e0
(Also available at http://amc-journal.eu)
Realisation of groups as automorphism groups
in permutational categories
Gareth A. Jones *
School of Mathematical Sciences, University of Southampton,
Southampton SO17 1BJ, UK
Received 31 October 2018, accepted 18 July 2020, published online 10 August 2021
Abstract
It is shown that in various categories, including many consisting of maps or hypermaps,
oriented or unoriented, of a given hyperbolic type, or of coverings of a suitable topological
space, every countable group A is isomorphic to the automorphism group of uncountably
many non-isomorphic objects, infinitely many of them finite if A is finite. In particular, the
latter applies to dessins d’enfants, regarded as finite oriented hypermaps.
Keywords: Permutation group, centraliser, automorphism group, map, hypermap, dessin d’enfant.
Math. Subj. Class. (2020): 05C10, 14H57, 20B25, 20B27, 52B15, 57M10
1 Introduction
In 1939 Frucht published his celebrated theorem [15] that every finite group is isomor-
phic to the automorphism group of a finite graph; in 1960, by allowing infinite graphs,
Sabidussi [46] extended this result to all groups. Similar results have been obtained, re-
alising all finite groups (or in some cases all groups, or all countable groups) as automor-
phism groups of various other mathematical structures. Examples include the following, in
chronological order: distributive lattices, by Birkhoff [4] in 1946; regular graphs of a given
degree, by Sabidussi [45] in 1957; Riemann surfaces, by Greenberg [18, 19] in 1960 and
1973; projective planes, by Mendelsohn [37] in 1972; Steiner triple and quadruple systems,
by Mendelsohn [38] in 1978; fields, by Fried and Kollár [14] in 1978; matroids of rank 3,
by Babai [1] in 1981; oriented maps and hypermaps, by Cori and Machı̀ [10] in 1982; finite
volume hyperbolic manifolds of a given dimension, by Belolipetsky and Lubotzky [3] in
*The author is grateful to Ernesto Girondo, Gabino González-Diez and Rubén Hidalgo for discussions about
dessins d’enfants which motivated this work, and also to Ashot Minasyan, Egon Schulte, David Singerman and
Alexander Zvonkin for some very helpful comments.
E-mail address: g.a.jones@maths.soton.ac.uk (Gareth A. Jones)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
2005; abstract polytopes, by Schulte and Williams [47] in 2015 and by Doignon [12] in
2016. Babai has given comprehensive surveys of this topic in [1, 2].
In many of these cases, each group is represented as the automorphism group of not
just one object, but infinitely many non-isomorphic objects. The aim of this paper is to ob-
tain results of this nature for certain ‘permutational categories’, introduced and discussed
in [24]. These are categories C which are equivalent to the category of permutation rep-
resentation of some ‘parent group’ Γ = ΓC: thus each object O in C can be identified
with a permutation representation θ : Γ → S := Sym(Ω) of Γ 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 Ωi. They include the categories of maps or hy-
permaps on surfaces, oriented or unoriented, and possibly of a given type. Other examples
include the category of coverings of a ‘suitably nice’ topological space; this includes the
category of dessins d’enfants, regarded as finite coverings of the thrice-punctured sphere,
or equivalently as finite oriented hypermaps.
The automorphism group AutC(O) of an object O in a permutational category C is
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 if G is transitive on Ω, as we will assume throughout
this paper. Such objects correspond to conjugacy classes of subgroups of Γ, the point-
stabilisers. An important result is the following:
Theorem 1.1. 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.
There are analogous results in various contexts, ranging from abstract polytopes to
covering spaces, which can be regarded as special cases of Theorem 1.1. Proofs of this
result for particular categories can be found in the literature: for instance, in [28] it is
deduced for oriented maps from a more general result about morphisms in that category;
in [29, Theorem 2.2 and Corollary 2.1] a proof for dessins is briefly outlined; similar results
for covering spaces are proved in [35, Appendix] and [39, Theorem 81.2], and for abstract
polytopes in [36, Propositions 2D8 and 2E23(a)]. Theorem 1.1 follows immediately from
the following ‘folklore’ result, proved in [26, Theorem 2(1)] (see also [44, Theorem 3.2]):
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 C ∼= NG(H)/H .
Of course, finite objects in any category have finite automorphism groups. In most of
the permutational categories we will consider, the parent groups are finitely generated, so
by Theorem 1.1 the automorphism groups of connected objects are all countable. Let us
define a category C to be countably (resp. finitely) abundant if every countable (resp. finite)
group A is isomorphic to AutC(O) for some connected object (resp. finite connected ob-
ject) O in C. Let us define C to be countably (resp. finitely) superabundant if there are 2ℵ0
(resp. ℵ0) isomorphism classes of such objects O realising each A.
In the case of a permutational category C, these properties follow immediately from
Theorem 1.1 if the associated parent group Γ has the corresponding abundance properties,
namely that every countable group A is isomorphic to NΓ(M)/M for the required number
of conjugacy classes of subgroups M of Γ, and these can be chosen to have finite index in
Γ if A is finite. If Γ is finitely generated, then the cardinalities 2ℵ0 (resp. ℵ0) are the best
that can be achieved, since they are upper bounds on the number of conjugacy classes of
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 3
subgroups (resp. subgroups of finite index) in Γ, and hence on the number of isomorphism
classes of objects (resp. finite objects) available.
We will be mainly concerned with permutational categories consisting of maps and
hypermaps of various types (p, q, r), where p, q, r ∈ N ∪ {∞}. For these, the parent
groups are either extended triangle groups ∆[p, q, r], generated by reflections in the sides of
a triangle with internal angles π/p, π/q and π/r, or (for subcategories of oriented objects)
their orientation-preserving subgroups, the triangle groups ∆(p, q, r). We say that a triple
(p, q, r) is spherical, euclidean or hyperbolic as p−1 + q−1 + r−1 > 1,= 1 or < 1
respectively (where by convention we take ∞−1 = 0), so that these groups act on the
sphere, euclidean plane, or hyperbolic plane. We will call the triple cocompact if these
groups act cocompactly, or equivalently p, q, r ∈ N. We will use Theorem 1.1 to prove:
Theorem 1.3.
(a) If (p, q, r) is a hyperbolic triple, where p, q, r ∈ N∪{∞}, then the groups ∆(p, q, r)
and ∆[p, q, r], together with their associated categories of oriented hypermaps and
of all hypermaps of type (p, q, r), are finitely superabundant.
(b) If, in addition, (p, q, r) is not cocompact then these groups and categories are all
countably superabundant.
By contrast, if we take Γ to be a Tarski monster [42], an infinite group in which every
subgroup M ̸= Γ, 1 has order p for some (very large) prime p, then NΓ(M) = M for each
M ̸= 1, and hence the only groups realised as automorphism groups in the corresponding
category are 1 and Γ.
The spherical and euclidean triples must be excluded from Theorem 1.3 since the cor-
responding triangle groups are either finite or solvable, so the same restriction applies to
the automorphism groups of connected objects in the associated categories. By taking
p = r = ∞ and q = 2 or ∞ we see that the categories M and H of all maps and hy-
permaps, together with their subcategories M+ and H+ of oriented maps and hypermaps,
satisfy Theorem 1.3. In the case of M+ and H+, Cori and Machı̀ [9] showed in 1982 that
every finite group arises as an automorphism group; they considered only finite groups,
but their proof extends to countable groups. In fact, by Theorem 1.3(a) the category of
Grothendieck’s dessins d’enfants [20] of any given hyperbolic type is finitely superabun-
dant. Of course these categories are not countably abundant. Nevertheless, in §8 we will
prove a result, based on work of Conder [7, 8] on alternating and symmetric quotients of
triangle groups, to support the following conjecture:
Conjecture 1.4. The non-cocompactness condition can be omitted from Theorem 1.3(b),
so that the triangle groups ∆(p, q, r) and ∆[p, q, r] of any hyperbolic type, and their asso-
ciated categories, are countably superabundant.
The proof of Theorem 1.3 is divided into several cases, depending on the particular
group Γ = ΓC involved and whether we wish to realise countable or finite groups. In each
case we construct a primitive permutation representation of Γ, of infinite or unbounded
finite degree, such that a point stabiliser N has an epimorphism onto a free group of count-
ably infinite or unbounded finite rank, and hence onto an arbitrary countable or finite group
A. By arranging that the kernel M is not normal in Γ we see from the maximality of N
in Γ that NΓ(M)/M = N/M ∼= A, so Theorem 1.1 gives AutC(O) ∼= A for the object
O ∈ C corresponding to M . Variations in the constructions yield 2ℵ0 or ℵ0 conjugacy
4 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
classes of such subgroups M ≤ Γ, and hence that number of objects O realising A. These
objects are regular coverings, with covering group A, of the object N ∼= O/AutC(O) in C
corresponding to N and its conjugates.
In fact a deep result of Belolipetsky and Lubotzky [3, Theorem 2.1] implies finite su-
perabundance for every finitely generated group which is large, that is, has a subgroup of
finite index with an epimorphism onto a non-abelian free group. This applies to every non-
elementary finitely generated Fuchsian group, and in particular to every hyperbolic triangle
group, as in Theorem 1.3(a). However, the proof of [3, Theorem 2.1] is long, delicate and
non-constructive, so here we offer a shorter, more direct argument, specific to the context
of this paper in using maps and hypermaps.
One should not confuse countable abundance with the SQ-universality of a group Γ,
a concept introduced by P. M. Neumann in [41], and proved there for (among others) all
hyperbolic triangle groups and extended triangle groups: this requires that every countable
group is isomorphic to a subgroup of a quotient of Γ, that is, to N/M where M ≤ N ≤ Γ
and M is normal in Γ, so that NΓ(M) = Γ, while countable abundance requires that
NΓ(M) = N . In terms of permutational categories, SQ-universality of the parent group
Γ means that every countable group A is embedded in the automorphism group of some
regular object O (one with a transitive monodromy group G, so that M is normal in Γ and
AutC(O) ∼= Γ/M ∼= G), whereas countable abundance means that A is isomorphic to the
automorphism group of some object, not necessarily regular. Both properties mean that any
phenomenon exhibited by some countable group, no matter how exotic or pathological, can
be realised within Γ, and hence within C: see [23] for some examples where C = H+.
Soon after this paper was submitted, a very interesting paper [5] by Bottinelli, Grave
de Peralta and Kolpakov appeared on the arXiv. It independently introduces some of the
concepts and proves some of the results presented here: for instance their concept of a
‘telescopic group’ coincides with our notion of finite abundance, and they prove this for all
free products of cyclic groups (except C2 ∗ C2). However, their methods of construction
differ substantially from ours, and they obtain asymptotic estimates for the number of finite
objects realising a given finite group as their automorphism group, a topic not considered
here.
2 Permutational categories
Following [24], let us define a permutational category C to be a category which is equiv-
alent to the category of permutation representations θ : Γ → S := Sym(Ω) of a parent
group Γ = ΓC. We then define the automorphism group Aut(O) = AutC(O) of an object
O in C to be the group of all permutations of Ω commuting with the action of Γ on Ω;
thus it is the centraliser CS(G) of the monodromy group G = θ(Γ) of O in the symmetric
group S. In this paper we will restrict our attention to the connected objects O in C, those
corresponding to transitive representations of Γ. We will pay particular attention to those
categories for which 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. We will now give some important examples of such
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 5
categories; for more details, see [24]. In what follows, Cn denotes a cyclic group of order
n, Fn denotes a free group of rank n, V4 denotes a Klein four-group C2×C2 and ∗ denotes
a free product.
1. The category M of all maps on surfaces (possibly non-orientable or with boundary) has
parent group
Γ = ΓM = ∆[∞, 2,∞] ∼= V4 ∗ C2.
This group acts on the set Ω of incident vertex-edge-face flags of a map (equivalently, the
faces of its barycentric subdivision), with each generator Ri (i = 0, 1, 2) changing the
i-dimensional component of each flag (whenever possible) while preserving the other two.
2. The subcategory M+ of M consists of the oriented maps, those in which the underlying
surface is oriented and without boundary. This category has parent group
Γ = ΓM+ = ∆(∞, 2,∞) ∼= C∞ ∗ C2,
the orientation-preserving subgroup of index 2 in ∆[∞, 2,∞]. This group acts on the
directed edges of an oriented map: X uses the local orientation to rotate them about their
target vertices, and the involution Y reverses their direction, so that Z rotates them around
incident faces. Here, and in the preceding example, ∆(p, 2, r) and ∆[p, 2, r] are the parent
groups for the subcategories of maps of type {r, p} in the notation of [10], meaning that
the valencies of all vertices and faces divide p and r respectively, so that Xp = Zr = 1.
(By convention, all positive integers divide ∞.)
3. Hypermaps are natural generalisation of maps, without the restriction that each edge is
incident with at most two vertices and faces which implies that Y 2 = 1. There are several
ways of defining or representing hypermaps. The most convenient way is via the Walsh
bipartite map [54], where the black and white vertices 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 (possibly unoriented
and with boundary) has parent group
Γ = ΓH = ∆[∞,∞,∞] ∼= C2 ∗ C2 ∗ C2.
This group acts on the incident edge-face pairs of the bipartite map, with R0 and R1 pre-
serving the face and the incident white and black vertex respectively, while R2 preserves
the edge. As in the case of maps, ∆[p, q, r] is the parent group for the subcategory of
hypermaps of type (p, q, r).
4. For the subcategory H+ of oriented hypermaps, where the underlying surface is oriented
and without boundary, the parent group is the even subgroup
Γ = ΓH+ = ∆(∞,∞,∞) ∼= C∞ ∗ C∞ ∼= F2
of index 2 in ∆[∞, 2,∞]. This acts on the edges of the bipartite map, 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. Again ∆(p, q, r) is the parent group for the
subcategory of oriented hypermaps of type (p, q, r). Hypermaps of type (p, 2, r) can be re-
garded as maps of type {r, p} by deleting their white vertices; conversely maps correspond
to hypermaps with q = 2.
6 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
5. One can regard the category D of dessins d’enfants, introduced by Grothendieck [20],
as the subcategory of H+ consisting of its finite objects, where the bipartite graph is finite
and the surface is compact. The parent group is Γ = ∆(∞,∞,∞) ∼= F2, and its action is
the same as for H+.
Here we briefly mention two other classes of permutational categories where Theo-
rem 1.1 applies.
6. Abstract polytopes [36] are higher-dimensional generalisations of maps. Those n-
polytopes associated with the Schläfli symbol {p1, . . . , pn−1} can be regarded as transitive
permutation representations of the string Coxeter group Γ with presentation
⟨R0, . . . , Rn | R2i = (Ri−1Ri)pi = (RiRj)2 = 1 (|i− j| > 1)⟩,
acting on flags. For instance maps, in Example 1, correspond to the symbol {∞,∞}.
However, in higher dimensions, not all transitive representations of Γ correspond to abstract
polytopes, since they need to satisfy the intersection property [36, Proposition 2B10].
7. Under suitable connectedness conditions (see [35, 39] for example), the connected, un-
branched 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 . The automorphism
group of an object Y → X in this category is its group of covering transformations, the
centraliser in S of the monodromy group θ(Γ) of the covering. For instance, dessins (see
Example 5 above) correspond to finite unbranched coverings of the thrice-punctured sphere
X = P1(C)\{0, 1,∞} = C\{0, 1}, and hence to transitive finite permutation representa-
tions of its fundamental group Γ = π1X ∼= F2 ∼= ∆(∞,∞,∞). If we compactify surfaces
by filling in punctures, then the unit interval [0, 1] ⊂ P1(C) lifts to a bipartite map on the
covering surface Y , with black and white vertices over 0 and 1, and face-centres over ∞.
See [17, 29, 31] for further details of these and other properties of dessins.
3 Preliminary results
In this section we will prove some general results which ensure that certain groups have
various automorphism realisation properties.
Lemma 3.1. Let θ : Γ → Γ′ be an epimorphism of groups. If Γ′ is finitely or countably
abundant or superabundant, then so is Γ.
Proof. If A ∼= N ′/M ′ where M ′ ≤ N ′ ≤ Γ′ and N ′ = NΓ′(M ′), then A ∼= N/M
where M = θ−1(M ′) and N = θ−1(N ′) = NΓ(M), with |Γ : M | = |Γ′ : M ′|, so Γ
inherits finite or countable abundance from Γ′. Moreover, non-conjugate subgroups M ′ lift
to non-conjugate subgroups M , so the superabundance properties are also inherited.
Our basic tool for proving finite superabundance will be the following:
Proposition 3.2. Let Γ be a group with a sequence {Nn | n ≥ n0} of maximal subgroups
Nn of finite index such that for each a, d ∈ N there is some n with |Nn : Kn| > a, where
Kn is the core of Nn in Γ, and there is an epimorphism Nn → Fd. Then Γ is finitely
superabundant.
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 7
Proof. Any finite group A is an d-generator group for some d ∈ N, so there is an epimor-
phism Fd → A. By hypothesis, for some maximal subgroup N = Nn of Γ there is an
epimorphism N → Fd, and the core K of N satisfies |N : K| > |A|. Composition gives
an epimorphism N → A, and hence a normal subgroup M of N with N/M ∼= A. Then
NΓ(M) ≥ N , so the maximality of N implies that either N = NΓ(M) or M is a normal
subgroup of Γ. If M is normal in Γ then M must be contained in the core K of N , so that
|N : M | ≥ |N : K|. But this is impossible, since |N : M | = |A| and we chose N = Nn
so that |N : K| > |A|. Hence N = NΓ(M), as required. Moreover, given A we can
find such subgroups N with |N : K| arbitrarily large, so infinitely many of them are mu-
tually non-conjugate, and hence so are their corresponding subgroups M , since conjugate
subgroups have conjugate normalisers.
In order to deal with countable abundance or superabundance we need an analogue
of Proposition 3.2 for countable groups A. Here we have the advantage that, instead of
an infinite sequence of maximal subgroups, which are finitely generated if Γ is, a single
infinitely generated maximal subgroup is sufficient. However, when A is infinite we cannot
ensure that M is not normal in Γ simply by comparing indices of subgroups, since these
are not finite; a new idea is therefore needed.
Proposition 3.3. Let Γ be a group with a non-normal maximal subgroup N and an epi-
morphism ϕ : N → F∞. Then Γ is countably abundant. Moreover, each countable group
A ̸= 1 is realised as NΓ(M)/M by 2ℵ0 conjugacy classes of subgroups M in Γ with
NΓ(M) = N .
Proof. Given any countable group A there exist epimorphisms α : F∞ → A; composing
any of these with the epimorphism ϕ : N → F∞ gives an epimorphism ϕ ◦ α : N → A,
and hence a normal subgroup M = ker(ϕ ◦ α) of N with N/M ∼= A. As before, the
maximality of N implies that either N = NΓ(M), as required, or M is a normal subgroup
of Γ. In the latter case M is contained in the core K of N in Γ, so to prove the result we
need to show that we can choose α so that M ̸≤ K. Since N is not normal in Γ we have
N \K ̸= ∅, so choose any element g ∈ N \K, and define f := gϕ ∈ F∞. Then we can
choose α : F∞ → A so that all of the (finitely many) free generators of F∞ appearing in f
are in ker(α), and hence g ∈ M . Thus M ̸≤ K, so Γ is countably abundant.
If A ̸= 1 we can choose such epimorphisms α with 2ℵ0 different kernels, lifting back
to distinct subgroups M of Γ; these all have normaliser N , which is its own normaliser in
Γ, so they are mutually non-conjugate in Γ, giving us 2ℵ0 conjugacy classes of subgroups
M realising A.
Remark 3.4. Unfortunately, if A = 1 then α is unique, so that M = N , and the subgroup
N yields only one conjugacy class of subgroups realising A. In this case, in order to prove
that Γ is countably superabundant by this construction we would need to find not one but
2ℵ0 conjugacy classes of non-normal maximal subgroups N . For certain specific groups Γ
we will be able to do this.
4 Finite superabundance of hyperbolic triangle and extended triangle
groups
In this section we will use Proposition 3.2 to prove Theorem 1.3(a).
8 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
Case 1: Γ = ∆(p, q, r), cocompact. First assume that Γ = ∆(p, q, r), acting cocompactly
on the hyperbolic plane, that is, with finite periods p, q and r. By Dirichlet’s Theorem on
primes in an arithmetic progression, there are infinitely many primes n ≡ −1 mod (l),
where l := lcm{2p, 2q, 2r}. For each such n there is an epimorphism θn : Γ → PSL2(n)
sending the standard generators X,Y and Z of Γ to elements x, y and z of PSL2(n) of
orders p, q and r (see [16, Corollary C], for example). This gives an action of Γ on the
projective line P1(Fn), which is doubly transitive and hence primitive, so the subgroup Nn
of Γ fixing ∞ is a non-normal maximal subgroup of index n+1. Since p, q and r all divide
(n+ 1)/2, the elements x, y and z are semi-regular permutations on P1(Fn), with all their
cycles of length p, q or r. Thus no non-identity powers of X , Y or Z have fixed points, so
by a theorem of Singerman [48] Nn is a surface group
Nn = ⟨Ai, Bi (i = 1, . . . , g) |
g∏
i=1
[Ai, Bi] = 1⟩
of genus g given by the Riemann–Hurwitz formula:
2(g − 1) = (n+ 1)
(
1− 1
p
− 1
q
− 1
r
)
. (4.1)
This shows that g → ∞ as n → ∞. Now we can map Nn onto the free group Fg by sending
the generators Ai to a free basis, and the generators Bi to 1. The core Kn = ker(θn) of
Nn in Γ satisfies |Nn : Kn| = |PSL2(n)|/(n+1) = n(n− 1)/2, so Proposition 3.2 gives
the result.
Case 2: Γ = ∆(p, q, r), not cocompact. Now assume that Γ has k infinite periods p, q, r
for some k = 1, 2 or 3. We can adapt the above argument by first choosing an infinite set of
primes n ≥ 13 such that any finite periods of Γ divide (n+ 1)/2, as before. For each such
n we can map Γ onto a cocompact triangle group Γn, where each infinite period of Γ is
replaced with (n+1)/2. Since (n+1)/2 ≥ 7, the triangle group Γn is also hyperbolic, so as
before there is an epimorphism Γn → PSL2(n), giving (by composition) a primitive action
of Γ on P1(Fn). Again, no non-identity powers of any elliptic generators among X,Y and
Z have fixed points, but any parabolic generator (one of infinite order) now induces two
cycles of length (n+ 1)/2 on P1(Fn), so by [48] it introduces two parabolic generators Pi
into the standard presentation of the point-stabiliser Nn in Γ. We therefore have
Nn = ⟨Ai, Bi (i = 1, . . . , g), Pi (i = 1, . . . , 2k) |
g∏
i=1
[Ai, Bi].
2k∏
i=1
Pi = 1⟩,
a free group of rank 2g + 2k − 1, where the Riemann–Hurwitz formula now gives
2(g − 1) + 2k = (n+ 1)
(
1− 1
p
− 1
q
− 1
r
)
(4.2)
with 1/∞ = 0. Since k ≤ 3 we have g → ∞ as n → ∞, so Proposition 3.2 again gives
the result.
Case 3: Γ = ∆[p, q, r]. The proof when Γ is an extended triangle group ∆[p, q, r] of
hyperbolic type is similar to that for ∆(p, q, r). If Γ is cocompact then, as before, we
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 9
consider epimorphisms θn : Γ+ = ∆(p, q, r) → PSL2(n) for primes n ≡ −1 mod (l),
where now l = lcm{2p, 2q, 2r, 4}; the stabilisers of ∞ form a series of maximal subgroups
Nn of index n + 1 in Γ+. By an observation of Singerman [50] the core Kn of Nn in
Γ+ is normal in Γ, with quotient Γ/Kn isomorphic to PSL2(n) × C2 or PGL2(n) as
the automorphism of PSL2(n) inverting x and y is inner or not. Thus θn extends to a
homomorphism θ∗n : Γ → PGL2(n); in the first case its image is PSL2(n) and its kernel
K∗n contains Kn with index 2, and in the second case it is an epimorphism with kernel
K∗n = Kn. In either case the action of Γ
+ on P1(Fn) extends to an action of Γ, and the
stabiliser N∗n of ∞ is a maximal subgroup of index n + 1 in Γ, containing Nn with index
2. In order to apply Proposition 3.2 to these subgroups N∗n it is sufficient to show that they
map onto free groups of unbounded rank.
Now N∗n is a non-euclidean crystallographic (NEC) group, and Nn is its canonical
Fuchsian subgroup of index 2. We can obtain the signature of N∗n by using Hoare’s exten-
sion to NEC groups [22] of Singerman’s results [48] on subgroups of Fuchsian groups. As
before, Nn is a surface group of genus g given by (4.1). There are no elliptic or parabolic
elements in Nn, and hence none in N∗n. The reflections Ri (i = 0, 1, 2) generating Γ in-
duce involutions on P1(Fn), each with at most two fixed points. If Γ/Kn ∼= PSL2(n)×C2
these involutions are elements of PSL2(n), so they are even permutations by the simplicity
of this group, and hence they have no fixed points since n+1 ≡ 0 mod (4). Thus N∗n con-
tains no reflections; however, it is not a subgroup of Γ+, so it is a non-orientable surface
group
N∗n = ⟨G1, . . . , Gg∗ | G21 . . . G2g∗ = 1⟩
generated by glide-reflections Gi, with its genus g∗ given by the Riemann–Hurwitz formula
2− 2g = 2(2− g∗)
for the inclusion Nn ≤ N∗n, so g∗ = g + 1. Thus there is an epimorphism N∗n → Fd =
⟨X1, . . . , Xd | −⟩ where d = ⌊g∗/2⌋, given by G2i−1 7→ Xi and G2i 7→ X−1i for
i = 1, . . . , d and Gg∗ 7→ 1 if g∗ is odd. Since g → ∞ as n → ∞, we have d ∼ g/2 → ∞
also, so Proposition 3.2 gives the result.
Similar arguments also deal with the case where Γ/Kn ∼= PGL2(n). Since n ≡ −1
mod (4), each generating reflection Ri of Γ induces an odd permutation of P1(Fn) with
two fixed points, contributing two reflections to the standard presentation of the NEC group
N∗n. The Riemann–Hurwitz formula for the inclusion Nn ≤ N∗n then takes the form
2− 2g = 2(2− h∗ + s),
where h∗ = 2g∗ or g∗ as N∗n has an orientable or non-orientable quotient surface of genus
g∗ with s boundary components for some s ≤ 6. Thus h∗ ∼ g → ∞ as n → ∞. We
obtain an epimorphism N∗n → Fd with d ∼ h∗/2 as in the orientable or non-orientable
cases above, this time by mapping the additional standard generators of N∗n, associated
with the boundary components, to 1, so Proposition 3.2 again gives the result. Finally, in
the non-cocompact case, any periods p, q, r = ∞ can be dealt with as above for ∆(p, q, r).
This completes the proof of Theorem 1.3(a) .
Remark 4.1. It seems plausible that an argument based on the Čebotarev Density Theorem
would show that, given Γ = ∆[p, q, r], the cases Γ/Kn ∼= PSL2(n) × C2 and PGL2(n)
each occur for infinitely many primes n ≡ −1 mod (l), so that only one case would need to
10 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
be considered; however, the resulting shortening of the proof would not justify the effort.
Nevertheless this dichotomy, for general prime powers n, is interesting in its own right and
deserves further study.
Remark 4.2. The restrictions on the prime n in the above proof are partly for convenience
of exposition, rather than necessity. Relaxing them would allow X,Y and Z to have one
or two fixed points on P1(Fn), thus adding extra standard generators to Nn and N∗n and
extra summands to the Riemann–Hurwitz formulae used. However, these extra terms are
bounded as n → ∞, so asymptotically they make no significant difference. One advantage
of these restrictions is that since x, y and z are semi-regular permutations, the hypermaps
realising A in Case 1 are uniform, that is, their hypervertices, hyperedges and hyperfaces
all have valencies p, q and r. If we choose these periods so that ∆(p, q, r) is cocompact,
maximal (see [49]) and non-arithmetic (see [52]), then by a result of Singerman and Syd-
dall [51, Theorem 12.1] each hypermap (regarded as a dessin) has the same automorphism
group as its underlying Riemann surface. By [49, 52] these conditions apply to ‘most’
hyperbolic triples, such as (2, 3, 13), so we have the following:
Corollary 4.3. The category of compact Riemann surfaces is finitely superabundant.
In fact, Greenberg [19, Theorem 6′] showed in 1973 that, given a compact Riemann
surface S and a finite group A ̸= 1, there is a normal covering T → S with covering
group and Aut(T ) both isomorphic to A, while Teichmüller theory yields uncountably
many compact Riemann surfaces realising A. Since the Riemann surfaces realising a finite
group A in Corollary 4.3 are uniformised by subgroups of finite index in triangle groups, by
Grothendieck’s reinterpretation [20] of Belyı̆’s Theorem they are all defined (as algebraic
curves with automorphism group A) over number fields.
5 Countable abundance of non-cocompact hyperbolic triangle groups
We now turn to Theorem 1.3(b) and consider countable abundance, starting with the hyper-
bolic triangle groups Γ = ∆(p, q, r). We would like to show that Γ satisfies the hypotheses
of Proposition 3.3, that is, it has a non-normal maximal subgroup which has an epimor-
phism onto F∞. Given Γ, it is easy to find maximal subgroups of finite index by mapping
Γ onto primitive permutation groups of finite degree; however, such subgroups are finitely
generated, so they do not map onto F∞; a maximal subgroup of infinite index is needed,
and these seem to be harder to find. They certainly exist: by a result of Ol’shanskiı̆ [43],
Γ has a quotient Q ̸= 1 with no proper subgroups of finite index; by Zorn’s Lemma, Q
has maximal subgroups, which must have infinite index, and these lift back to maximal
subgroups of infinite index in Γ. These are not normal (otherwise they would have prime
index), but does one of them map onto F∞? Conceivably, they could be generated by ellip-
tic elements, which have finite order, in which case they would not map onto a free group
of any rank. As a first step we consider the case where Γ is not cocompact, that is, it has an
infinite period, so it is a free product of two cyclic groups. For simplicity of exposition we
first consider countable abundance, postponing superabundance until the next section.
Theorem 5.1. If Γ is a non-cocompact hyperbolic triangle group ∆(p, q, r), then Γ and
the corresponding category of oriented hypermaps are countably abundant.
Proof. By Proposition 3.3 it is sufficient to show that Γ has a non-normal maximal sub-
group N with an epimorphism N → F∞. Using the usual isomorphisms between triangle
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 11
groups, we may assume that r = ∞, and that p ≥ 3 and q ≥ 2, so that Γ ∼= Cp ∗ Cq with
p, q ∈ N ∪ {∞}.
Case 1: p = 3, q = 2. First we consider the case where p = 3 and q = 2, so that Γ is iso-
morphic to the modular group PSL2(Z) ∼= C3 ∗C2. We can construct a maximal subgroup
N of infinite index in Γ as the point stabiliser in a primitive permutation representation of
Γ of infinite degree. Since Γ is the parent group
∆(3, 2,∞) = ⟨X,Y, Z | X3 = Y 2 = XY Z = 1⟩
for the category C = M+3 of oriented trivalent maps, we can take N to be the subgroup of
Γ corresponding to an infinite map N3 in C.
−3−2−1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
4n− 2
4n− 1
4n
4n+ 1
1− n
F
F1 F2 F3 F4 Fn (n ≥ 3)
Figure 1: The trivalent map N3.
We will take N3 to be the infinite planar trivalent map shown in Figure 1, oriented with
the positive (anticlockwise) orientation of the plane. The monodromy group G = ⟨x, y⟩ of
this map gives a transitive permutation representation θ : Γ → G,X 7→ x, Y 7→ y, Z 7→ z
of Γ on the set Ω of directed edges of N3, with x rotating them anticlockwise around their
target vertices, and y reversing their direction. The vertices, all of valency 3, correspond
to the 3-cycles of x (it has no fixed points). The edges correspond to the cycles of y, with
three free edges corresponding to its fixed points and the other edges corresponding to its
2-cycles. The faces correspond to the cycles of z = yx−1, and in particular, the directed
edge α labelled 0 and fixed by y is in an infinite cycle C = (. . . , αz−1, α, αz, . . .) of z,
corresponding to the unbounded face F of N3; the directed edges αzi in C are indicated
by integers i in Figure 1. The unlabelled directed edges are fixed points of z, one incident
with each 1-valent face. The pattern seen in Figure 1 repeats to the right in the obvious
way. The ‘flowers’ Fn (n ≥ 1) above the horizontal axis continue indefinitely to the right,
with Fn an identical copy of F1 for each n ≥ 3; we will later need the fact that for each
n ≥ 2 the ‘stem’ of Fn (the vertical edge connecting it to the horizontal axis) carries two
directed edges in C, with only one of their two labels divisible by n.
Lemma 5.2. The group Γ acts primitively on Ω.
Proof. Suppose that ∼ is a Γ-invariant equivalence relation on Ω; we need to show that it
is either the identity or the universal relation. Since αy = α, the equivalence class E = [α]
containing α satisfies Ey = E. Since ⟨Z⟩ acts regularly on C we can identify C with Z by
identifying each αzi ∈ C with the integer i, so that Z acts by i 7→ i+ 1. Then ∼ restricts
to a translation-invariant equivalence relation on Z, which must be congruence mod (n) for
12 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
some n ∈ N∪{∞}, where we include n = 1 and ∞ for the universal and identity relations
on Z.
Suppose first that n ∈ N, so E ∩ C is the subgroup (n) of Z. If n = 1 then C ⊆ E.
Now Ex−1 is an equivalence class, and it contains αx−1 = 1; this is in C, and hence in E,
so Ex−1 = E. We have seen that Ey = E, so E = Ω since G = ⟨x−1, y⟩, and hence ∼ is
the universal relation on Ω.
We may therefore assume that n > 1. The vertical stem of the flower Fn is an edge
carrying two directed edges in C, with only one of its two labels divisible by n, so one
is in E whereas the other is not. However, these two directed edges are transposed by y,
contradicting the fact that Ey = E.
Finally suppose that n = ∞, so that all elements of C are in distinct equivalence
classes, and hence the same applies to Cy. In particular, since α ∈ C ∩ Cy we have
E ∩ C = {α} = E ∩ Cy. By inspection of Figure 1, Ω = C ∪ Cy and hence E = {α}.
It follows that all equivalence classes for ∼ are singletons, so ∼ is the identity relation, as
required.
We now return to the proof of Theorem 5.1. It follows from Lemma 5.2 that the sub-
group N = Γα of Γ fixing α is maximal. Clearly N is not normal in Γ, since G is not a
regular permutation group, so it sufficient to find an epimorphism N → F∞. One could
use the Reidemeister–Schreier algorithm to find a presentation for N : truncation converts
N3 into a coset diagram for N in Γ, and then deleting edges to form a spanning tree yields
a Schreier transversal. In fact a glance at Figure 1 shows that N is a free product of cyclic
groups: three of these, corresponding to the fixed points of y and generated by conjugates
of Y , have order 2, and there are infinitely many of infinite order, generated by conjugates
of Z and corresponding to the fixed points of z, that is, the 1-valent faces of N3, one in
each flower Fn for n ̸= 2. By mapping the generators of finite order to the identity we
obtain the required epimorphism N → F∞.
−1
0
1 2
3 4
p+ 2
t
t+ 1
t+ 3 t+ 4
2t− 1 nt
nt+ 1
nt+ 3 nt+ 4
(n+ 1)t− 1
F
F0 F1 Fn (n ≥ 1)
Figure 2: The p-valent map Np, with t := p+ 3.
Case 2: Finite p ≥ 4, q = 2. We now assume that Γ is a Hecke group Cp ∗ C2 for some
finite p ≥ 4. Let Np be the infinite p-valent planar map in Figure 2. Apart from F0, the
flowers are all identical copies of F1, with a ‘leaf’ growing out of its base and leading to
a vertex of valency 1, representing a fixed point of x. The ‘fans’ indicated by short dashed
lines represent however many free edges are needed in order that the incident vertex should
have valency p, that is, p − 3 free edges for vertices at the top of a stem, and p − 4 for
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 13
those at the base. As before, the elements αzi of the cycle C of z containing the directed
edge α = 0 are labelled with integers i; to save space in the diagram only a few labels are
shown. We define t = p + 3, since a translation from a flower Fn (n ≥ 1) to the next
flower Fn+1 adds that number to all labels.
The proof that the monodromy group G = ⟨x, y⟩ of Np is primitive is very similar to
that in Lemma 5.2 for N3. Any Γ-invariant equivalence relation ∼ on Ω restricts to C as
congruence mod (n) for some n ∈ N ∪ {∞}. The equivalence class E = [α] satisfies
Ey = E, so if n ̸= 1,∞ then the fact that y transposes the directed edges labelled nt and
nt + 1, with the first but not the second in E = (n), gives a contradiction. If n = 1 then
C ⊆ E, so both x and y preserve E and hence ∼ is the universal relation. If n = ∞ then
E ∩ C = {α}, and hence also E ∩ Cy = {α}; but Ω = C ∪ Cy and hence E = {α} and
∼ is the identity relation.
This shows that the subgroup N of Γ fixing α is maximal. As before, it is not normal,
and it is a free product of cyclic groups, now of order p, 2 or ∞, corresponding to the fixed
points of x, y and z (infinitely many in each case). Sending the generators of finite order to
the identity gives the required epimorphism N → F∞.
Case 3: Finite p, q ≥ 3. We modify the map Np in the proof of Case 2 by removing the leaf
attached to the base of each flower Fn (n ≥ 1), adding a white vertex to every remaining
edge (including one at the free end of each free edge), and finally adding edges incident
with 1-valent black vertices where necessary to ensure that all white vertices have valency
q or 1. The resulting map Np,q is shown in Figure 3.
−1
0
1
2
3 q
q + 1
t− 1
t t+ 1
t+ 2
t+ q
t+ q + 1
2t− 1
2t nt nt+ 1
nt+ 2
nt+ q
nt+ q + 1
(n+ 1)t− 1
F
F0
W1
F1
W2 Wn
Fn (n ≥ 1)
Figure 3: The bipartite map Np,q , with t := p+ 2q − 2.
Note that while the flowers Fn have grown since Case 2 was proved, small ‘weeds’
Wn (n ≥ 1) have grown between them. This bipartite map is the Walsh map for an oriented
hypermap of type (p, q,∞). Its monodromy group G is generated by permutations x and
y, of order p and q, which rotate edges around their incident black and white vertices. It
is sufficient to show that G acts primitively on the set Ω of edges, and that in the induced
action of Γ on Ω, the subgroup N fixing an edge has an epimorphism onto F∞.
The proof is similar to that for Case 2. The elements αzi of the cycle C of z containing
14 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
the edge α = 0 are labelled with integers i. (To save space in Figure 3, only a few signifi-
cant labels are shown.) Any Γ-invariant equivalence relation ∼ restricts to C as congruence
mod (n) for some n ∈ N ∪ {∞}. If E = [α] then since αy = α we have Ey = E. If
n ̸= ∞ then Ex = E since the edge β ∈ E labelled nt is fixed by x, so that EΓ = E
and hence E = Ω. Thus we may assume that n = ∞, so all elements of C are in distinct
conjugacy classes and hence E∩C = {α}. Similarly E∩Cy = {α}. But Ω = C∪Cy, so
E = {α} and ∼ is the identity relation. Thus G is primitive, so the subgroup N of Γ fixing
α is maximal. It is a free product of cyclic groups, of orders p, q and ∞, corresponding
to the fixed points of x, y and z. There are infinitely many of each, and mapping those of
finite order to the identity gives an epimorphism N → F∞.
Case 4: p or q = ∞. If p = ∞ or q = ∞ we can use the natural epimorphism from
Γ = ∆(p, q,∞) to a hyperbolic triangle group Γ′ = ∆(p′, q′,∞) with p′ and q′ both finite,
use Case 1, 2 or 3 to establish countable abundance for Γ′, and finally use Lemma 3.1 to
deduce it for Γ.
Remark 5.3. Constructions similar to those in the proof of Theorem 5.1 have been used
in [25] to prove that if Γ is a non-cocompact hyperbolic triangle group then Γ has uncount-
ably many conjugacy classes of maximal subgroups of infinite index. This strengthens
and generalises results of B. H. Neumann [40], Magnus [33, 34], Tretkoff [53], and Bren-
ner and Lyndon [6] on maximal nonparabolic subgroups of the modular group, and has
some overlap with work of Kulkarni [30] on maximal subgroups of free products of cyclic
groups.
6 Countable superabundance of non-cocompact hyperbolic triangle
groups
In order to prove countable superabundance for non-cocompact hyperbolic triangle groups
Γ = ∆(p, q, r), we need 2ℵ0 objects realising each countable group A. The proofs of
countable abundance for the various cases in Theorem 5.1 all used Proposition 3.3, and by
Remark 3.4 this yields the required number of objects in all cases except when A ̸= 1.
In fact, for any countable group A these proofs can be adapted (as in Remark 5.3) to
produce not just one but 2ℵ0 conjugacy classes of subgroups N satisfying the conditions of
Proposition 3.3. We thus obtain 2ℵ0 non-isomorphic objects N , each with 2ℵ0 coverings
O realising any countable group A ̸= 1, and with one covering (namely O = N ) realising
A = 1; each O has the property that O/Aut(O) ∼= N , so A is realised by 2ℵ0 non-
isomorphic objects.
We will give the required details for Case 1 of Theorem 5.1, where p = 3, q = 2 and
Γ is the modular group; the argument is similar in the other cases. We can modify the map
N3 in Figure 1 by adding ‘stalks’ between the flowers Fn, each consisting of a new vertex
on the horizontal axis, and a new free edge pointing upwards. Adding a stalk between Fm
and Fm+1 adds 2 to the value of all labels on flowers Fn for n > m. For the proof of
Theorem 5.1 to work we need to preserve the property that only one of the two labels on
the stem of each flower Fn (n > 1) is divisible by n. This can be done, in 2ℵ0 different
ways, by ensuring that for each n > 1 the total number of stalks added between F1 and Fn
is a multiple of n. The proof for Case 1 then proceeds as before, except that it now yields
2ℵ0 conjugacy classes of maximal subgroups N .
In the remaining cases of Theorem 5.1 we could use similar modifications to the maps
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 15
Np and Np,q in Figures 2 and 3, or alternatively add extra vertices and edges to those below
the horizontal axis, so that the non-negative labels above the axis are unaltered.
7 Countable superabundance of non-cocompact extended triangle
groups
We now consider countable superabundance for extended triangle groups Γ = ∆[p, q, r]
and their associated categories of unoriented hypermaps, again restricting attention to non-
cocompact groups. Earlier we realised countable groups A as automorphism groups in
various categories C+ of oriented hypermaps of a given type by constructing specific ob-
jects N = Np (p ≥ 3) or Np,q (p, q ≥ 3) in those categories, and then forming regular
coverings M of N , with covering group A, constructed so that M has only those auto-
morphisms induced by A. These objects M and N correspond to subgroups M and N of
the parent group Γ+ = ∆(p, q, r) for C+ with N = NΓ+(M). We can also regard M and
N as objects in the corresponding category C of unoriented maps or hypermaps of type
(p, q, r), for which the parent group is the extended triangle group Γ.
Lemma 7.1. For these objects M we have AutC(M) = AutC+(M) ∼= A.
Proof. We have AutC(M) ∼= NΓ(M)/M and AutC+(M) ∼= NΓ+(M)/M ∼= A by Theo-
rem 1.1, so it is sufficient to show that NΓ(M) = NΓ+(M). Clearly NΓ(M) ≥ NΓ+(M).
If this inclusion is proper then since NΓ+(M) = NΓ(M) ∩ Γ+ with |Γ : Γ+| = 2 we have
|NΓ(M) : NΓ+(M)| = 2, so the subgroup N = NΓ+(M) is normalised by some elements
of Γ \ Γ+. This is impossible, since in all cases the map or hypermap N corresponding
to N is chiral (without orientation-reversing automorphisms), by the proof of Theorem 5.1
and by inspection of Figures 1, 2 and 3. The same applies to the modified maps required to
produce 2ℵ0 such objects O.
Corollary 7.2. Each non-cocompact hyperbolic extended triangle group ∆[p, q, r] and its
associated category of all hypermaps of type (p, q, r) are countably superabundant.
This completes the proof of Theorem 1.3(b).
Remark 7.3. It would not have been possible to use Lemma 7.1 also in the proof of The-
orem 1.3(a) in §4, since the maximal subgroups Nn of Γ+ = ∆(p, q, r) constructed there
are normalised by orientation-reversing elements of Γ = ∆[p, q, r]. Instead of the natural
representation of PSL2(n), we could have used its representation on the cosets of a maxi-
mal subgroup H ∼= A5 for n ≡ ±1 mod (5), or H ∼= S4 for n ≡ ±1 mod (8): in both of
these cases there are two conjugacy classes of subgroups H , transposed by conjugation in
PGL2(n) (see [11, Ch. XII]) and corresponding to a chiral pair of hypermaps. However, in
either case the point stabilisers H have constant order as n → ∞, whereas Proposition 3.2
requires |Nn : Kn| = |H| to be unbounded, so we would need an alternative argument to
show that M is not normal in Γ+, as in the proof of Proposition 3.3.
8 Countable superabundance of some cocompact triangle groups
Theorem 1.3(b) proves countable superabundance only for non-cocompact hyperbolic tri-
angle groups ∆(p, q, r) and ∆[p, q, r]. We would like to extend to this property to the
cocompact case. The arguments we used to prove Theorem 1.3(b) depend on a standard
generator of ∆(p, q, r) (Z, without loss of generality) having infinite order, so that it can
16 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
have a cycle C of infinite length in some permutation representation, which is then proved
to be primitive by identifying C with Z. Clearly this is impossible in the cocompact case,
so a different approach is needed. The following is a first step in this direction.
Proposition 8.1. If one of p, q and r is even, another is divisible by 3, and the third is at
least 7, then the cocompact triangle groups ∆(p, q, r) and ∆[p, q, r] and their associated
categories are countably superabundant.
Proof. By permuting periods and applying Lemma 3.1 we may assume that p = 3, q = 2
and r ≥ 7. First suppose that r = 7. We will construct an infinite transitive permutation
representation of the group
Γ = ∆[3, 2, 7] = ⟨X,Y, T | X3 = Y 2 = T 2 = (XY )7 = (XT )2 = (Y T )2 = 1⟩
(where T = R2) in which the subgroup Γ+ = ∆(3, 2, 7) = ⟨X,Y ⟩ acts primitively, and
we will then apply Proposition 3.3 to a point-stabiliser in Γ+. This representation is con-
structed by adapting the Higman–Conder technique of ‘sewing coset diagrams together’,
used in [7] to realise finite alternating and symmetric groups as quotients of Γ+ and Γ. We
refer the reader to [7] for full technical details of this method. (Note that we have changed
Conder’s notation, which has X2 = Y 3 = 1, by transposing the symbols X and Y ; this
has no significant effect on the following proof.)
G H
Figure 4: The maps G and H.
Conder gives 14 coset diagrams A, . . . , N for subgroups of index n = 14, . . . , 108 in
Γ+, with respect to the generators X and Y ; these can be interpreted as describing transitive
representations of Γ+ of degree n. Each diagram is bilaterally symmetric, so this action
of Γ+ extends to a transitive representation of Γ of degree n, with T fixing vertices on the
vertical axis of symmetry, and transposing pairs of vertices on opposite sides of it. Although
Conder does not do this, in the spirit of the proof of Theorem 5.1 we can convert each of
his diagrams into a planar map of type {7, 3} (equivalently a hypermap of type (3, 2, 7))
by contracting the small triangles representing 3-cycles of X to trivalent vertices, so that
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 17
the cycles of X,Y and Z on directed edges correspond to its vertices, edges and faces.
(Warning: although Γ+ acts as the monodromy group of this oriented map, permuting
directed edges as described in Example 2 of §2, Γ does not act as the monodromy group
of the unoriented map, as in Example 1: the latter permutes flags, whereas T uses the
symmetry of the map to extend the action of Γ+ on directed edges to an action of Γ on
directed edges.)
We will construct an infinite coset diagram from Conder’s diagrams G and H of de-
gree n = 42; the corresponding maps G and H are shown in Figure 4. Conder defines
a (1)-handle in a diagram to be a pair α, β of fixed points of Y with β = αX = αT ,
represented in the corresponding map by two free edges incident with the same vertex
on the axis of symmetry. Thus G has three (1)-handles, while H has one. If diagrams
Di (i = 1, 2) of degree ni have (1)-handles αi, βi then one can form a new diagram,
called a (1)-join D1(1)D2, by replacing these four fixed points of Y with transpositions
(α1, α2) and (β1, β2), and leaving the permutations representing X,Y and T in D1 and D2
otherwise unchanged; the result is a new coset diagram giving a transitive representation
of Γ of degree n1 + n2. In terms of the corresponding maps Di, this is a connected sum
operation, in which the two surfaces are joined across cuts between the free ends of the
free edges representing the fixed points αi and βi; in particular, if Di has genus gi then
D1(1)D2 has genus g1+g2. This is illustrated in Figure 5, where the (1)-handle at the bot-
tom of G is joined to that at the top of H by two dashed edges to form G(1)H; these edges
can be carried by a tube connecting the two surfaces, showing the additivity of the genera
(both equal to 0 here). (Further details about this and more general joining operations on
dessins can be found in [27].)
G
H
Figure 5: Joining G and H to form G(1)H.
Using (1)-handles in G and H , we first form an infinite diagram
H(1)G(1)G(1)G(1)G · · · .
corresponding to an infinite planar map H(1)G(1)G(1)G(1)G · · · of type {7, 3}: the (1)-
handle at the top of each map H or G is joined, as in Figure 5, to that at the bottom of
the next map G. In this chain, each copy of G has an unused (1)-handle; we join these
18 Ars Math. Contemp. 21 (2021) #P1.01 / 1–22
arbitrarily in pairs, using (1)-compositions. Each such join adds a bridge to the underlying
surface, increasing the genus by 1, so the result is an oriented trivalent map N of type {7, 3}
and of infinite genus. This gives an infinite transitive permutation representation X 7→ x,
Y 7→ y, T 7→ t of Γ on the directed edges of N , with Γ+ again acting as its monodromy
group, and T acting as a reflection.
We need to prove that Γ and Γ+ act primitively. As shown by Conder [7] the permu-
tation w = yxt (= xyt in his notation) induced by Y XT has cycle structures 13133 and
1131101111171 in G and H . In each of the (1)-compositions we have used, two fixed
points of w are paired to form a cycle of length 2 of w, and a cycle of w of length 13 in G is
merged with one of length 13 or 10 in G or H to form a cycle of length 26 or 23. All other
cycles of w are unchanged, so in particular its cycle C of length 17 in H remains a cycle in
the final diagram. Since all other cycles of w have finite length coprime to 17, some power
of w acts on C as w and fixes all other points. Since 17 is prime, it follows that if Γ+ acts
imprimitively, then all points in C must lie in the same equivalence class E. Now C is
what Conder calls a ‘useful cycle’, since it contains a fixed point of y not in a (1)-handle
(the right-hand free edge β in the central circle in H in Figure 4) and a pair of points from
a 3-cycle of x (namely β and βx = βw8). It follows that X and Y leave E invariant,
which is impossible since they generate the transitive group Γ+. Thus Γ+ acts primitively
(as therefore does Γ), so the point-stabilisers N = Γα and N+ = Γ+α of a directed edge
α are maximal subgroups of Γ and Γ+. By the Reidemeister–Schreier algorithm, N+ is a
free product of four cyclic groups of order 2 (arising from fixed points of y in H not in the
(1)-handle), and infinitely many of infinite order (two arising from each bridge between a
pair of copies of G). Thus N+ admits an epimorphism onto F∞, so Proposition 3.3 shows
that Γ+ is countably abundant.
We can choose α to be fixed by t, so that T ∈ N , and hence N is a semidirect product
of N+ by ⟨T ⟩. The action of t is to reflect H and all the copies of G in the diagram,
together with the bridges added between pairs of them. Acting by conjugation on N+,
T therefore induces two transpositions on the elliptic generators of order 2. Each bridge
contributes a pair of free generators to N+, one of them (represented by a loop crossing
the bridge and returning ‘at ground level’), centralised by T , the other (represented by
a loop transverse to the first, following a cross-section of the bridge) inverted by T ; by
sending T , together with the inverted generators and the four elliptic generators of N+, to
the identity, we can map N onto the free group of countably infinite rank generated by the
centralised generators, so Proposition 3.3 shows that Γ is countably abundant. In fact, there
are 2ℵ0 ways of pairing the copies of G to produce bridges, giving mutually inequivalent
permutation representations and hence mutually non-conjugate subgroups N and N+ of Γ
and Γ+, so this argument establishes countable superabundance.
The extension to the case r ≥ 7 is essentially the same, but based on the coset dia-
grams in Conder’s later paper [8] on alternating and symmetric quotients of ∆(3, 2, r) and
∆[3, 2, r]. In this case his coset diagrams S(h, d) and U(h, d) play the roles of G and H ,
where r = h+ 6d with d ∈ N and h = 7, . . . , 12.
Proposition 8.1 accounts for a proportion 121/216 of all hyperbolic triples. It seems
plausible that coset diagrams of Everitt [13] and others, constructed to extend Conder’s
results on alternating group quotients to all finitely generated non-elementary Fuchsian
groups, could be used to prove that all cocompact hyperbolic triangle groups ∆(p, q, r)
and ∆[p, q, r], together with their associated categories, are countably superabundant, thus
proving Conjecture 1.4.
G. A. Jones: Realisation of groups as automorphism groups in permutational categories 19
9 Realisation of other groups
Finally, although this paper is mainly about triangle groups and their associated categories
of maps and hypermaps, we can deduce realisation properties for many other groups and
categories.
Theorem 9.1. If a group Γ has an epimorphism onto a non-abelian free group then it is
finitely and countably superabundant.
Proof. By Theorem 1.3 the free group F2 = ∆(∞,∞,∞) has these properties. Since
F2 is an epimorphic image of every other non-abelian free group, the result follows from
Lemma 3.1.
For example, Theorem 9.1 applies to the fundamental groups Γ of many topological
spaces, so their categories of coverings are finitely and countably superabundant.. Exam-
ples include compact orientable surfaces of genus g with k punctures, where 2g + k ≥ 3.
Taking g = 0, k = 3 shows that the category D of all dessins is finitely superabundant (see
Theorem 1.3(a) for a more specific result); in fact, Cori and Machı̀ [9] proved that every
finite group is the automorphism group of a finite oriented hypermap, two years before
Grothendieck introduced dessins in [20].
Hidalgo [21] has proved the stronger result that every action of a finite group A by
orientation-preserving self-homeomorphisms of a compact oriented surface S is topologi-
cally equivalent to the automorphism group of a dessin. One way to see this is to triangulate
S/A, with all critical values of the projection π : S → S/A among the vertices, none of
which has valency 1, and then to add an edge to an additional 1-valent vertex v in the in-
terior of a face. This gives a map (that is, a dessin with q = 2) on S/A which lifts via
π to a dessin D on S with A ≤ Aut(D). The only 1-valent vertices in D are the |A|
vertices in π−1(v). These are permuted by Aut(D), with A and hence Aut(D) acting
transitively; however, the stabiliser of a 1-valent vertex (in any dessin) must be the identity,
so Aut(D) = A, as required. By starting with inequivalent triangulations of S/A one can
obtain infinitely many non-isomorphic dessins realising this action of A.
ORCID iD
Gareth A. Jones https://orcid.org/0000-0002-7082-7025
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.02 / 23–32
https://doi.org/10.26493/1855-3974.2288.a20
(Also available at http://amc-journal.eu)
Complex uniformly resolvable
decompositions of Kv
Csilla Bujtás *
Faculty of Mathematics and Physics, University of Ljubljana,
Jadranska 19, Ljubljana 1000, Slovenia
Mario Gionfriddo, Elena Guardo, Lorenzo Milazzo, Salvatore Milici †
Dipartimento di Matematica e Informatica, Università di Catania,
Viale A. Doria, 6, 95125 - Catania, Italia
Zsolt Tuza ‡
Alfréd Rényi Institute of Mathematics,
1053 Budapest, Reáltanoda u. 13–15, Hungary, and
Department of Computer Science and Systems Technology, University of Pannonia,
8200 Veszprém, Egyetem u. 10, Hungary
Dedicated to the good friend and colleague Lorenzo Milazzo
who passed away in March 2019.
Received 22 March 2020, accepted 2 February 2021, published online 10 August 2021
Abstract
In this paper we consider complex uniformly resolvable decompositions of the com-
plete graph Kv into subgraphs such that each resolution class contains only blocks isomor-
phic to the same graph from a given set H and at least one parallel class is present from
each graph of H. We completely determine the spectrum for the cases H = {K2, P3,K3},
H = {P4, C4}, and H = {K2, P4, C4}.
Keywords: Resolvable decomposition, complex uniformly resolvable decomposition, path, cycle.
Math. Subj. Class. (2020): 05C51, 05C38, 05C70
*Supported by the Slovenian Research Agency under the project N1-0108.
†Supported by MIUR and I.N.D.A.M. (G.N.S.A.G.A.), Italy and by Università degli Studi di Catania, “Piano
della Ricerca 2016/2018 Linea di intervento 2” and “PIACERI 2020/22”.
‡Corresponding author. Research supported by the National Research, Development and Innovation Office –
NKFIH under the grant SNN 129364.
E-mail addresses: bujtas@fmf.uni-lj.si (Csilla Bujtás), gionfriddo@dmi.unict.it (Mario Gionfriddo),
guardo@dmi.unict.it (Elena Guardo), — (Lorenzo Milazzo), milici@dmi.unict.it (Salvatore Milici),
tuza@dcs.uni-pannon.hu (Zsolt Tuza)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
24 Ars Math. Contemp. 21 (2021) #P1.02 / 23–32
1 Introduction and definitions
Given a set H of pairwise non-isomorphic graphs, an H-decomposition (or H-design) of a
graph G is a decomposition of the edge set of G into subgraphs (called blocks) isomorphic
to some element of H. We say that an H-decomposition is complex1 if all H ∈ H are
present in it.
An H-factor of G is a spanning subgraph of G which is a vertex-disjoint union of some
copies of graphs belonging to H. If H = {H}, we will briefly speak of an H-factor. An
H-decomposition of G is resolvable if its blocks can be partitioned into H-factors (H-
factorization or resolution of G). An H-factor in an H-factorization is referred to as a
parallel class. Note that the parallel classes are mutually edge-disjoint, by definition.
An H-factorization F of G is called uniform if each factor of F is an H-factor for
some graph H ∈ H. A K2-factorization of G is known as a 1-factorization and its fac-
tors are called 1-factors; it is well known that a 1-factorization of Kv exists if and only
if v is even ([10]). If H = {F1, . . . , Fk} and ri ≥ 0 for i = 1, . . . , k, we denote by
(F1, . . . , Fk)-URD(v; r1, . . . , rk) a uniformly resolvable decomposition of the complete
graph Kv having exactly ri Fi-factors. A complex (F1, . . . , Fk)-URD(v; r1, . . . , rk) is a
uniformly resolvable decomposition of the complete graph Kv into r1 + · · · + rk parallel
classes with the requirement that at least one parallel class is present for each Fi ∈ H, i.e.,
ri > 0 for i = 1, . . . , k.
Recently, the existence problem for H-factorizations of Kv has been studied and a lot
of results have been obtained, especially on the following types of uniformly resolvable
H-decompositions: for a set H consisting of two complete graphs of orders at most five in
[3, 13, 14, 15]; for a set H of two or three paths on two, three, or four vertices in [5, 6, 9];
for H = {P3,K3 + e} in [4]; for H = {K3,K1,3} in [8]; for H = {C4, P3} in [11]; for
H = {K3, P3} in [12]; for 1-factors and n-stars in [7]; and for H = {P2, P3, P4} in [9].
In connection with our current studies the following cases are most relevant:
• perfect matchings and parallel classes of triangles or 4-cycles (that is, {K2,K3} or
{K2, C4}, Rees [13]);
• perfect matchings and parallel classes of 3-paths ({K2, P3}, Bermond et al. [1],
Gionfriddo and Milici [5]);
• parallel classes of 3-paths and triangles ({K3, P3}, Milici and Tuza [12]).
In this paper we give a complete characterization of the spectrum (the set of all admissible
combinations of the parameters) for the following two triplets of graphs and for the pair
contained in one of them which is not covered by the cases known so far:
• complex {K2, P3,K3}-decompositions of Kv (Section 3, Theorem 3.1);
• complex {K2, P4, C4}-decompositions of Kv (Section 4, Theorem 4.1);
• complex {P4, C4}-decompositions of Kv (Section 5, Theorem 5.1).
We summarize the formulation of those results in the concluding section, where a conjec-
ture related to the method of “metamorphosis” of parallel classes is also raised. We provide
the basis for this approach by applying linear algebra in Section 2.
1In the terminology of [2] one may say that in a complex decomposition all H ∈ H are essential (page 131)
or mandatory (page 232). In Chapter II.7.9 on uniformly resolvable designs the default is that all block sizes are
essential, while in Chapter IV.1 on partially balanced designs no block size is required to be mandatory.
Cs. Bujtás et al.: Complex uniformly resolvable decompositions of Kv 25
2 Local metamorphosis
In this section we prove three relations between uniform parallel classes of 4-cycles and
4-paths that will be used in the proofs of our main theorems. Before presenting the new
statements, we recall the Milici–Tuza–Wilson Lemma from [12].
It will be assumed throughout, without further mention, that all parallel classes are
meant on the same vertex set.
Theorem 2.1 ([12]). The union of two parallel classes of 3-cycles can be decomposed into
three parallel classes of P3.
The next two results, Theorems 2.2 and 2.3, will directly imply Theorem 2.4 which
states a pure metamorphosis from 4-cycles to 4-paths.
Theorem 2.2. The union of two parallel classes of C4 is decomposable into two parallel
classes of P4 and one perfect matching.
Proof. Let the vertices be v1, . . . , vn where n is a multiple of 4. The union of two parallel
classes of C4 forms a 4-regular graph G with 2n edges, say e1, . . . , e2n. We associate
a Boolean variable xi with each edge ei (1 ≤ i ≤ 2n) and construct a system of linear
equations over GF (2), which has 3n2 − 1 equations over the 2n variables. Let us set
xi1 + xi2 + xi3 + xi4 = 1 (mod 2)
for each 4-tuple of indices such that ei1 , ei2 , ei3 , ei4 are either the edges of a C4 in a parallel
class (call this a C-equation) or are the four edges incident with a vertex vi (a V -equation).
This gives 32n equations, but the V -equation for vn can be omitted since the n V -equations
sum up to 0 (as each edge is counted twice in the total sum) and therefore the one for vn
follows from the others.
We claim that this system of equations is contradiction-free over GF (2). To show this,
we need to prove that if the left sides of a subcollection E of the equations sum up to 0,
then also the right sides have zero sum; that is, the number |E| of its equations is even.
Observe that each variable is present in precisely three equations: in one C-equation
and two V -equations. Hence, to have zero sum on the left side, any xi should either not
appear in any equations of E or be present in precisely two. This means one of the following
two situations.
(T1) If (ei1 , ei2 , ei3 , ei4) is a 4-cycle (in this cyclic order of edges) and its C-equation
belongs to E , then precisely two related V -equations must be present in E , namely
either those for the vertices ei1 ∩ ei2 and ei3 ∩ ei4 or those for ei2 ∩ ei3 and ei1 ∩ ei4 .
(T2) If (ei1 , ei2 , ei3 , ei4) is a 4-cycle such that its C-equation does not belong to E but
some xij (1 ≤ j ≤ 4) is involved in E , then all the four V -equations for vi1 , vi2 , vi3 ,
vi4 must be present in E .
In the first and second parallel class of 4-cycles, respectively, let us denote the number
of cycles of type (T1) by a1 and b1, and that of type (T2) by a2 and b2. Then the number
of V -equations in E is equal to both 2a1 + 4a2 and 2b1 + 4b2, which is the same as the
average of these two numbers. Thus, the number |E| of equations is equal to
a1 + b1 +
1
2
((2a1 + 4a2) + (2b1 + 4b2)) = 2(a1 + a2 + b1 + b2)
26 Ars Math. Contemp. 21 (2021) #P1.02 / 23–32
that is even, as needed.
Since the system of equations is non-contradictory, it has a solution ξ ∈ {0, 1}2n over
GF (2). We observe further that in any C-equation xi1 + xi2 + xi3 + xi4 = 1 we may
switch the values from ξ(xij ) to 1− ξ(xij ) simultaneously for all 1 ≤ j ≤ 4, and doing so
the modified values remain a solution because the parities of sums in the V -equations do
not change either. In this way, we can transform ξ to a basic solution ξ0 in which every C-
equation contains precisely one 1 and three 0s. Since each V -equation contains precisely
two or zero variables from each C-equation, it follows that in the basic solution ξ0 each
V -equation, too, contains precisely one 1 and three 0s.
As a consequence, the variables which have xi = 1 in the basic solution define a perfect
matching in G (since at most two 1s may occur at each vertex, and then the corresponding
V -equation implies that there is precisely one). Moreover, removing those edges from G,
each cycle of each parallel class becomes a P4. In this way we obtain two parallel classes
of P4, and one further class which is a perfect matching.
Theorem 2.3. The edge-disjoint union of a perfect matching and a parallel class of C4 is
decomposable into two parallel classes of P4.
Proof. We apply several ideas from the previous proof, but in a somewhat different way.
We now introduce Boolean variables x1, . . . , xn for the edges e1, . . . , en of the 4-cycles
only; but still there will be two kinds of linear equations, namely n/4 of them for 4-cycles
(called C-equations) and n/2 of them for the edges of the perfect matching (M -equations).
They are of the same form as before:
xi1 + xi2 + xi3 + xi4 = 1 (mod 2)
The C-equations require ei1 , ei2 , ei3 , ei4 to be the edges of a C4 in the parallel class. The
M -equations take ei1 , ei2 , ei3 , ei4 as the four edges incident with a matching edge. If a
matching edge is the diagonal of a 4-cycle, then their equations coincide; and if a matching
edge shares just one vertex with a 4-cycle then the M -equation and the C-equation share
two variables which correspond to consecutive edges on the cycle. Further, we recall that
the matching is edge-disjoint from the cycles, therefore each variable associated with a
cycle-edge occurs in precisely two distinct M -equations.
These facts imply that only two types of C-equations can occur in a subcollection E of
equations whose left sides sum up to 0 over GF (2).
(T1) If (ei1 , ei2 , ei3 , ei4) is a 4-cycle and its C-equation belongs to E , then the corre-
sponding C4 has precisely two (antipodal) vertices for which the M -equations of the
incident matching edges are present in E . (At the moment it is unimportant whether
those two vertices form a matching edge or not.)
(T2) If (ei1 , ei2 , ei3 , ei4) is a 4-cycle whose C-equation does not belong to E but some xij
(1 ≤ j ≤ 4) is involved in E , then each M -equation belonging to a matching edge
incident with some of the four vertices vi1 , vi2 , vi3 , vi4 is present in E . (It is again
unimportant whether one or both or none of the diagonals of the C4 in question is a
matching edge.)
Let now a1 and a2 denote the number of cycles with type (T1) and type (T2), re-
spectively. By what has been said, the number of vertices requiring an M -equation is
equal to 2a1 + 4a2. Since each of those equations is now counted at both ends of the
Cs. Bujtás et al.: Complex uniformly resolvable decompositions of Kv 27
corresponding matching edge, we obtain that E contains exactly a1 + 2a2 M -equations;
moreover it has a1 C-equations, by definition. Thus, the number |E| of equations is equal
to a1 + (a1 + 2a2) = 2(a1 + a2) which is even. Thus, if the left sides in E sum up to
zero, then also the right sides have sum 0 in GF (2). It proves that the system of the 3n/4
equations is contradiction-free and has a solution ξ ∈ {0, 1}n over GF (2).
Now, we observe that in any C-equation xi1 + xi2 + xi3 + xi4 = 1 we may switch the
values from ξ(xij ) to 1− ξ(xij ) simultaneously for all 1 ≤ j ≤ 4. Doing so, the modified
values remain a solution as the parities of sums in the M -equations do not change either.
In this way we can transform ξ to a basic solution ξ0 in which every C-equation contains
precisely one 1 and three 0s. Since each M -equation has precisely two or four or zero
variables from any C-equation, it follows that in the basic solution ξ0 each M -equation,
too, contains precisely one 1 and three 0s.
As a consequence, the variables (cycle-edges) which have xi = 1 in the basic solution
establish a pairing between the edges of the original matching. Hence the set I = {ei :
ξ0(xi) = 1} together with the edges of the given matching factor determines a P4-factor.
Moreover, removing the edges of I from the 4-cycles, we obtain another parallel class of
paths P4.
These two types of metamorphosis can be combined to obtain the following third one.
Theorem 2.4. The union of three parallel classes of C4 is decomposable into four parallel
classes of P4.
Proof. Applying Theorem 2.2 we transform the union of the first and the second C4-class
into two P4-classes and a perfect matching. After that we combine the third C4-class with
the perfect matching just obtained into two further P4-classes, by Theorem 2.3.
3 The spectrum for H = {K2, P3,K3}
In this section we consider complex uniformly resolvable decompositions of the complete
graph Kv into m classes containing only copies of 1-factors (perfect matchings), p classes
containing only copies of paths P3 and t classes containing only copies of triangles K3. The
current problem is to determine the set of feasible triples (m, p, t) such that m · p · t ̸= 0,
for which there exists a complex (K2, P3,K3)-URD(v;m, p, t). A little more than that, for
v = 6 and v = 12 we shall list all the possible (m, p, t) with m, p, t ≥ 0 such that there
exists a uniformly resolvable decomposition (K2, P3,K3)-URD(v;m, p, t).
Theorem 3.1. The necessary and sufficient conditions for the existence of a complex
(K2, P3,K3)-URD(v;m, p, t) are:
(i) v ≥ 12 and v is a multiple of 6;
(ii) 3m+ 4p+ 6t = 3v − 3.
Moreover, the parameters m, p, t are in the following ranges:
(iii) 1 ≤ m ≤ v − 7 and m is odd,
3 ≤ p ≤ 3 · ⌊ v4 − 1⌋,
1 ≤ t ≤ ⌊v2 − 3⌋.
28 Ars Math. Contemp. 21 (2021) #P1.02 / 23–32
Proof. We first prove that the conditions are necessary. Divisibility of v by 6 is immediately
seen, due to the presence of K3-classes and 1-factors. We observe further that the number
of edges in a parallel class is v for a triangle-class, 2v/3 for a P3-class and v/2 for a
matching. Thus, in any (K2, P3,K3)-URD(v;m, p, t) we must have
mv
2
+
2pv
3
+ tv =
(
v
2
)
.
Dividing it by v/6, the assertion of (ii) follows.
As (ii) implies, p is a multiple of 3, say p = 3x. Then we obtain
m+ 4x+ 2t = v − 1
and also conclude that m is odd. Since m ≥ 1, t ≥ 1, and x ≥ 1, this equation yields
m ≤ v − 7, x ≤ v/4− 1, t ≤ v/2− 3,
implying the conditions listed in (iii), and the first one also excludes v = 6. This completes
the proof that the conditions (i) – (iii) are necessary.
To prove the sufficiency of (i) – (ii), we consider v ≥ 18 first. Since v is a multiple of
6 according to (i), there exists a Nearly Kirkman Triple System of order v, which means
m = 1 perfect matching and t = v2 − 1 parallel classes of triangles. More generally,
for every odd m in the range 1 ≤ m ≤ v − 7, there exists a collection of m perfect
matchings and t = v−1−m2 parallel classes of triangles, which together decompose Kv;
this was proved in [13]. From such a system, for every 0 < x < v−1−m4 , we can take
2x parallel classes of triangles. Applying Theorem 2.1 [12], also proved independently
by Wilson (unpublished), we obtain 3x parallel classes of paths P3. This gives a com-
plex (K2, P3,K3)-URD(v;m, 3x, v−1−m2 − 2x). For v = 12, the statement follows by
Proposition 3.3 below.
3.1 Small cases
Obviously, the proofs of the necessary conditions that v is a multiple of 6, and that the
equality 3m+4p+6t = 3v− 3 must be satisfied by every (K2, P3,K3)-URD(v;m, p, t),
do not use the assumption m · p · t ̸= 0.
Proposition 3.2. There exists a (K2, P3,K3)-URD(6;m, p, t) if and only if (m, p, t) ∈
{(5, 0, 0), (3, 0, 1), (1, 3, 0)}.
Proof. Putting v = 6, the equation 3m+ 4p+ 6t = 15 has exactly four solutions (m, p, t)
over the nonnegative integers. The case (1, 0, 2) would correspond to an NKTS(6) which
is known not to exist [13]. The case (5, 0, 0) corresponds to a 1-factorization of the com-
plete graph K6 which is known to exist [10]. The case of (1, 3, 0) is just the same as a
(K2, P3)-URD(6; 1, 3) that is known to exist [6]. To see the existence for (3, 0, 1), consider
V (K6) = Z6 and the following classes: {{1, 4}, {2, 5}, {3, 6}}, {{1, 5}, {2, 6}, {3, 4}},
{{1, 6}, {2, 4}, {3, 5}}, {(1, 2, 3), (4, 5, 6)}.
Proposition 3.3. There exists a (K2, P3,K3)-URD(12;m, p, t) if and only if (m, p, t) ∈
{(11, 0, 0), (9, 0, 1), (7, 3, 0), (7, 0, 2), (5, 0, 3), (5, 3, 1), (3, 6, 0), (3, 3, 2), (3, 0, 2),
(1, 6, 1), (1, 3, 3)}.
Cs. Bujtás et al.: Complex uniformly resolvable decompositions of Kv 29
Proof. Checking the nonnegative integer solutions of 3m + 4p + 6t = 33, the case of
(1, 0, 5) would correspond to an NKTS(12) which is known not to exist [13]. The case of
(11, 0, 0) corresponds to a 1-factorization of the complete graph K12 that is known to exist
[2]. The result for the cases (9, 0, 1), (7, 0, 2), (5, 0, 3), (3, 0, 4) follows by [13]. Applying
Theorem 2.1 to (7, 0, 2), (5, 0, 3), (3, 0, 4), we obtain the existence for (7, 3, 0), (5, 3, 1),
(3, 3, 2), and (3, 6, 0). The existence for the case (1, 3, 3) is shown by the following con-
struction. Let V (K12) = Z12, and consider the following parallel classes:
• matching: {{1, 6}, {2, 4}, {3, 0}, {5, 11}, {7, 9}, {8, 10}};
• paths: {{5, 0, 11}, {8, 1, 7}, {9, 2, 3}, {6, 4, 10}}, {{1, 3, 8}, {7, 5, 10}, {4, 9, 0},
{2, 11, 6}}, {{0, 6, 5}, {2, 7, 4}, {9, 8, 11}, {3, 10, 1}};
• triangles: {{0, 1, 2), {3, 4, 5}, {6, 7, 8}, {9, 10, 11}}, {{0, 4, 8}, {3, 7, 11},
{2, 6, 10}, {1, 5, 9}}, {{0, 7, 10}, {3, 6, 9}, {2, 5, 8}, {1, 4, 11}}.
Finally, we apply Theorem 2.1 to the case (1, 3, 3) and infer that a (K2, P3,K3)-
URD(12; 1, 6, 1) exists, too.
4 The spectrum for F = {K2, P4, C4}
In this section we consider complex uniformly resolvable decompositions of the complete
graph Kv into m parallel 1-factors, p parallel classes of 4-paths, and c parallel classes of
4-cycles. The current problem is to determine the set of feasible triples (m, p, c) such that
m · p · c ̸= 0, for which there exists a complex (K2, P3, C4)-URD(v;m, p, c). The case
{P4, C4} will be discussed in Section 5.
Theorem 4.1. The necessary and sufficient conditions for the existence of a complex
(K2, P4, C4)-URD(v;m, p, c) are:
(i) v ≥ 8 and v is a multiple of 4;
(ii) 2m+ 3p+ 4c = 2v − 2.
Moreover, the parameters m, p, c are in the following ranges:
(iii) 1 ≤ c ≤ v2 − 3,
1 ≤ m ≤ v − 6,
2 ≤ p ≤ 2 ⌊ v−43 ⌋;
(iv) p is even; and if p ≡ 2 (mod 4), then also m is even.
Proof. We first show that the conditions are necessary. Since Kv has a C4-factor, v must
be a multiple of 4. Further, as a C4-, K2-, and P4-factor respectively cover exactly v, v/2,
and 3v/4 edges, in a (K2, P4, C4)-URD(v;m, p, c) we have
mv
2
+
3pv
4
+ cv =
(
v
2
)
.
This equality directly implies (ii) and we may also conclude that p is even and, further, if
p ≡ 2 (mod 4), then m must be even as well. Putting p = 2x we obtain
m+ 3x+ 2c = v − 1.
30 Ars Math. Contemp. 21 (2021) #P1.02 / 23–32
By our condition, all the three types of parallel classes are present in the decomposition,
i.e. we have c ≥ 1, m ≥ 1, and x ≥ 1. These, together with the equality above, imply the
necessity of (iii).
Next we prove the sufficiency of (i) – (ii). We first take a 1-factorization of Kv/2 into
v/2 − 1 perfect matchings, which exists because v is a multiple of 4. Now, replace each
vertex of Kv/2 with two non-adjacent vertices, and each edge with a copy of the complete
bipartite graph K2,2, i.e. a 4-cycle whose missing diagonals are the non-adjacent vertex
pairs.2 This blow-up results in v/2 − 1 parallel classes of C4 inside Kv , and the missing
edges can be taken as a perfect matching. Let C be the set of the parallel classes of C4 and
x be a nonnegative integer such that 0 < x ≤ ⌊v−43 ⌋. The construction splits into two cases
depending on the parity of x.
If x is even, take 3x2 parallel classes from C. Applying Theorem 2.4, we transform the
3x
2 parallel classes of C4 into 2x = p parallel classes of paths P4. For any given y = c
in the range 0 < y ≤ ⌊v−22 −
3x
2 ⌋, keep y classes of C4 and transform the remaining
v−2
2 −
3x
2 − y classes of C4 into 2(
v−2
2 −
3x
2 − y) = m − 1 classes of 1-factors. In this
way we obtain a complex (K2, P4, C4)-URD(v; v − 3x− 2y − 1, 2x, y).
If x is odd, take 3(x−1)2 + 2 parallel classes from C. By Theorems 2.2 and 2.4, we
can transform the 3(x−1)2 + 2 parallel classes of C4 into 2x = p parallel classes of paths
P4 and a 1-factor. For any given y = c in the range 0 < y ≤ v−22 −
3(x−1)
2 − 2, keep
y classes of C4 and transform the remaining v−22 −
3(x−1)
2 − 2 − y classes of C4 into
2( v−22 −
3(x−1)
2 − 2− y) = m− 2 classes of 1-factors. In this way, we obtain a complex
(K2, P4, C4)-URD(v; v − 3x− 2y − 1, 2x, y).
The result, for every v ≡ 0 (mod 4), 0 < x ≤ ⌊ v−43 ⌋, and 0 < y ≤ ⌊
v−2
2 −
3x
2 ⌋,
is a uniformly resolvable decomposition of Kv into into v − 1 − 3x − 2y = m classes
containing only copies of 1-factors, 2x = p classes containing only copies of paths P4,
and y = c classes containing only copies of 4-cycles C4. This finishes the proof of the
theorem.
5 The spectrum for H = {P4, C4}
Finally, we consider complex uniformly resolvable decompositions of the complete graph
Kv into p classes containing only copies of paths P4 and c classes containing only copies
of 4-cycles C4.
Theorem 5.1. The necessary and sufficient conditions for the existence of a complex
(P4, C4)-URD(v; p, c) are:
(i) v ≥ 8 and v is a multiple of 4;
(ii) 3p+ 4c = 2v − 2.
Proof. Necessity is a consequence of Theorem 4.1, since we did not need to assume m > 0
in that part of its proof. Turning to sufficiency, the condition 3p+4c = 2v− 2 implies that
3p ≡ 2v − 2 (mod 4). This gives p = 2 + 4x and c = v−42 − 3x. For a construction, we
start with a decomposition of Kv into a perfect matching F and v−22 parallel classes of C4
as in the proof of Theorem 4.1. By Theorem 2.3, we can transform one class of C4 and F
2With another terminology, this “blow-up” is the lexicographic product Kv/2[2K1].
Cs. Bujtás et al.: Complex uniformly resolvable decompositions of Kv 31
into two classes of paths P4. Then, by Theorem 2.4, we transform 3x parallel classes of
C4 into 4x parallel classes of P4. The result, for every x such that 0 ≤ x ≤ ⌊ v−66 ⌋, is a
uniformly resolvable decomposition of Kv into 2 + 4x classes containing only copies of
paths P4 and v−42 − 3x classes containing only copies of 4-cycles C4. This completes the
proof.
6 Conclusion
Combining Theorems 3.1, 4.1, and 5.1, we obtain the main result of this paper.
Theorem 6.1.
(i) A complex (K2, P3,K3)-URD(v;m, p, t) exists if and only if v ≥ 12, v is a multiple
of 6, and 3m+ 4p+ 6t = 3v − 3.
(ii) A complex (K2, P4, C4)-URD(v;m, p, t) exists if and only if v ≥ 8, v is a multiple
of 4, and 2m+ 3p+ 4t = 2v − 2.
(iii) A complex (P4, C4)-URD(v; p, t) exists if and only if v ≥ 8, v is a multiple of 4, and
3p+ 4t = 2v − 2.
Concerning the local metamorphosis studied in Section 2, we pose the following con-
jecture as a common generalization of Theorems 2.1 and 2.4.
Conjecture 6.2. The union of k−1 parallel classes of Ck is decomposable into k parallel
classes of Pk.
ORCID iDs
Csilla Bujtás https://orcid.org/0000-0002-0511-5291
Zsolt Tuza https://orcid.org/0000-0003-3235-9221
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.03 / 33–44
https://doi.org/10.26493/1855-3974.2384.77d
(Also available at http://amc-journal.eu)
General d-position sets*
Sandi Klavžar
Faculty of Mathematics and Physics, University of Ljubljana, Slovenia, and
Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia, and
Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Douglas F. Rall
Department of Mathematics, Furman University, Greenville, SC, USA
Ismael G. Yero †
Departamento de Matemáticas, Universidad de Cádiz, Algeciras, Spain
Received 17 July 2020, accepted 14 February 2021, published online 10 August 2021
Abstract
The general d-position number gpd(G) of a graph G is the cardinality of a largest set
S for which no three distinct vertices from S lie on a common geodesic of length at most
d. This new graph parameter generalizes the well studied general position number. We first
give some results concerning the monotonic behavior of gpd(G)with respect to the suitable
values of d. We show that the decision problem concerning finding gpd(G) is NP-complete
for any value of d. The value of gpd(G) when G is a path or a cycle is computed and a
structural characterization of general d-position sets is shown. Moreover, we present some
relationships with other topics including strong resolving graphs and dissociation sets. We
finish our exposition by proving that gpd(G) is infinite wheneverG is an infinite graph and
d is a finite integer.
Keywords: General d-position sets, dissociation sets, strong resolving graphs, computational com-
plexity, infinite graphs.
Math. Subj. Class. (2020): 05C12, 05C63, 05C69
*We acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-
0297 and projects J1-9109, J1-1693, N1-0095, N1-0108). This research was initiated while the third author was
visiting the University of Ljubljana, Slovenia, supported by “Ministerio de Educación, Cultura y Deporte”, Spain,
under the “José Castillejo” program for young researchers (reference number: CAS18/00030).
†Corresponding author.
E-mail addresses: sandi.klavzar@fmf.uni-lj.si (Sandi Klavžar), doug.rall@furman.edu (Douglas F. Rall),
ismael.gonzalez@uca.es (Ismael G. Yero)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
34 Ars Math. Contemp. 21 (2021) #P1.03 / 33–44
1 Introduction
A general position set of a graph G is a set of vertices S ⊆ V (G) such that no three
vertices from S lie on a common shortest path of G. The order of a largest general po-
sition set, shortly called a gp-set, is the general position number gp(G) of G (also writ-
ten gp-number). This concept was recently and independently introduced in [6, 16]. We
should mention though that the same concept was studied on hypercubes already in 1995
by Körner [14]. Following [16] and its notation and terminology, the concept received a lot
of attention, see the series of papers [8, 13, 11, 12, 17, 18, 21, 22]. In particular, in [18] the
general position problem was studied on complementary prisms. In order to characterize
an extremal case for the general position number of these graphs, the concept of general
3-position was introduced as an essential ingredient of the characterization. In this paper
we extend this idea as follows.
Let d ∈ N and let G be a (connected) graph. Then S ⊆ V (G) is a general d-position set
if the following holds:
{u, v,w} ∈ (S
3
), v ∈ IG(u,w)⇒ dG(u,w) > d , (1.1)
where dG(u,w) denotes the shortest-path distance in G between u and w, and IG(u,w) =
{x ∈ V (G) ∶ dG(u,w) = dG(u,x)+dG(x,w)} is the interval between u and w. In words,
S is a general d-position set if no three different vertices from S lie on a common geodesic
of length at most d. We will say that vertices u, v,w that fulfill condition (1.1) lie in general
d-position. The cardinality of a largest general d-position set in a graph G is the general
d-position number of G and is denoted by gpd(G).
We proceed as follows. In the rest of this section we recall needed definitions and state
some basic facts and results on the general d-position number. Then, in Section 2, we
demonstrate that in the inequality chain gpdiam(G)(G) ≤ gpdiam(G)−1(G) ≤ ⋯ ≤ gp2(G)
all kinds of equality and strict inequality cases are possible. Using one of the correspond-
ing constructions we also prove that the problem of determining the gpd number is NP-
complete. In Section 3 we determine the gpd number of paths and cycles and give a general
upper bound on the gpd number in term of the diameter of a given graph. In the subsequent
section we prove a structural characterization of general d-position sets. In Section 5 we
report on the connections between general d-position sets and two well-established con-
cepts, the dissociation number and strong resolving graphs. In the concluding section we
consider the gpd number of infinite graphs and pose several open questions.
1.1 Preliminaries
For a positive integer k we will use the notation [k] = {1, . . . , k}. The clique number and
the independence number ofG are denoted by ω(G) and α(G). If S ⊆ V (G), then the sub-
graph ofG induced by S is denoted by ⟨S⟩ and (S
k
) denotes the set of all subsets of S having
cardinality k. A subgraph H of a graph G is isometric if dH(u, v) = dG(u, v) holds for all
u, v ∈ V (H). If H1 and H2 are subgraphs of G, then the distance dG(H1,H2) between
H1 and H2 is defined as min{dG(h1, h2) ∶ h1 ∈ V (H1), h2 ∈ V (H2)}. In particular, if
H1 is the one vertex graph with u being its unique vertex, then we will write dG(u,H2) for
dG(H1,H2). We say that the subgraphs H1 and H2 are parallel, denoted by H1 ∥ H2, if
for every pair of vertices h1 ∈ V (H1) and h2 ∈ V (H2)we have dG(h1, h2) = dG(H1,H2).
If H1 and H2 are not parallel, we will write H1 ∦ H2. The open neighborhood and the
S. Klavžar et al.: General d-position sets 35
closed neighborhood of a vertex v of G will be denoted by NG(x) and NG[x], respec-
tively. Vertices x and y of G are true twins if NG[u] = NG[v]. We may omit the subscript
G in the above definitions if the graph G is clear from the context.
Clearly, if d = 1, then every subset of vertices of G is a general 1-position set, and if
d ≥ diam(G), then S is a general d-position set if and only if S is a general position set.
Moreover, note that
gpdiam(G)(G) ≤ gpdiam(G)−1(G) ≤ ⋯ ≤ gp2(G) . (1.2)
We conclude the preliminaries with the following useful property.
Proposition 1.1. Let G be a graph and let 2 ≤ d ≤ diam(G) − 1 be a positive integer.
If H1, . . . ,Hr are isometric subgraphs of G such that dG(Hi,Hj) ≥ d for i ≠ j, then
gpd(G) ≥ ∑ri=1 gpd(Hi).
Proof. For each i ∈ [r], let Si be a general d-position set of Hi such that ∣Si∣ = gpd(Hi).
We claim that S = ⋃ri=1 Si is a general d-position set of G. Suppose {x, y, z} ∈ (S3) such
that y ∈ IG(x, z) and dG(x, z) ≤ d. That is, there exists a shortest x, z-path of length
at most d in G that contains y. Since dG(u, v) ≥ d for any two vertices u ∈ V (Hi) and
v ∈ V (Hj) with i ≠ j, there exists k ∈ [r] such that {x, y, z} ⊆ V (Hk). Now, since Hk
is an isometric subgraph of G, it follows that dHk(x, y) = dG(x, y), dHk(y, z) = dG(y, z)
and dHk(x, z) = dG(x, z). This implies that there is a x, z-geodesic in Hk that contains
y. Hence, y ∈ IHk(x, z), and since Sk is a general d-position set of Hk, we infer that
dG(x, z) = dHk(x, z) > d, which is a contradiction. Therefore, S is a general d-position
set of G, and it follows that gpd(G) ≥ ∑ri=1 gpd(Hi).
2 On the inequality chain (1.2) and computational complexity
In this section we investigate the inequality chain (1.2) by constructing different classes of
graphs which demonstrate that all kinds of equality and strict inequality cases can happen.
We conclude the section by applying one of these constructions to prove that the GENERAL
d-POSITION PROBLEM is NP-complete.
Equality in (1.2) simultaneously.
For n ≥ 2, let S be a star with center x and leaves u1, . . . , un, v1, . . . , vn. Construct a graph
Gn of order 2n + 3 by taking the disjoint union of S and an independent set of vertices
{u, v} together with the set of edges {uui, vvi ∶ i ∈ [n]}. The diameter of Gn is 4, and we
have gp4(Gn) = gp3(Gn) = gp2(Gn) = 2n.
Equality in (1.2) simultaneously again.
Let Tr, r ≥ 2, be the tree obtained from the path Pr+1 on r + 1 vertices, by attaching two
leaves to each of its internal vertices. Then we claim that
gpr(Tr) = gpr−1(Tr) = ⋯ = gp2(Tr) .
Indeed, first note that diam(Tr) = r. Since the gp-number of a tree is the number of its
leaves (cf. [16, Corollary 3.7]), we have gpr(Tr) = 2r. Let next S be a general 2-position
set. If u is a vertex of Tr adjacent to exactly two leaves, say v andw, then ∣S∩{u, v,w}∣ ≤ 2.
Moreover, if u is a vertex of Tr adjacent to exactly three leaves, say v, w, and z, then
∣S ∩ {u, v,w, z}∣ ≤ 3. It follows that gp2(Tr) ≤ 2(r − 3) + 2 ⋅ 3 = 2r. In conclusion,
2r = gpr(Tr) ≤ gpr−1(Tr) ≤ ⋯ ≤ gp2(Tr) ≤ 2r, hence equality holds throughout.
36 Ars Math. Contemp. 21 (2021) #P1.03 / 33–44
Strict inequality in (1.2) in exactly one case.
Let k, ℓ ≥ 4 and let Gk,ℓ be a graph defined as follows. Its vertex set is
V (Gk,ℓ) =
k
⋃
j=1
{uj ,wj , xj,1, . . . , xj,ℓ} ∪ {xk+1,1} .
For j ∈ [k], each of the vertices uj and wj is adjacent to xj,1, . . . , xj,ℓ and to xj+1,1. There
are no other edges in Gk,ℓ. Note that ∣V (Gk,ℓ)∣ = k(ℓ + 2) + 1 and that diam(Gk,ℓ) = 2k.
It is straightforward to see that the set X = ⋃kj=1{xj,1, . . . , xj,ℓ} ∪ {xk+1,1} is a largest
independent set of Gk,ℓ. Moreover, X is also a largest general 2-position set and a largest
general 3-position set. Furthermore, it is not difficult to infer that the setX∖{x2,1, . . . , xk,1}
is a largest general d-position set for each d ∈ {4, . . . ,2k}. In conclusion,
gp2k(Gk,ℓ) = gp2k−1(Gk,ℓ) = ⋯ = gp4(Gk,ℓ) < gp3(Gk,ℓ) = gp2(Gk,ℓ) = α(G) .
Strict inequality in (1.2) in every case.
Given a positive integer t, construct the graph Ht as follows. Begin with a complete graph
K4t with vertex set V (K4t) = A ∪ B where ∣A∣ = ∣B∣ = 2t. Next, add a path Pt−1 =
v1 . . . vt−1, and join with an edge every vertex of B with the leaf v1 of Pt−1. Then, add a
pendant vertex ui to every vertex vi ∈ {v2, . . . , vt−1}, and finally, for every i ∈ {2, . . . , t−1},
add the edge uivi−1. As an example, the graph H8 is represented in Figure 1.
A B
v7v6v5v4v3v2v1
u7u6u5u4u3u2
Figure 1: The graph H8. Edges joining the sets A and B, as well as joining B with the
vertex v1 are indicated with dotted lines.
Notice that the graph Ht has diameter t. The general d-position number of Ht for all
possible d is given in the following result.
Proposition 2.1. If 2 ≤ d ≤ t, then
gpd(Ht) =
⎧⎪⎪⎪⎨⎪⎪⎪⎩
4t; d = t,
4t + 2; d = t − 1,
5t − d + 1; otherwise.
Proof. We first note that the set A ∪ B is a general position set of Ht, or equivalently a
general t-position set. Thus, gp(Ht) = gpt(Ht) ≥ 4t. Suppose gp(Ht) = gpt(Ht) > 4t
and let S be a general t-position set. Hence, there exists at least one vertex not in A ∪ B
which is in S. Since every shortest path joining a vertex of A with a vertex not in A ∪B
passes through a vertex in B, it follows that S ∩ A = ∅ or S ∩ B = ∅. This implies that
∣S∣ ≤ 2t+2t−3 = 4t−3, and this is not possible. Therefore gp(Ht) = gpt(Ht) = 4t = 5t−d,
when d = t.
S. Klavžar et al.: General d-position sets 37
We next consider the case d = t − 1. The set A ∪B ∪ {vt−1, ut−1} is a general (t − 1)-
position set of Ht, and so, gpt−1(Ht) ≥ 4t+2. If we suppose that gpt−1(Ht) > 4t+2, then
a similar argument to that above for d = t leads to a contradiction. Therefore, gpt−1(Ht) =
4t + 2.
We finally consider d = t − k with 2 ≤ k ≤ t − 2. Notice that the set A ∪ B ∪
{vt−1, ut−1, ut−2, . . . , ut−k} is a general d-position set of Ht of cardinality 4t + k + 1 =
4t + (t − d + 1) = 5t − d + 1, and so, gpd(Ht) ≥ 5t − d + 1. Again, an argument similar to
the two cases above leads to gpd(Ht) = 5t − d + 1.
Proposition 2.1 yields strict inequalities in the chain (1.2), that is, for any graphHt with
t ≥ 3, we have
gpt(Ht) < gpt−1(Ht) < ⋯ < gp2(Ht) . (2.1)
We shall finish this section by considering the computational complexity of the decision
problem related to finding the general d-position number of graphs, in which we also show
the usefulness of the above graphs Ht.
GENERAL d-POSITION PROBLEM
Input: A graph G, an integer d ≥ 2, and a positive integer r.
Question: Is gpd(G) larger than r?
We first remark that the GENERAL d-POSITION PROBLEM is known to be NP-complete
for every d ≥ diam(G) (see [16]). Hence, we may center our attention on the cases d ∈
{2, . . . ,diam(G) − 1}, although our reduction also works for the case d = diam(G).
Theorem 2.2. If d ≥ 2, then the GENERAL d-POSITION PROBLEM is NP-complete.
Proof. First, we can readily observe that the problem belongs to the class NP, since check-
ing that a given set is indeed a general d-position set can be done in polynomial time. From
now on, we make a reduction from the MAXIMUM CLIQUE PROBLEM to the GENERAL
d-POSITION PROBLEM.
In order to present the reduction, for a given graph G of order t ≥ 3, we shall construct
a graph G′ by using the above graphs Ht. We construct G′ from the disjoint union of G
and Ht, by adding all possible edges between A ∪ B ∪ {v1} and V (G). It is then easily
observed that ω(G′) = ∣A∣+ ∣B∣+ω(G). Moreover, using similar arguments as in the proof
of Proposition 2.1, we deduce that gpd(G′) = gpd(Ht) + ω(G). From this fact, since the
value gpd(Ht) is known from Proposition 2.1, the reduction is completed, and the theorem
is proved.
3 Paths and cycles
In this section we determine the general d-position number of paths and cycles. The first
result in turn implies a general upper bound on the general d-position number in term of
the diameter of a given graph.
Proposition 3.1. If n ≥ 3 and 2 ≤ d ≤ n − 1, then
gpd(Pn) =
⎧⎪⎪⎨⎪⎪⎩
2 ⌈ n
d+1⌉ − 1; n ≡ 1 (mod d + 1),
2 ⌈ n
d+1⌉ ; otherwise.
38 Ars Math. Contemp. 21 (2021) #P1.03 / 33–44
Proof. Let d ∈ {2, . . . , n − 1} and Pn = v1v2 . . . vn. If n ≡ 1 (mod d + 1), then let
S = {v(d+1)i+1, v(d+1)i+2 ∶ 0 ≤ i ≤ ⌊n/(d + 1)⌋ − 1} ∪ {vn} ,
and if n /≡ 1 (mod d + 1), then let
S = {v(d+1)i+1, v(d+1)i+2 ∶ 0 ≤ i ≤ ⌈n/(d + 1)⌉ − 1} .
It can be readily seen that S is a general d-position set of Pn, which gives the lower bound
gpd(Pn) ≥
⎧⎪⎪⎨⎪⎪⎩
2 ⌈ n
d+1⌉ − 1; n ≡ 1 (mod d + 1),
2 ⌈ n
d+1⌉ ; otherwise.
On the other hand, suppose
gpd(Pn) >
⎧⎪⎪⎨⎪⎪⎩
2 ⌈ n
d+1⌉ − 1; n ≡ 1 (mod d + 1),
2 ⌈ n
d+1⌉ ; otherwise,
and let S′ be a general d-position set of cardinality gpd(Pn). By the pigeonhole principle,
we deduce that there exists a subpath in Pn of length d that contains at least three elements
of S′, but this is not possible. Therefore, the desired equality follows.
Specializing to n = 14 in Proposition 3.1, we next show a table with the values of
gpd(Pn) for every possible value of d. Notice that, equalities and inequalities occur in
distinct positions with respect to the chain (1.2).
d 2 3 4 5 6 7 8 9 10 11 12 13
gpd(P14) 10 8 6 6 4 4 4 4 4 4 3 2
Table 1: The values of gpd(P14) for every 2 ≤ d ≤ 13.
The result for paths gives the following general lower bound.
Corollary 3.2. Let G be a connected graph of diameter d. If 2 ≤ k ≤ d, then
gpk(G) ≥
⎧⎪⎪⎨⎪⎪⎩
2 ⌈ d+1
k+1⌉ − 1; d ≡ 0 (mod k + 1),
2 ⌈ d+1
k+1⌉ ; otherwise.
Proof. Shortest paths are isometric subgraphs; in particular, this holds for diametrical
paths. Hence G contains an isometric Pd+1, and therefore gpk(G) ≥ gpk(Pd+1) by Propo-
sition 1.1 with r = 1. Applying Proposition 3.1 yields the result.
In a similar manner as done for paths, we can compute the general d-position number
for cycles. It is easy to show that gpd(C3) = 3 for any d, gp1(C4) = 4, and gpd(C4) = 2
for d ≥ 2.
Proposition 3.3. If n ≥ 5 and 2 ≤ d < ⌊n
2
⌋, then
gpd(Cn) =
⎧⎪⎪⎨⎪⎪⎩
2 ⌊ n
d+1⌋ + 1; n ≡ d (mod d + 1),
2 ⌊ n
d+1⌋ ; otherwise.
If d ≥ ⌊n
2
⌋, then gpd(Cn) = 3.
S. Klavžar et al.: General d-position sets 39
Proof. Let Cn = v1v2 . . . vnv1. Note that diam(Cn) = ⌊n2 ⌋, and the argument naturally
splits into two cases.
First assume that 2 ≤ d < ⌊n
2
⌋. Let m = ⌊ n
d+1⌋ and for each k ∈ [m] we define Xk
by Xk = {vi ∶ (k − 1)(d + 1) + 1 ≤ i ≤ k(d + 1)}. Let X = V (Cn) ∖ ⋃mk=1Xk. Note
that ∣X ∣ = x where n ≡ x (mod d + 1) and x is the unique integer such that 0 ≤ x ≤ d.
If x ≠ d, then let S = {v(k−1)(d+1)+1, v(k−1)(d+1)+2 ∶ 1 ≤ k ≤ m}. If x = d, then let
S = {v(k−1)(d+1)+1, v(k−1)(d+1)+2 ∶ 1 ≤ k ≤ m} ∪ {vm(d+1)+1}. It is straightforward to
check that in both cases S is a general d-position set, which shows that the claimed value
is a lower bound for gpd(Cn). As in the proof of Proposition 3.1, an application of the
pigeonhole principle establishes the upper bound.
Since diam(Cn) = ⌊n2 ⌋, to prove the second statement it is sufficient to show that
gpd(Cn) = 3 for d = ⌊n2 ⌋. For this purpose, let S = {v1, v3, v⌈n2 ⌉+2}. For n = 2r, we see that
S = {v1, v3, vr+2} and d = r. On the other hand, for n = 2r + 1, we have S = {v1, v3, vr+3}
and d = r. In both cases an easy computation shows that none of the three vertices lies on a
shortest path in Cn between the other two vertices. Therefore, S is a general d-position set,
and it follows that gpd(Cn) ≥ 3. Suppose T is an arbitrary general d-position set ofCn. We
may assume without loss of generality that v1 ∈ T . It follows that ∣T ∩ {v2, . . . , vr+1}∣ ≤ 1
and ∣T ∩ {vr+2, . . . vn}∣ ≤ 1, for otherwise T contains three vertices that lie on a path of
length at most d. Therefore, gpd(Cn) ≤ ∣T ∣ ≤ 3.
4 A characterization of general d-position sets
In [1, Theorem 3.1] a structural characterization of general position sets of a given graph
was proved. In this section we give such a characterization for general d-position sets and
as a consequence deduce the characterization from [1].
Theorem 4.1. Let G be a connected graph and let d ≥ 2 be an integer. Then S ⊆ V (G) is
a general d-position set if and only if the following conditions hold:
(i) ⟨S⟩ is a disjoint union of complete graphs Q1, . . . ,Qℓ.
(ii) If Qi ∦ Qj , i ≠ j, then dG(Qi,Qj) ≥ d.
(iii) If dG(Qi,Qj) + dG(Qj ,Qk) = dG(Qi,Qk) for {i, j, k} ∈ ([ℓ]3 ), then dG(Qi,Qk) >
d.
Proof. Let S be a general d-position set of G and let H be a connected component of ⟨S⟩.
If H is not complete, then it contains an induced P3. The vertices of this P3 are on a
geodesic of length 2 which is not possible since they belong to S and d ≥ 2. Hence H must
be complete.
Consider next two cliques Qi and Qj that are not parallel. Let dG(Qi,Qj) = p and let
u ∈ Qi and v ∈ Qj be vertices with dG(u, v) = p. Since Qi ∦ Qj , we may assume without
loss of generality that there is a vertex w ∈ Qi such that dG(w,Qj) = p + 1. Then u lies on
a w, v-geodesic of length p+1 which implies that p+1 ≥ d+1 and so, dG(Qi,Qj) = p ≥ d.
Assume next that dG(Qi,Qj)+dG(Qj ,Qk) = dG(Qi,Qk) for some {i, j, k} ∈ ([ℓ]3 ). If
Qi ∦ Qj , then by the already proved condition (ii) we immediately get that dG(Qi,Qj) ≥ d
and thus dG(Qi,Qk) > d. The same holds if Qj ∦ Qk. Hence assume next that Qi ∥
Qj and Qj ∥ Qk. Let u ∈ Qi and w ∈ Qk be vertices with dG(u,w) = dG(Qi,Qk).
Since dG(Qi,Qj) + dG(Qj ,Qk) = dG(Qi,Qk), Qi ∥ Qj , and Qj ∥ Qk, it follows that
40 Ars Math. Contemp. 21 (2021) #P1.03 / 33–44
dG(u,w) = dG(u, v)+dG(v,w) for every vertex v ofQj . We conclude that dG(Qi,Qk) >
d.
To prove the converse, assume that conditions (i), (ii), and (iii) are fulfilled for a given
set S and let {u, v,w} ∈ (S
3
). We need to show that u, v,w lie in general d-position.
If u, v,w lie in the same connected component of ⟨S⟩, then by (i), this component is
complete and the assertion is clear. Suppose next that u, v,w lie in the union of cliques Qi
and Qj . If Qi ∥ Qj , then u, v,w are clearly in general d-position. And if Qi ∦ Qj , then
u, v,w lie in general d-position by (ii).
In the last case to consider the three vertices lie in different cliques, say u ∈ Qi, v ∈ Qj ,
andw ∈ Qk. If the assertion does not hold, then the three vertices lie on a common geodesic
and we may assume without loss of generality that dG(u,w) = dG(u, v) + dG(v,w). If
Qi ∦ Qj , then by (ii), we get dG(Qi,Qj) ≥ d and hence dG(u,w) = dG(u, v)+dG(v,w) ≥
dG(Qi,Qj)+dG(Qj ,Qk) ≥ d+1 > d. Analogously, ifQj ∦ Qk, we also get dG(u,w) > d.
Suppose then that Qi ∥ Qj and Qj ∥ Qk. If also Qi ∥ Qk, then dG(u,w) = dG(u, v) +
dG(v,w) implies that dG(Qi,Qj) + dG(Qj ,Qk) = dG(Qi,Qk) and so dG(Qi,Qk) > d
by (iii). Again using the fact that Qi ∥ Qk, it follows that dG(u,w) > d. We are left with
the case that Qi ∥ Qj , Qj ∥ Qk, and Qi ∦ Qk. If dG(u,w) = dG(Qi,Qk), then by (iii),
we get that dG(u,w) > d. Otherwise we may assume without loss of generality that there
exists a vertex u′ ∈ Qi, u′ ≠ u, such that dG(Qi,Qk) = dG(u′,Qk) < dG(u,w). Since
Qi ∦ Qk, (ii) implies that dG(u′,Qk) ≥ d. But then dG(u,w) > dG(Qi,Qk) ≥ d.
Corollary 4.2 ([1, Theorem 3.1]). Let G be a connected graph. Then S ⊆ V (G) is a
general position set if and only if the following conditions hold:
(i) ⟨S⟩ is a disjoint union of complete graphs Q1, . . . ,Qℓ.
(ii) Qi ∥ Qj for every i ≠ j.
(iii) dG(Qi,Qj) + dG(Qj ,Qk) ≠ dG(Qi,Qk) for every {i, j, k} ∈ ([ℓ]3 ).
Proof. Set d = diam(G), so that general d-position sets are precisely general position sets.
Condition (ii) of Theorem 4.1 implies that in cliques Qi and Qj , which are not parallel,
we can find a pair of vertices at distance larger than diam(G). Since this is not possible,
every two cliques must be parallel. Similarly, if the assumption of condition (iii) would be
fulfilled for some cliques Qi, Qj , and Qk, then we would again have vertices at distance
larger than diam(G). Therefore, dG(Qi,Qj) + dG(Qj ,Qk) ≠ dG(Qi,Qk) must hold for
every {i, j, k} ∈ ([ℓ]
3
).
5 Connections with other topics
In this section we connect general d-position sets with the dissociation number and with
strong resolving graphs.
Strong resolving graphs
A vertex u of a connected graph G is maximally distant from a vertex v if every w ∈ N(u)
satisfies dG(v,w) ≤ dG(u, v). If u is maximally distant from v, and v is maximally distant
from u, then u and v are mutually maximally distant (MMD for short). Given an integer
d ≥ 2, the strong d-resolving graph GdSR of G has vertex set V (G), and two vertices u, v
S. Klavžar et al.: General d-position sets 41
are adjacent in GdSR if either u, v are MMD in G, or dG(u, v) ≥ d. The terminology used
in this construction comes from the notion of the strong resolving graph introduced in [19]
as a tool to study the strong metric dimension of graphs. See also [15].
The following observation will be useful in the proof of Theorem 5.1.
Observation 1. If G is connected and a vertex u of G is maximally distant from a vertex
v of G, then u ∉ I(v,w) for every w ∈ V (G) ∖ {u}.
Proof. For the sake of contradiction suppose there exists such a vertex w ∈ V (G) ∖ {u}
such that u ∈ I(v,w). Suppose that v = v0 . . . vi−1u = vivi+1 . . . vk = w is a v,w-geodesic.
Since this is a geodesic, it follows that d(v, u) = i. But u is maximally distant from v,
and thus d(v, vi+1) ≤ d(v, u) = i. Now, by following a shortest v, vi+1-path with the path
vi+2 . . . vk = w we arrive at a v,w-path of length less than k, which is a contradiction.
From Observation 1 it follows immediately that if three vertices x, y, z are pairwise
MMD, then x ∉ I(y, z), y ∉ I(x, z), and z ∉ I(x, y). From this we infer that x, y, z lie in
general d-position.
Theorem 5.1. If G is a connected graph and d ≥ 2 is an integer, then gpd(G) ≥ ω(GdSR).
Proof. We consider a set S ⊆ V (GdSR) that induces a (largest) complete subgraph of GdSR.
Then every two vertices x, y ∈ S are MMD in G, or dG(x, y) ≥ d. We now consider three
vertices x, y, z of S in the graph G. If they are pairwise MMD in G, then as above, x, y, z
lie in general d-position. Suppose then that two of them, say x and y, are not MMD in G.
Since x, y are adjacent in GdSR, it follows that dG(x, y) ≥ d. Suppose for instance that x, z
are MMD inG. By Observation 1, it follows that x ∉ I(z, y) and z ∉ I(x, y). If y ∈ I(x, z),
then dG(x, z) = dG(x, y)+dG(y, z) ≥ d+1, and hence x, y, z lie in general d-position. On
the other hand, if y ∉ I(x, z), then by definition, x, y, z lie in general d-position.
It remains only to consider the case in which no pair of x, y, z is MMD in G. This
means that the distance between any two of them is at least d, and this clearly means that
x, y, z are in general d-position.
Note that if d = diam(G), then GdSR is the standard strong resolving graph GSR as
defined in [19]. In this case Theorem 5.1 reduces to gp(G) ≥ ω(GSR), a result earlier
obtained in [12, Theorem 3.1].
Dissociation number and independence number
If G is a graph and S ⊆ V (G), then S is a dissociation set if ⟨S⟩ has maximum degree at
most 1. The dissociation number diss(G) of G is the cardinality of a largest dissociation
set in G. This concept was introduced by Yanakkakis [23]; see also [3, 4, 10]. Further, a
k-path vertex cover of G is a subset S of vertices of G such that every path of order k in
G contains at least one vertex from S. The minimum cardinality of a k-path vertex cover
in G is denoted by ψk(G). The minimum 3-path vertex cover is a dual problem to the
dissociation number because diss(G) = ∣V (G)∣ − ψ3(G); see [10]. For the algorithmic
state of the art on the 3-path vertex cover problem see [2].
Proposition 5.2. IfG is a triangle-free graph, d ≥ 2, and S ⊆ V (G) is a general d-position
set, then S is a dissociation set. Moreover, if d = 2, then S is a general 2-position set if and
only if S is a dissociation set.
42 Ars Math. Contemp. 21 (2021) #P1.03 / 33–44
Proof. Let d be a positive integer such that d ≥ 2. Suppose that S is a general d-position
set in a triangle-free graph G. By Theorem 4.1 every component of the subgraph ⟨S⟩ of G
induced by S is a complete graph. Since G is triangle-free, we conclude that each of these
components has order 1 or 2. Therefore, S is a dissociation set. Now assume that d = 2
and S is a dissociation set in G. The components C1, . . . ,Ck of ⟨S⟩ each have order 1 or 2
and are thus complete graphs. For every pair of distinct indices i, j in [k], the fact that Ci
and Cj are distinct components of the induced subgraph ⟨S⟩ implies that dG(Ci,Cj) ≥ 2.
Therefore, conditions (ii) and (iii) of Theorem 4.1 follow immediately, and hence S is a
general 2-position set.
Proposition 5.2 immediately gives the following result for triangle-free graphs.
Corollary 5.3. If G is a triangle-free graph and d ≥ 2, then gpd(G) ≤ diss(G). Moreover,
gp2(G) = diss(G).
We next relate the particular case of general 2-position number with the independence
number of graphs.
Proposition 5.4. If G is a connected graph without true twins, then gp2(G) ≥ α(G).
Proof. Let x, y ∈ V (G). Suppose first that xy ∈ E(G). Since x and y are not true twins, it
follows that x and y are not MMD. By definition, we infer that xy ∉ E(G2SR). On the other
hand, if xy ∉ E(G), then dG(x, y) ≥ 2 and by definition xy ∈ E(G2SR). Consequently,
G2SR is the complement G of G. Then by using Theorem 5.1, we have gp2(G) ≥ ω(G) =
α(G).
It is straighforward to see that if 2 ≤ m ≤ n, then gp2(Km,n) = n = α(Km,n). Hence
the bound of Proposition 5.4 is sharp. For another such family consider the grid graphs
P2r ◻ P2s. (For the definition of the Cartesian product operation ◻ see, for instance, [9].)
As already mentioned, ψ3(G) = n− diss(G) holds for any graph G of order n. Also, from
[5] it is known that ψ3(P2r ◻ P2s) = 2rs. Moreover, from Corollary 5.3, we have that
gp2(P2r ◻ P2s) = diss(P2r ◻ P2s). Thus,
gp2(P2r ◻ P2s) = diss(P2r ◻ P2s) = 4rs − ψ3(P2r ◻ P2s) = 2rs = α(P2r ◻ P2s) .
6 Infinite graphs and some open problems
The general position problem has been partially studied also on infinite graphs. In [17] it
was proved that gp(P 2∞) = 4, where P 2∞ is the 2-dimensional grid graph (alias the Cartesian
product of two copies of the two way infinite path). The general position number of the 2-
dimensional strong grid graph was also determined, and it was shown that 10 ≤ gp(P 3∞) ≤
16. In [13] the latter lower bound was improved to 14. All these efforts were recently
rounded off in [11] where it is proved that if n ∈ N, then gp(Pn∞) = 22
n−1
. On the other
hand, the following result reduces the study of the general d-position number of infinite
graphs to the case d =∞.
Proposition 6.1. If G is an infinite graph and d <∞, then gpd(G) =∞.
Proof. Let d <∞ be a fixed positive integer. There is nothing to be proved if d = 1, hence
assume that d ≥ 2.
S. Klavžar et al.: General d-position sets 43
Suppose first that diam(G) =∞. In this case G contains an infinite isometric path P =
v1v2 . . .. It is clear that {vdi ∶ i ∈ N} is a general d-position set, and hence gpd(G) =∞.
Suppose second that diam(G) < ∞. Considering an arbitrary vertex of G and its
distance levels we infer that G contains a vertex x with deg(x) = ∞. Let H = ⟨N[x]⟩.
Since H is an infinite graph, Erdős-Dushnik-Miller theorem [7] implies that H contains
a (countably) infinite independent set I or an infinite clique Q (of the same cardinality as
H). If H contains Q, then Q is also a clique of G, and hence G contains an infinite general
d-position set. On the other hand, if H contains I , then I is also an independent set of G.
Moreover, having in mind that H = ⟨N[x]⟩, we infer that each pair of vertices of I is at
distance 2 in G. This fact in turn implies that I is an infinite general d-position set of G.
We conclude that gpd(G) =∞.
6.1 Open questions
In this section we point out several questions that, in our opinion, are worthy of considera-
tion.
• In [20, Lemma 5.1] there is a polynomial algorithm for the dissociation number
of trees T and hence for gp2(T ). On the other hand, gpdiam(T )(T ) can also be
efficiently computed. Hence, is it possible to compute in polynomial time gpd(T )
for any 2 < d < diam(T )? More generally, what can be done for the case of block
graphs? We know that the simplicial vertices of a block graph form a gp-set. Can the
algorithm of Papadimitriou and Yannakakis be modified for block graphs?
• Compare diss(G) with gp2(G) for graphs G with ω(G) ≥ 3. Our guess is that these
invariants are incomparable in such graphs. Is there some relationship when G is a
block graph?
• What is gpd(G) whenever G is a grid-like graph? Note that by applying Corol-
lary 5.3 together with Theorem 4.1 in [5], one can find the value of gp2(Pn ◻ Pm)
for any n and m. Find gpd(Pn ◻ Pm) for d ≥ 3. Find the general d-position number
of a partial grid graph for d ≥ 2.
ORCID iDs
Sandi Klavžar https://orcid.org/0000-0002-1556-4744
Douglas F. Rall https://orcid.org/0000-0002-5482-756X
Ismael G. Yero https://orcid.org/0000-0002-1619-1572
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.04 / 45–55
https://doi.org/10.26493/1855-3974.2358.3c9
(Also available at http://amc-journal.eu)
On Hermitian varieties in PG(6, q2)
Angela Aguglia *
Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari,
Via Orabona 4, I-70125 Bari, Italy
Luca Giuzzi †
DICATAM, University of Brescia, Via Branze 53, I-25123 Brescia, Italy
Masaaki Homma
Department of Mathematics and Physics, Kanagawa University,
Hiratsuka 259-1293, Japan
Received 8 June 2020, accepted 15 February 2021, published online 10 August 2021
Abstract
In this paper we characterize the non-singular Hermitian variety H(6, q2) of PG(6, q2),
q ̸= 2 among the irreducible hypersurfaces of degree q + 1 in PG(6, q2) not containing
solids by the number of its points and the existence of a solid S meeting it in q4 + q2 + 1
points.
Keywords: Unital, Hermitian variety, algebraic hypersurface.
Math. Subj. Class. (2020): 51E21, 51E15, 51E20
1 Introduction
The set of all absolute points of a non-degenerate unitary polarity in PG(r, q2) determines
the Hermitian variety H(r, q2). This is a non-singular algebraic hypersurface of degree
q + 1 in PG(r, q2) with a number of remarkable properties, both from the geometrical and
the combinatorial point of view; see [6, 16]. In particular, H(r, q2) is a 2-character set with
respect to the hyperplanes of PG(r, q2) and 3-character blocking set with respect to the
*Corresponding author. The author was supported by the Italian National Group for Algebraic and Geometric
Structures and their Applications (GNSAGA - INdAM).
†The author was supported by the Italian National Group for Algebraic and Geometric Structures and their
Applications (GNSAGA - INdAM).
E-mail addresses: angela.aguglia@poliba.it (Angela Aguglia), luca.giuzzi@unibs.it (Luca Giuzzi),
homma@kanagawa-u.ac.jp (Masaaki Homma)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
46 Ars Math. Contemp. 21 (2021) #P1.04 / 45–55
lines of PG(r, q2) for r > 2. An interesting and widely investigated problem is to provide
combinatorial descriptions of H(r, q2).
First, we observe that a condition on the number of points and the intersection numbers
with hyperplanes is not in general sufficient to characterize Hermitian varieties; see [1,
2]. On the other hand, it is enough to consider in addition the intersection numbers with
codimension 2 subspaces in order to get a complete description; see [7].
In general, a hypersurface H of PG(r, q) is viewed as a hypersurface over the algebraic
closure of GF(q) and a point of PG(r, qi) in H is called a GF(qi)-point. A GF(q)-point
of H is also said to be a rational point of H. Throughout this paper, the number of GF(qi)-
points of H will be denoted by Nqi(H). For simplicity, we shall also use the convention
|H| = Nq(H).
In the present paper, we shall investigate a combinatorial characterization of the Her-
mitian hypersurface H(6, q2) in PG(6, q2) among all hypersurfaces of the same degree
having also the same number of GF(q2)-rational points.
More in detail, in [12, 13] it has been proved that if X is a hypersurface of degree q+1
in PG(r, q2), r ≥ 3 odd, with |X | = |H(r, q2)| = (qr+1 + (−1)r)(qr − (−1)r)/(q2 − 1)
GF(q2)-rational points, not containing linear subspaces of dimension greater than r−12 ,
then X is a non-singular Hermitian variety of PG(r, q2). This result generalizes the char-
acterization of [8] for the Hermitian curve of PG(2, q2), q ̸= 2.
The case where r > 4 is even is, in general, currently open. A starting point for
a characterization in arbitrary even dimension can be found in [3] where the case of a
hypersurface X of degree q + 1 in PG(4, q2), q > 3 is considered. There, it is shown that
when X has the same number of rational points as H(4, q2), does not contain any subspaces
of dimension greater than 1 and meets at least one plane π in q2+1GF(q2)-rational points,
then X is a Hermitian variety.
In this article we deal with hypersurfaces of degree q + 1 in PG(6, q2) and we prove
that a characterization similar to that of [3] holds also in dimension 6. We conjecture that
this can be extended to arbitrary even dimension.
Theorem 1.1. Let S be a hypersurface of PG(6, q2), q > 2, defined over GF(q2), not
containing solids. If the degree of S is q + 1 and the number of its rational points is
q11 + q9 + q7 + q4 + q2 + 1, then every solid of PG(6, q2) meets S in at least q4 + q2 + 1
rational points. If there is at least a solid Σ3 such that |Σ3 ∩ S| = q4 + q2 + 1, then S is a
non-singular Hermitian variety of PG(6, q2).
Furthermore, we also extend the result of [3] to the case q = 3.
2 Preliminaries and notation
In this section we collect some useful information and results that will be crucial to our
proof.
A Hermitian variety in PG(r, q2) is the algebraic variety of PG(r, q2) whose points ⟨v⟩
satisfy the equation η(v, v) = 0 where η is a sesquilinear form GF(q2)r+1×GF(q2)r+1 →
GF(q2). The radical of the form η is the vector subspace of GF(q2)r+1 given by
Rad(η) := {w ∈ GF(q2)r+1 : ∀v ∈ GF(q2)r+1, η(v, w) = 0}.
The form η is non-degenerate if Rad(η) = {0}. If the form η is non-degenerate, then the
corresponding Hermitian variety is denoted by H(r, q2) and it is a non-singular algebraic
A. Aguglia et al.: On Hermitian varieties in PG(6, q2) 47
variety, of degree q + 1 containing
(qr+1 + (−1)r)(qr − (−1)r)/(q2 − 1)
GF(q2)-rational points. When η is degenerate we shall call vertex Rt of the degenerate Her-
mitian variety associated to η the projective subspace Rt := PG(Rad(η)) := {⟨w⟩ : w ∈
Rad(η)} of PG(r, q2). A degenerate Hermitian variety can always be described as a cone
of vertex Rt and basis a non-degenerate Hermitian variety H(r − t, q2) disjoint from Rt
where t = dim(Rad(η)) is the vector dimension of the radical of η. In this case we shall
write the corresponding variety as RtH(r − t, q2). Indeed,
RtH(r − t, q2) := {X ∈ ⟨P,Q⟩ : P ∈ Rt, Q ∈ H(r − t, q2)}.
Any line of PG(r, q2) meets a Hermitian variety (either degenerate or not) in either
1, q + 1 or q2 + 1 points (the latter value only for r > 2). The maximal dimension of
projective subspaces contained in the non-degenerate Hermitian variety H(r, q2) is (r −
2)/2, if r is even, or (r − 1)/2, if r is odd. These subspaces of maximal dimension are
called generators of H(r, q2) and the generators of H(r, q2) through a point P of H(r, q2)
span a hyperplane P⊥ of PG(r, q2), the tangent hyperplane at P .
It is well known that this hyperplane meets H(r, q2) in a degenerate Hermitian variety
PH(r − 2, q2), that is in a Hermitian cone having as vertex the point P and as base a
non-singular Hermitian variety of Θ ∼= PG(r − 2, q2) contained in P⊥ with P ̸∈ Θ.
Every hyperplane of PG(r, q2) that is not tangent meets H(r, q2) in a non-singular
Hermitian variety H(r−1, q2), and is called a secant hyperplane of H(r, q2). In particular,
a tangent hyperplane contains
1 + q2(qr−1 + (−1)r)(qr−2 − (−1)r)/(q2 − 1)
GF(q2)-rational points of H(r, q2), whereas a secant hyperplane contains
(qr + (−1)r−1)(qr−1 − (−1)r−1)/(q2 − 1)
GF(q2)-rational points of H(r, q2).
We now recall several results which shall be used in the course of this paper.
Lemma 2.1 ([15]). Let d be an integer with 1 ≤ d ≤ q + 1 and let C be a curve of degree
d in PG(2, q) defined over GF(q), which may have GF(q)-linear components. Then the
number of its rational points is at most dq + 1 and Nq(C) = dq + 1 if and only if C is a
pencil of d lines of PG(2, q).
Lemma 2.2 ([10]). Let d be an integer with 2 ≤ d ≤ q + 2, and C a curve of degree d
in PG(2, q) defined over GF(q) without any GF(q)-linear components. Then Nq(C) ≤
(d− 1)q+ 1, except for a class of plane curves of degree 4 over GF(4) having 14 rational
points.
Lemma 2.3 ([11]). Let S be a surface of degree d in PG(3, q) over GF(q). Then
Nq(S) ≤ dq2 + q + 1
Lemma 2.4 ([8]). Suppose q ̸= 2. Let C be a plane curve over GF(q2) of degree q + 1
without GF(q2)-linear components. If C has q3 + 1 rational points, then C is a Hermitian
curve.
48 Ars Math. Contemp. 21 (2021) #P1.04 / 45–55
Lemma 2.5 ([7]). A subset of points of PG(r, q2) having the same intersection numbers
with respect to hyperplanes and spaces of codimension 2 as non-singular Hermitian vari-
eties is a non-singular Hermitian variety of PG(r, q2).
From [9, Theorem 23.5.1, Theorem 23.5.3] we have the following.
Lemma 2.6. If W is a set of q7 + q4 + q2 + 1 points of PG(4, q2), q > 2 such that every
line of PG(4, q2) meets W in 1, q + 1 or q2 + 1 points, then W is a Hermitian cone with
vertex a line and base a unital.
Finally, we recall that a blocking set with respect to lines of PG(r, q) is a point set
which blocks all the lines, i.e., intersects each line of PG(r, q) in at least one point.
3 Proof of Theorem 1.1
We first provide an estimate on the number of points of a curve of degree q+1 in PG(2, q2),
where q is any prime power.
Lemma 3.1. Let C be a plane curve over GF(q2), without GF(q2)-lines as components
and of degree q + 1. If the number of GF(q2)-rational points of C is N < q3 + 1, then
N ≤
 q
3 − (q2 − 2) if q > 3
24 if q = 3
8 if q = 2.
(3.1)
Proof. We distinguish the following three cases:
(a) C has two or more GF(q2)-components;
(b) C is irreducible over GF(q2), but not absolutely irreducible;
(c) C is absolutely irreducible.
Suppose first q ̸= 2.
Case (a) Suppose C = C1 ∪ C2. Let di be the degree of Ci, for each i = 1, 2. Hence
d1 + d2 = q + 1. By Lemma 2.2,
N ≤ Nq2(C1) +Nq2(C2) ≤ [(q + 1)− 2]q2 + 2 = q3 − (q2 − 2)
Case (b) Let C′ be an irreducible component of C over the algebraic closure of GF(q2). Let
GF(q2t) be the minimum defining field of C′ and σ be the Frobenius morphism of GF(q2t)
over GF(q2). Then
C = C′ ∪ C′σ ∪ C′σ
2
∪ . . . ∪ C′σ
t−1
,
and the degree of C′, say e, satisfies q + 1 = te with e > 1. Hence any GF(q2)-rational
point of C is contained in ∩t−1i=0C′σ
i
. In particular, N ≤ e2 ≤ ( q+12 )
2 by Bezout’s Theorem
and ( q+12 )
2 < q3 − (q2 − 2).
Case (c) Let C be an absolutely irreducible curve over GF(q2) of degree q + 1. Either C
has a singular point or not.
In general, an absolutely irreducible plane curve M over GF(q2) is q2-Frobenius non-
classical if for a general point P (x0, x1, x2) of M the point P q
2
= P q
2
(xq
2
0 , x
q2
1 , x
q2
2 ) is
A. Aguglia et al.: On Hermitian varieties in PG(6, q2) 49
on the tangent line to M at the point P . Otherwise, the curve M is said to be Frobenius
classical. A lower bound of the number of GF(q2)-points for q2-Frobenius non-classical
curves is given by [4, Corollary 1.4]: for a q2-Frobenius non-classical curve C′ of degree
d, we have Nq2(C′) ≥ d(q2 − d + 2). In particular, if d = q + 1, the lower bound is just
q3 + 1.
Going back to our original curve C, we know that C is Frobenius classical because
N < q3 + 1. Let F (x, y, z) = 0 be an equation of C over GF(q2). We consider the curve
D defined by ∂F∂x x
q2 + ∂F∂y y
q2 + ∂F∂z z
q2 = 0. Then C is not a component of D because C is
Frobenius classical. Furthermore, any GF(q2)-point P lies on C ∩ D and the intersection
multiplicity of C and D at P is at least 2 by Euler’s theorem for homogeneous polynomials.
Hence by Bézout’s theorem, 2N ≤ (q + 1)(q2 + q). Hence
N ≤ 1
2
q(q + 1)2.
This argument is due to Stöhr and Voloch [18, Theorem 1.1]. This Stöhr and Voloch’s
bound is lower than the estimate for N in case (a) for q > 4 and it is the same for q = 4.
When q = 3 the bound in case (a) is smaller than the Stöhr and Voloch’s bound.
Finally, we consider the case q = 2. Under this assumption, C is a cubic curve and neither
case (a) nor case (b) might occur. For a degree 3 curve over GF(q2) the Stöhr and Voloch’s
bound is loose, thus we need to change our argument. If C has a singular point, then C is a
rational curve with a unique singular point. Since the degree of C is 3, singular points are
either cusps or ordinary double points. Hence N ∈ {4, 5, 6}. If C is nonsingular, then it is
an elliptic curve and, by the Hasse-Weil bound, see [19], N ∈ I where I = {1, 2, . . . , 9}
and for each number N belonging to I there is an elliptic curve over GF(4) with N points,
from [14, Theorem 4.2]. This completes the proof.
Henceforth, we shall always suppose q > 2 and we denote by S an algebraic hypersur-
face of PG(6, q2) satisfying the following hypotheses of Theorem 1.1:
(S1) S is an algebraic hypersurface of degree q + 1 defined over GF(q2);
(S2) |S| = q11 + q9 + q7 + q4 + q2 + 1;
(S3) S does not contain projective 3-spaces (solids);
(S4) there exists a solid Σ3 such that |S ∩ Σ3| = q4 + q2 + 1.
We first consider the behavior of S with respect to the lines.
Lemma 3.2. An algebraic hypersurface T of degree q + 1 in PG(r, q2), q ̸= 2, with
|T | = |H(r, q2)| is a blocking set with respect to lines of PG(r, q2)
Proof. Suppose on the contrary that there is a line ℓ of PG(r, q2) which is disjoint from
T . Let α be a plane containing ℓ. The algebraic plane curve C = α ∩ T of degree q + 1
cannot have GF(q2)-linear components and hence it has at most q3 + 1 points because of
Lemma 2.2. If C had q3 +1 rational points, then from Lemma 2.4, C would be a Hermitian
curve with an external line, a contradiction since Hermitian curves are blocking sets. Thus
Nq2(C) ≤ q3. Since q > 2, by Lemma 3.1, Nq2(C) < q3−1 and hence every plane through
r meets T in at most q3−1 rational points. Consequently, by considering all planes through
r, we can bound the number of rational points of T by Nq2(T ) ≤ (q3 − 1) q
2r−4−1
q2−1 =
50 Ars Math. Contemp. 21 (2021) #P1.04 / 45–55
q2r−3 + · · · < |H(r, q2)|, which is a contradiction. Therefore there are no external lines to
T and so T is a blocking set w.r.t. lines of PG(r, q2).
Remark 3.3. The proof of [3, Lemma 3.1] would work perfectly well here under the
assumption q > 3. The alternative argument of Lemma 3.2 is simpler and also holds for
q = 3.
By the previous Lemma and assumptions (S1) and (S2), S is a blocking set for the
lines of PG(6, q2) In particular, the intersection of S with any 3-dimensional subspace Σ
of PG(6, q2) is also a blocking set with respect to lines of Σ and hence it contains at least
q4 + q2 + 1 GF(q2)-rational points; see [5].
Lemma 3.4. Let Σ3 be a solid of PG(6, q2) satisfying condition (S4), that is Σ3 meets S
in exactly q4 + q2 + 1 points. Then, Π := S ∩ Σ3 is a plane.
Proof. S∩Σ3 must be a blocking set for the lines of PG(3, q2); also it has size q4+q2+1.
It follows from [5] that Π := S ∩ Σ3 is a plane.
Lemma 3.5. Let Σ3 be a solid of satisfying condition (S4). Then, any 4-dimensional
projective space Σ4 through Σ3 meets S in a Hermitian cone with vertex a line and basis a
Hermitian curve.
Proof. Consider all of the q6 + q4 + q2 + 1 subspaces Σ3 of dimension 3 in PG(6, q2)
containing Π = S ∩ Σ3.
From Lemma 2.3 and condition (S3) we have |Σ3 ∩ S| ≤ q5 + q4 + q2 + 1. Hence,
|S| = (q7 + 1)(q4 + q2 + 1) ≤ (q6 + q4 + q2)q5 + q4 + q2 + 1 = |S|.
Consequently, |Σ3 ∩ S| = q5 + q4 + q2 + 1 for all Σ3 ̸= Σ3 such that Π ⊂ Σ3.
Let C := Σ4 ∩ S . Counting the number of rational points of C by considering the
intersections with the q2+1 subspaces Σ′3 of dimension 3 in Σ4 containing the plane Π we
get
|C| = q2 · q5 + q4 + q2 + 1 = q7 + q4 + q2 + 1.
In particular, C ∩Σ′3 is a maximal surface of degree q+ 1; so it must split in q+ 1 distinct
planes through a line of Π; see [17]. So C consists of q3 + 1 distinct planes belonging to
distinct q2 pencils, all containing Π ; denote by L the family of these planes. Also for each
Σ′3 ̸= Σ3, there is a line ℓ′ such that all the planes of L in Σ′3 pass through ℓ′. It is now
straightforward to see that any line contained in C must necessarily belong to one of the
planes of L and no plane not in L is contained in C.
In order to get the result it is now enough to show that a line of Σ4 meets C in either 1,
q + 1 or q2 + 1 points. To this purpose, let ℓ be a line of Σ4 and suppose ℓ ̸⊆ C. Then, by
Bezout’s theorem,
1 ≤ |ℓ ∩ C| ≤ q + 1.
Assume |ℓ ∩ C| > 1. Then we can distinguish two cases:
1. ℓ∩Π ̸= ∅. If ℓ and Π are incident, then we can consider the 3-dimensional subspace
Σ′3 := ⟨ℓ,Π⟩. Then ℓ must meet each plane of L in Σ′3 in different points (otherwise
ℓ passes through the intersection of these planes and then |ℓ ∩ C| = 1). As there are
q + 1 planes of L in Σ′3, we have |ℓ ∩ C| = q + 1.
A. Aguglia et al.: On Hermitian varieties in PG(6, q2) 51
2. ℓ ∩ Π = ∅. Consider the plane Λ generated by a point P ∈ Π and ℓ. Clearly
Λ ̸∈ L. The curve Λ∩S has degree q+1 by construction, does not contain lines (for
otherwise Λ ∈ L) and has q3 + 1 GF(q2)-rational points (by a counting argument).
So from Lemma 2.4 it is a Hermitian curve . It follows that ℓ is a q + 1 secant.
We can now apply Lemma 2.6 to see that C is a Hermitian cone with vertex a line.
Lemma 3.6. Let Σ3 be a space satisfying condition (S4) and take Σ5 to be a 5-dimensional
projective space with Σ3 ⊆ Σ5. Then S ∩ Σ5 is a Hermitian cone with vertex a point and
basis a Hermitian hypersurface H(4, q2).
Proof. Let
Σ4 := Σ
1
4,Σ
2
4, . . . ,Σ
q2+1
4
be the 4-spaces through Σ3 contained in Σ5. Put Ci := Σi4 ∩ S , for all i ∈ {1, . . . , q2 +1}
and Π = Σ3 ∩ C1. From Lemma 3.5 Ci is a Hermitian cone with vertex a line, say ℓi.
Furthermore Π ⊆ Σ3 ⊆ Σi4 where Π is a plane. Choose a plane Π′ ⊆ Σ14 such that
m := Π′ ∩ C1 is a line m incident with Π but not contained in it. Let P1 := m ∩ Π. It is
straightforward to see that in Σ14 there are exactly 1 plane through m which is a (q
4+q2+1)-
secant, q4 planes which are (q3 + q2 + 1)-secant and q2 planes which are (q2 + 1)-secant.
Also P1 belongs to the line ℓ1. There are now two cases to consider:
(a) There is a plane Π′′ ̸= Π′ not contained in Σi4 for all i = 1, . . . , q2+1 with m ⊆ Π′′ ⊆
S ∩ Σ5.
We first show that the vertices of the cones Ci are all concurrent. Consider mi :=
Π′′ ∩ Σi4. Then {mi : i = 1, . . . , q2 + 1} consists of q2 + 1 lines (including m) all
through P1. Observe that for all i, the line mi meets the vertex ℓi of the cone Ci in
Pi ∈ Π. This forces P1 = P2 = · · · = Pq2+1. So P1 ∈ ℓ1, . . . , ℓq2+1.
Now let Σ4 be a 4-dimensional space in Σ5 with P1 ̸∈ Σ4; in particular Π ̸⊆ Σ4. Put
also Σ3 := Σ14 ∩ Σ4. Clearly, r := Σ3 ∩ Π is a line and P1 ̸∈ r. So Σ3 ∩ S cannot be
the union of q + 1 planes, since if this were to be the case, these planes would have to
pass through the vertex ℓ1. It follows that Σ3∩S must be a Hermitian cone with vertex
a point and basis a Hermitian curve. Let W := Σ4 ∩ S . The intersection W ∩ Σi4, as i
varies, is a Hermitian cone with basis a Hermitian curve, so, the points of W are
|W| = (q2 + 1)q5 + q2 + 1 = (q2 + 1)(q5 + 1);
in particular, W is a hypersurface of Σ4 of degree q + 1 such that there exists a plane
of Σ4 meeting W in just one line (such planes exist in Σ3). Also suppose W to contain
planes and let Π′′′ ⊆ W be such a plane. Since Σi4∩W does not contain planes, all Σi4
meet Π′′′ in a line ti. Also Π′′′ must be contained in
⋃q2+1
i=1 ti. This implies that the set
{ti}i=1,...,q2+1 consists of q2 + 1 lines through a point P ∈ Π \ {P1}.
Furthermore each line ti passing through P must meet the radical line ℓi of the Hermi-
tian cone S ∩Σi4 and this forces P to coincide with P1, a contradiction. It follows that
W does not contain planes.
So by the characterization of H(4, q2) of [3] we have that W is a Hermitian variety
H(4, q2).
52 Ars Math. Contemp. 21 (2021) #P1.04 / 45–55
We also have that |S∩Σ5| = |P1H(4, q2)|. Let now r be any line of H(4, q2) = S∩Σ4
and let Θ be the plane ⟨r, P1⟩. The plane Θ meets Σi4 in a line qi ⊆ S for each
i = 1, . . . , q2 + 1 and these lines are concurrent in P1. It follows that all the points of
Θ are in S. This completes the proof for the current case and shows that S ∩ Σ5 is a
Hermitian cone P1H(4, q2).
(b) All planes Π′′ with m ⊆ Π′′ ⊆ S ∩Σ5 are contained in Σi4 for some i = 1, . . . , q2+1.
We claim that this case cannot happen. We can suppose without loss of generality
m ∩ ℓ1 = P1 and P1 ̸∈ ℓi for all i = 2, . . . , q2 + 1. Since the intersection of the
subspaces Σi4 is Σ3, there is exactly one plane through m in Σ5 which is (q
4+ q2+1)-
secant, namely the plane ⟨ℓ1,m⟩. Furthermore, in Σ14 there are q4 planes through m
which are (q3+q2+1)-secant and q2 planes which are (q2+1)-secant. We can provide
an upper bound to the points of S ∩ Σ5 by counting the number of points of S ∩ Σ5
on planes in Σ5 through m and observing that a plane through m not in Σ5 and not
contained in S has at most q3 + q2 + 1 points in common with S ∩ Σ5. So
|S ∩ Σ5| ≤ q6 · q3 + q7 + q4 + q2 + 1.
As |S∩Σ5| = q9+q7+q4+q2+1, all planes through m which are neither (q4+q2+1)-
secant nor (q2+1)-secant are (q3+q2+1)-secant. That is to say that all of these planes
meet S in a curve of degree q + 1 which must split into q + 1 lines through a point
because of Lemma 2.1.
Take now P2 ∈ Σ24 ∩ S and consider the plane Ξ := ⟨m,P2⟩. The line ⟨P1, P2⟩ is
contained in Σ24; so it must be a (q+1)-secant, as it does not meet the vertex line ℓ2 of
C2 in Σ24. Now, Ξ meets every of Σ
i
4 for i = 2, . . . , q
2 + 1 in a line through P1 which
is either a 1-secant or a q + 1-secant; so
|S ∩ Ξ| ≤ q2(q) + q2 + 1 = q3 + q2 + 1.
It follows that |S ∩ Ξ| = q3 + q2 + 1 and S ∩ Ξ is a set of q + 1 lines all through the
point P1. This contradicts our previous construction.
Lemma 3.7. Every hyperplane of PG(6, q2) meets S either in a non-singular Hermitian
variety H(5, q2) or in a cone with vertex a point over a Hermitian hypersurface H(4, q2).
Proof. Let Σ3 be a solid satisfying condition (S4). Denote by Λ a hyperplane of PG(6, q2).
If Λ contains Σ3 then, from Lemma 3.6 it follows that Λ∩S is a Hermitian cone PH(4, q2).
Now assume that Λ does not contain Σ3. Denote by S
j
5 , with j = 1, . . . , q
2 + 1
the q2 + 1 hyperplanes through Σ14, where as before, Σ
1
4 is a 4-space containing Σ3. By
Lemma 3.6 again we get that Sj5 ∩ S = P jH(4, q2). We count the number of rational
points of Λ∩ S by studying the intersections of Sj5 ∩ S with Λ for all j ∈ {1, . . . , q2 + 1}.
Setting Wj := Sj5 ∩ S ∩ Λ, Ω := Σ14 ∩ S ∩ Λ then
|S ∩ Λ| =
∑
j
|Wj \ Ω|+ |Ω|.
If Π is a plane of Λ then Ω consists of q + 1 planes of a pencil. Otherwise let m be the
line in which Λ meets the plane Π. Then Ω is either a Hermitian cone P0H(2, q2), or q+1
A. Aguglia et al.: On Hermitian varieties in PG(6, q2) 53
planes of a pencil, according as the vertex P j ∈ Π is an external point with respect to m or
not.
In the former case Wj is a non singular Hermitian variety H(4, q2) and thus |S ∩Λ| =
(q2 + 1)(q7) + q5 + q2 + 1 = q9 + q7 + q5 + q2 + 1.
In the case in which Ω consists of q+1 planes of a pencil then Wj is either a P0H(3, q2)
or a Hermitian cone with vertex a line ℓ and basis a Hermitian curve H(2, q2).
If there is at least one index j such that Wj = ℓH(2, q2), then there must be a 3-
dimensional space Σ′3 of S
j
5 ∩Λ meeting S in a generator. Hence, from Lemma 3.6 we get
that S ∩ Λ is a Hermitian cone P ′H(4, q2).
Assume that for all j ∈ {1, . . . , q2 + 1}, Wj is a P0H(3, q2). In this case
|S ∩ Λ| = (q2 + 1)q7 + (q + 1)q4 + q2 + 1 = q9 + q7 + q5 + q4 + q2 + 1 = |H(5, q2)|.
We are going to prove that the intersection numbers of S with hyperplanes are only two
that is q9 + q7 + q5 + q4 + q2 + 1 or q9 + q7 + q4 + q2 + 1.
Denote by xi the number of hyperplanes meeting S in i rational points with i ∈ {q9 +
q7 + q4 + q2 + 1, q9 + q7 + q5 + q2 + 1, q9 + q7 + q5 + q4 + q2 + 1}. Double counting
arguments give the following equations for the integers xi:
∑
i xi = q
12 + q10 + q8 + q6 + q4 + q2 + 1∑
i ixi = |S|(q10 + q8 + q6 + q4 + q2 + 1)∑
i=1 i(i− 1)xi = |S|(|S| − 1)(q8 + q6 + q4 + q2 + 1).
(3.2)
Solving (3.2) we obtain xq9+q7+q5+q2+1 = 0. In the case in which |S ∩ Λ| = |H(5, q2)|,
since S ∩ Λ is an algebraic hypersurface of degree q + 1 not containing 3-spaces, from
[19, Theorem 4.1] we get that S ∩Λ is a Hermitian variety H(5, q2) and this completes the
proof.
Proof of Theorem 1.1. The first part of Theorem 1.1 follows from Lemma 3.4. From
Lemma 3.7, S has the same intersection numbers with respect to hyperplanes and 4-spaces
as a non-singular Hermitian variety of PG(6, q2), hence Lemma 2.5 applies and S turns
out to be a H(6, q2).
Remark 3.8. The characterization of the non-singular Hermitian variety H(4, q2) given
in [3] is based on the property that a given hypersurface is a blocking set with respect to
lines of PG(4, q2), see [3, Lemma 3.1]. This lemma holds when q > 3. Since Lemma 3.2
extends the same property to the case q = 3 it follows that the result stated in [3] is also
valid in PG(4, 32).
4 Conjecture
We propose a conjecture for the general 2n-dimensional case.
Let S be a hypersurface of PG(2d, q2), q > 2, defined over GF(q2), not containing d-
dimensional projective subspaces. If the degree of S is q+1 and the number of its rational
points is |H(2d, q2)|, then every d-dimensional subspace of PG(2d, q2) meets S in at least
θq2(d − 1) := (q2d−2 − 1)/(q2 − 1) rational points. If there is at least a d-dimensional
54 Ars Math. Contemp. 21 (2021) #P1.04 / 45–55
subspace Σd such that |Σd ∩ S| = |PG(d − 1, q2)|, then S is a non-singular Hermitian
variety of PG(2d, q2).
Lemma 3.1 and Lemma 3.2 can be a starting point for the proof of this conjecture since
from them we get that S is a blocking set with respect to lines of PG(2d, q2).
ORCID iDs
Angela Aguglia https://orcid.org/0000-0001-6854-8679
Luca Giuzzi https://orcid.org/0000-0003-3975-7281
Masaaki Homma https://orcid.org/0000-0003-4568-6408
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.05 / 57–69
https://doi.org/10.26493/1855-3974.2357.373
(Also available at http://amc-journal.eu)
Achromatic numbers of Kneser graphs*
Gabriela Araujo-Pardo †
Instituto de Matemáticas, Unidad Juriquilla, Universidad Nacional Autónoma de México,
76230 Campus Juriquilla, Querétaro City, Mexico
Juan Carlos Dı́az-Patiño
Instituto de Matemáticas, Unidad Juriquilla, Universidad Nacional Autónoma de México,
76230 Campus Juriquilla, Querétaro City, Mexico
Christian Rubio-Montiel ‡
División de Matemáticas e Ingenierı́a, FES Acatlán,
Universidad Nacional Autónoma de México,
53150 Acatlán, Naucalpan de Juárez, Mexico
Received 8 June 2020, accepted 21 March 2021, published online 10 August 2021
Abstract
Complete vertex colorings have the property that any two color classes have at least
an edge between them. Parameters such as the Grundy, achromatic and pseudoachromatic
numbers come from complete colorings, with some additional requirement. In this paper,
we estimate these numbers in the Kneser graph K(n, k) for some values of n and k. We
give the exact value of the achromatic number of K(n, 2).
Keywords: Achromatic number, pseudoachromatic number, Grundy number, block designs, geometric
type Kneser graphs.
Math. Subj. Class. (2020): 05C15, 05B05, 05C62
*The authors wish to thank the anonymous referees of this paper for their suggestions and remarks.
Part of the work was done during the I Taller de Matemáticas Discretas, held at Campus-Juriquilla, Universidad
Nacional Autónoma de México, Querétaro City, Mexico on July 28 – 31, 2014. Part of the results of this paper
was announced at Discrete Mathematics Days – JMDA16 in Barcelona, Spain on July 6 – 8, 2016, see [4].
†Araujo-Pardo was partially supported by PAPIIT of Mexico grants IN107218, IN106318 and CONACyT of
Mexico grant 282280.
‡Corresponding author.
E-mail addresses: garaujo@matem.unam.mx (Gabriela Araujo-Pardo), juancdp@im.unam.mx (Juan Carlos
Dı́az-Patiño), christian.rubio@acatlan.unam.mx (Christian Rubio-Montiel)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
58 Ars Math. Contemp. 21 (2021) #P1.05 / 57–69
1 Introduction
Since the beginning of the study of colorings in graph theory, many interesting results
have appeared in the literature, for instance, the chromatic number of Kneser graphs. Such
graphs give an interesting relation between finite sets and graphs.
Let V be the set of all k-subsets of [n] := {1, 2, . . . , n}, where 1 ≤ k ≤ n/2. The
Kneser graph K(n, k) is the graph with vertex set V such that two vertices are adjacent if
and only if the corresponding subsets are disjoint. Lovász [18] proved that χ(K(n, k)) =
n− 2(k − 1) via the Borsuk-Ulam theorem, see Chapter 38 of [2].
Some results on the Kneser graphs and parameters of colorings have appeared since
then, for instance [4, 8, 10, 13, 16, 19].
An l-coloring of a graph G is a surjective function ς that assigns a number from the
set [l] to each vertex of G. An l-coloring of G is proper if any two adjacent vertices have
different colors. An l-coloring ς is complete if for each pair of different colors i, j ∈ [l]
there exists an edge xy ∈ E(G) such that ς(x) = i and ς(y) = j.
The largest value of l for which G has a complete l-coloring is called the pseudoachro-
matic number of G [12], denoted ψ(G). A similar invariant, which additionally requires
an l-coloring to be proper, is called the achromatic number of G and denoted by α(G)
[15]. Note that α(G) is at least χ(G) since the chromatic number χ(G) of G is the smallest
number l for which there exists a proper l-coloring of G and then such an l-coloring is also
complete. Therefore, for any graph G, χ(G) ≤ α(G) ≤ ψ(G).
In this paper, we estimate these parameters arising from complete colorings of Kneser
graphs. The paper is organized as follows. In Section 2 we recall notions of block designs.
Section 3 is devoted to the achromatic number α(K(n, 2)) of the Kneser graphK(n, 2).
It is proved that α(K(n, 2)) =
⌊(
n+1
2
)
/3
⌋
for n ̸= 3.
In Section 4 it is shown that ψ(K(n, 2)) satisfies
⌊(
n
2
)
/2
⌋
≤ ψ(K(n, 2)) ≤
⌊
(
(
n
2
)
+
⌊n
2
⌋
)/2
⌋
for n ≥ 7 and that the upper bound is tight.
The Section 5 establishes that the Grundy number Γ(K(n, 2)) equals α(K(n, 2)). The
Grundy number Γ(G) of a graph G is determined by the worst-case result of a greedy
proper coloring applied on G. A greedy l-coloring technique operates as follows. The ver-
tices (listed in some particular order) are colored according to the algorithm that assigns to a
vertex under consideration the smallest available color. Therefore, greedy proper colorings
are also complete.
Section 6 gives a natural upper bound for the pseudoachromatic number ofK(n, k) and
a lower bound for the achromatic number of K(n, k) in terms of the b-chromatic number
of K(n, k), another parameter arising from complete colorings.
Section 7 is about the achromatic numbers of some geometric type Kneser graphs. A
complete geometric graph of n points is an embedding of the complete graph Kn in the
Euclidean plane such that its vertex set is a set V of points in general position, and its
edges are straight-line segments connecting pairs of points in V . We study the achromatic
numbers of graphs DV (n) whose vertex set is the set of edges of a complete geometric
graph of n points and adjacency is defined in terms of geometric disjointness.
To end, in Section 8, we discuss the case of the odd graphs K(2k + 1, k).
G. Araujo-Pardo et al.: Achromatic numbers of Kneser graphs 59
2 Preliminaries
All graphs in this paper are finite and simple. Note that the complement of the line graph
of the complete graph on n vertices is the Kneser graph K(n, 2). We use this model of the
Kneser graph K(n, 2) in Sections 3, 4, 5 and 7.
Let n, b, k, r and λ be positive integers with n > 1. Let D = (P,B, I) be a triple
consisting of a set P of n distinct objects, called points of D, a set B of b distinct objects,
called blocks of D (with P ∩ B = ∅), and an incidence relation I , a subset of P × B. We
say that v is incident to u if exactly one of the ordered pairs (u, v) and (v, u) is in I; then
v is incident to u if and only if u is incident to v. D is called a 2-(n, b, k, r, λ) block design
(for short, 2-(n, b, k, r, λ) design) if it satisfies the following axioms.
1. Each block of D is incident to exactly k distinct points of D.
2. Each point of D is incident to exactly r distinct blocks of D.
3. If u and v are distinct points of D, then there are exactly λ blocks of D incident to
both u and v.
A 2-(n, b, k, r, λ) design is called a balanced incomplete block design BIBD; it is called an
(n, k, λ)-design, too, since the parameters of a 2-(n, b, k, r, λ) design are not all indepen-
dent. The two basic equations connecting them are nr = bk and r(k− 1) = λ(n− 1). For
a detailed introduction to block designs we refer to [5, 6].
A design is resolvable if its blocks can be partitioned into r sets so that b/r blocks of
each part are point-disjoint and each part is called a parallel class.
A Steiner triple system STS(n) is an (n, 3, 1)-design. It is well-known that an STS(n)
exists if and only if n ≡ 1, 3 mod 6. A resolvable STS(n) is called a Kirkman triple system
and denoted by KTS(n) and exists if and only if n ≡ 3 mod 6, see [21].
An (n, 5, 1)-design exists if and only if n ≡ 1, 5 mod 20, see [6].
An (n, k, 1)-design can naturally be regarded as an edge partition into Kk subgraphs,
of the complete graph Kn.
Finally, we recall that the concepts of a 1-factor and a 1-factorization represent, for the
case of Kn, a parallel class and a resolubility of an (n, 2, 1)-design, respectively.
3 The exact value of α(K(n, 2))
In this section, we prove that α(K(n, 2)) =
⌊(
n+1
2
)
/3
⌋
for every n ̸= 3. The proof is about
the upper bound and the lower bound have the same value.
Theorem 3.1. The achromatic number α(K(n, 2)) ofK(n, 2) equals
⌊(
n+1
2
)
/3
⌋
for n ̸= 3
and α(K(3, 2)) = 1.
Proof. First, we prove the upper bound α(K(n, 2)) ≤
⌊(
n+1
2
)
/3
⌋
.
Let ς be a proper and complete coloring of K(n, 2). Consider the graph K(n, 2) as the
complement of L(Kn). Note that vertices corresponding to a color class of ς of size two
induce a P3 subgraph, say abc, of the complete graphKn with V (Kn) = [n]; then no color
class of ς of size one is a pair containing b. Therefore, if ς has x color classes of size one
(they form a matching in Kn of size x) and y color classes of size two, then y ≤ n− 2x,
α(K(n, 2)) ≤
(
n
2
)
− x− 2(n− 2x)
3
+x+(n− 2x) =
(
n
2
)
+ 2x+ (n− 2x)
3
=
(
n
2
)
+ n
3
60 Ars Math. Contemp. 21 (2021) #P1.05 / 57–69
and we get α(K(n, 2)) ≤
⌊(
n+1
2
)
/3
⌋
. For the case of n = 3, K(3, 2) is an edgeless graph,
hence α(K(3, 2)) = 1.
Next, we exhibit a proper and complete edge coloring of the complement of L(Kn) that
uses
⌊(
n+1
2
)
/3
⌋
colors. We remark that in order to obtain such a tight coloring it suffices
to achieve that all color classes are of size at most three, while the number of exceptional
vertices of Kn (that are involved neither in a color class of size one nor in the role of the
“center” of a color class of size two) is at most one. We shall refer to this condition as the
condition (C).
Figure 1 documents the equality for n ≤ 5. For the remainder of this proof, we need to
Figure 1: α(K(n, 2)) =
⌊(
n+1
2
)
/3
⌋
for n = 2, 4, 5 and α(K(3, 2)) = 1.
distinguish four cases, namely, when n = 6k, 6k+2; n = 6k+3, 6k+5; n = 6k+4 and
n = 6k + 1 for k ≥ 1.
1. Case n = 6k or n = 6k + 2. Since n + 1 ≡ 1, 3 mod 6 there exists an STS(n +
1). We can think of K(n, 2) as having the vertex set equal to the set of points of
STS(n + 1) other than v. Then each vertex of K(n, 2) is a subset of exactly one
block of STS(n+ 1)− v; the blocks of STS(n+ 1)− v are (3-element) blocks of
STS(n+1) not containing v, and (2-element) blocksB \{v}, whereB is a block of
STS(n+1) with v ∈ B. Consider a vertex coloring of K(n, 2) that is defined in the
following way: Color classes of size three are triangles of STS(n+ 1)− v (we use
this simplified expression to indicate that all vertices of K(n, 2), that are subsets of a
fixed triangle of STS(n+1)−v, receive the same color). All remaining color classes
are of size one; they are formed by 2-element blocks of STS(n+ 1)− v. (They can
also be regarded as edges of a perfect matching of the “underlying” complete graph
on points of STS(n+ 1)− v.) The coloring is obviously proper. It is complete, too
(see Figure 2), and satisfies the condition (C), hence α(K(n, 2)) =
⌊(
n+1
2
)
/3
⌋
.
Figure 2: Every two classes have two disjoint edges in the complement of L(Kn).
2. Case n = 6k + 3 or n = 6k + 5. Add two points u, v to the points of STS(n− 2),
and colorK(n, 2) as follows. Color classes of size three are triangles of STS(n−2)
except for one with points a, b, c. The remaining color classes are of size two. Five
of them correspond to the optimum coloring of K(5, 2) depicted in Figure 1 (with
G. Araujo-Pardo et al.: Achromatic numbers of Kneser graphs 61
points a, b, c, u, v). Finally, every point x of STS(n− 2), x /∈ {a, b, c}, gives rise to
the color class {ux, xv}, see Figure 3 (Left). The coloring is proper and complete,
see Figure 3 (Right), and it satisfies the condition (C).
STS
u
v
a b
c x
Figure 3: (Left) Color classes of size two in the proof of Case 2. (Right) Every two distinct
color classes of size two contain disjoint edges in the complement of L(Kn).
3. Case n = 6k + 4. Add a point v to the points of a resolvable STS(n − 1), for
instance a KTS(n − 1). Color classes of size three are triangles of STS(n − 1)
except for the triangles of a parallel class P = {Ti : i = 0, 1, . . . , n−43 }, where
Ti = {v3i+1, v3i+2, v3i+3}. For each triangle Ti of P color vertices of K(4, 2) with
the vertex set Ti ∪ {v} according to the optimum coloring of Figure 1, see Figure
4. The resulting coloring is proper and complete, and it fulfills the condition (C).
Indeed, the number of color classes of size two is n − 1; since “centers” of those
color classes are pairwise disjoint, v is the only exceptional vertex.
STS
v
v1
v2
v3 vn−1
vn−2
vn−3
Figure 4: Color classes of size two in the proof of Case 3.
4. Case n = 6k + 1. First, we analyze the case of k = 1. Delete two points of
STS(9) presented in Figure 5 (Left) to finish with points v1, v2, . . . , v7. The “sur-
vived” triangles are color classes of size three, see Figure 5 (Center). The remain-
ing six pairs of points are divided into four color classes {v1v2, v2v3}, {v3v4},
62 Ars Math. Contemp. 21 (2021) #P1.05 / 57–69
{v4v5, v5v6} and {v6v1}, see Figure 5 (Right). The obtained coloring shows that
α(K(7, 2)) = 9 =
⌊(
7+1
2
)
/3
⌋
.
v1
v2
v3
v4
v5
v6
Figure 5: (Left) STS(9). (Center) The 5 color classes of size three of K7. (Right) Color
classes of size one and two in K(7, 2).
If k ≥ 2, consider an STS(n − 4) with points v1, v2, . . . , vn−4 that has a paral-
lel class P = {Ti : i = 0, 1, . . . , n−73 }, where Ti = {v3i+1, v3i+2, v3i+3}. Add
to points of STS(n − 4) the points a, b, c, d. Every triangle of STS(n − 4) ex-
cept for the triangles of P is a color class of size three. Let Hi denote the join
of Ti with the complement of K4 on vertices a, b, c, d; the join of two vertex dis-
joint graphs G and H has the vertex set V (G) ∪ V (H) and the edge set E(G) ∪
E(H) ∪ {xy : x ∈ V (G), y ∈ V (H)}. Pairs of points corresponding to edges of
Hi, i = 0, 1, . . . , n−103 , form color classes of size three determined by point triples
{v3i+1, v3i+2, a}, {v3i+2, v3i+3, b} and {v3i+1, v3i+3, c}, and color classes of size
two {av3i+3, v3i+3d}, {bv3i+1, v3i+1d} and {cv3i+2, v3i+2d}, see Figure 6. Finally,
pairs of points from the set S = {vn−6, vn−5, vn−4, a, b, c, d} are colored so that
nine color classes are created just as in the coloring of K(7, 2) described above for
the case k = 1. The coloring is proper and complete, and the condition (C) is
fulfilled, since the number of exceptional vertices in the “underlying” Kn is one (ex-
ceptional is the vertex of S that is involved only in color classes of size three); so,
α(K(n, 2)) =
⌊(
n+1
2
)
/3
⌋
in this case, too.
a
b c
d
v3i+3
v3i+2
v3i+1
Figure 6: The 6-coloring of Hi.
By the four cases, the theorem follows.
G. Araujo-Pardo et al.: Achromatic numbers of Kneser graphs 63
4 About the value of ψ(K(n, 2))
In this section, we determine bounds for ψ(K(n, 2)). The gap between the bounds is Θ(n),
however, the upper bound is tight for an infinite number of values of n.
Theorem 4.1. ψ(K(n, 2)) = α(K(n, 2)) for 2 ≤ n ≤ 6 and⌊(
n
2
)
2
⌋
≤ ψ(K(n, 2)) ≤
⌊(
n
2
)
+
⌊
n
2
⌋
2
⌋
for n ≥ 7. Moreover, the upper bound is tight if n ≡ 0, 4 mod 20.
Proof. For n = 2, 3, the graph K(n, 2) is edgeless and then ψ(K(n, 2)) = α(K(n, 2)) =
1.
For n = 4, 5, 6, α(K(n, 2)) is 3, 5, 7, respectively (by Theorem 3.1). Note that any
complete coloring having a color class of size one uses at most k = 2, 4, 7 colors, re-
spectively. And any complete coloring without color classes of size one uses at most
k = 3, 5, 7 colors, respectively. Therefore, ψ(K(n, 2)) is at most 3, 5, 7, respectively.
Hence ψ(K(n, 2)) = α(K(n, 2)).
For n ≥ 7, any complete coloring of K(n, 2) has at most ω(K(n, 2)) =
⌊
n
2
⌋
classes
of size 1 (ω(G) is the clique number of the garph G, that is, the largest order of a complete
subgraph of G), then
ψ(K(n, 2)) ≤
⌊(
n
2
)
−
⌊
n
2
⌋
2
+
⌊n
2
⌋⌋
=
⌊(
n
2
)
+
⌊
n
2
⌋
2
⌋
.
Such an upper bound is proved in [1].
To see the lower bound, we use a 1-factorization F of K2t such that no component
induced by two distinct 1-factors of F is a 4-cycle, see [17, 20]. We need to distinguish
four cases, namely, when n = 4k − 1, 4k, n = 4k + 1 and n = 4k + 2 for k ≥ 1.
1. Case n = 4k. Consider F for t = 2k. Since each 1-factor contains t edges, we have
k color classes of size two for each 1-factor, therefore the lower bound follows.
2. Case n = 4k + 1. Consider F for t = 2k + 1 and delete a vertex of K4k+2. Since
each maximal matching arising from a 1-factor of F contains t − 1 edges, we have
k color classes of size two for each such maximal matching, hence the lower bound
follows.
3. Case n = 4k + 2. Consider F for t = 2k and add two new vertices a and b to
V (K4k) to obtain K4k+2. Color the subgraph K4k as above, and the remaining
edges as follows. For each vertex x of K4k, we have the classes {ax, xb}. Finally,
color the edge ab in a greedy way and the result follows.
4. Case n = 4k − 1. Consider F for t = 2k − 1 and adding a new vertex b to obtain
K4k−1. Color the subgraph K4k−3 = K4k−1 − {a, b} as in the case n ≡ 1 mod 4,
and form for each vertex x of K4k−3 the color class {ax, xb}. Finally, choose for
the edge ab greedily a color that is already used; the result then follows.
Now, to verify that the upper bound is tight, consider an (n + 1, 5, 1)-design D, see
[6]. Therefore n + 1 ≡ 1, 5 mod 20. Choose a point v of D and let P = {Qi : i =
64 Ars Math. Contemp. 21 (2021) #P1.05 / 57–69
0, 1, . . . , n−44 } with Qi = {v4i+1, v4i+2, v4i+3, v4i+4} be the set of 4-blocks of D − v.
Pairs of points of every 5-block of D − v are colored so that five color classes of size two
are created, see Figure 7; the coloring is not proper, since all those color classes induce a
K2 subgraph of K(n, 2).
Figure 7: A complete coloring of K(5, 2) using five colors that is not proper.
Label the edges of each block Qi of P as f2i = v4i+1v4i+2, f2i+1 = v4i+3v4i+4,
ei, en/4+i, en/2+i and e3n/4+i. Remaining vertices of K(n, 2) are colored to form color
classes {fi} of size one for i = 0, 1, . . . , n/2 − 1, and color classes {e2i, e2i+1} of size
two for i = 0, 1, . . . . , n/4− 1. The coloring is complete, hence the result follows.
5 On the Grundy number ofK(n, 2)
In this section, we observe that the coloring used in Theorem 3.1 is also a greedy coloring.
An l-coloring of G is called Grundy, if it is a proper coloring having the property that
for every two colors i and j with i < j, every vertex colored j has a neighbor colored i
(consequently, every Grundy coloring is a complete coloring). Moreover, a coloring ς of
a graph G is a Grundy coloring of G if and only if ς is a greedy coloring of G, see [7].
Therefore, the Grundy number Γ(G) is the largest l for which a Grundy l-coloring of G
exists. Any graph G satisfies, χ(G) ≤ Γ(G) ≤ α(G) ≤ ψ(G).
Consider the coloring used in Theorem 3.1. Divide colors into small, medium and high
(recall that colors used in Theorem 3.1 are positive integers), and use them for color classes
of size three, two and one, respectively. We only need to verify that if i and j are colors
with i < j, then for every edge e of color j there exists an edge of color i that is disjoint
with e. This is certainly true if j is a high color, since the coloring is complete. If the
color j is not high, the required condition is satisfied because of the following facts: (i)
(3-element) vertex sets corresponding to color classes i and j have at most one vertex in
common; (ii) the centers of involved P3 subgraphs are distinct if both i and j are medium
colors.
Consider the coloring used in Theorems 3.1. Taking the highest colors as the color
class of size 1 and the smallest colors as the color classes of size 3. We only need to verify
that for every two color classes with colors i and j, i < j, and every edge of color j there
always exist a disjoint edge of color i. This is true if the color classes are triangles because
they only share at most one vertex. If the color classes are an triangle K3 with color i and
a path P3 with color j this is also true.
Theorem 5.1. Γ(K(n, 2)) =
⌊(
n+1
2
)
/3
⌋
for n ̸= 3 and Γ(K(3, 2)) = 1.
G. Araujo-Pardo et al.: Achromatic numbers of Kneser graphs 65
6 About general upper bounds
The known upper bound for the pseudoachromatic number states, for K(n, k) (see [7]),
that
ψ(K(n, k)) ≤ 1
2
+
√
1
4
+
(
n
k
)(
n− k
k
)
= O
(
nk/2(n− k)k/2
k!
)
. (6.1)
A slightly improved upper bound is the following. Let ς be a complete coloring of
K(n, k) using l colors with l = ψ(K(n, k)). Let x = min{
∣∣ς−1(i)∣∣ : i ∈ [l]}, that is, x is
the cardinality of the smallest color class of ς; without loss of generality we may suppose
that x =
∣∣ς−1(l)∣∣. Since ς defines a partition of the vertex set of K(n, k) it follows that
l ≤ f(x) :=
(
n
k
)
/x.
Additionally, since K(n, k) is
(
n−k
k
)
-regular, there are at most
(
n−k
k
)
vertices adjacent
in K(n, k) to a vertex of ς−1(l). With X :=
⋃
X∈ς−1(l)X we have |[n] \ X | ≥ n− kx. If
n−kx ≥ k and Y ⊆ [n]\X , |Y | = k, then each of x edgesXY , X ∈ ς−1(l), corresponds
to the pair of colors l, ς(Y ). Therefore, ψ(K(n, k)) ≤ g(x), where g(x) := 1+ x
(
n−k
k
)
−
(x−1)
(
n−kx
k
)
, if n−kx ≥ k, and g(x) := 1+x
(
n−k
k
)
otherwise. Consequently, we have:
ψ(K(n, k)) ≤ max {min{f(x), g(x)} : x ∈ N} .
Hence, we conclude that:
ψ(K(n, k)) ≤ ⌊max {min{f(x), g(x)} : x ∈ N}⌋
and then
ψ(K(n, k)) ≤
⌊
max
{
min{f(x), g(x)} : x ∈ R+
}⌋
.
It is not hard to see that max {min{f(x), g(x)} : x ∈ R+} ≤ 12 +
√
1
4 +
(
n
k
)(
n−k
k
)
.
On a general lower bound. An l-coloring ς is called dominating if every color class
contains a vertex that has a neighbor in every other color class. The b-chromatic number
φ(G) of G is defined as the largest number l for which there exists a dominating l-coloring
of G (see [16]). Since a dominating coloring is also complete, hence, for any graph G,
φ(G) ≤ α(G). The following theorem was proved in [13]:
Theorem 6.1 (Hajiabolhassan [13]). Let k ≥ 3 an integer. If n ≥ 2k, then 2
(⌊n2 ⌋
k
)
≤
φ(K(n, k)).
In consequence, for any n, k satisfying n ≥ 2k ≥ 6, we have
α(K(n, k)) ≥ 2
(⌊n
2
⌋
k
)
= Ω
(
nk
2k−1kk
)
.
7 The achromatic numbers ofDV (n)
Let V be a set of n points in general position in the plane, i.e., no three points of V are
collinear. The segment disjointness graph DV (n) has the vertex set equal to the set of
all straight line segments with endpoints in V , and two segments are adjacent in DV (n)
if and only if they are disjoint. Each graph DV (n) is a spanning subgraph of K(n, 2).
The chromatic number of the graph DV (n) is bounded in [3] where it is proved that
χ(DV (n)) = Θ(n).
66 Ars Math. Contemp. 21 (2021) #P1.05 / 57–69
In this subsection, we prove bounds for α(DV (n)) and ψ(DV (n)). Having in mind the
fact that ψ(H) ≤ ψ(G) if H is a subgraph of G, Theorem 4.1 yields
ψ(DV (n)) ≤
⌊(
n
2
)
+
⌊
n
2
⌋
2
⌋
≤ n
2
4
.
For the lower bound, we use the following results. A straight line thrackle is a set S of
straight line segments such that any two distinct segments of S either meet at a common
endpoint or they cross each other (see [9]).
Theorem 7.1 (Erdős [9] (see also the proof of Theorem 1 of [22])). If d1(n) denotes the
maximum number of edges of a straight line thrackle of n vertices then d1(n) = n.
Lemma 7.2. Any two triangles T1 and T2 with points in V , that share at most one point,
contain two disjoint edges.
Proof. Case 1. T1 has a point in common with T2: Since T1 ∪ T2 have five points and six
edges, then two of its edges are disjoint due to d1(5) = 5.
Case 2. T1 has no points in common with T2: Let e be an edge of T2. Let us suppose
that T1 ∪ T2 does not contain two disjoint edges, then T1 and e is a straight line thrackle.
Therefore, a vertex of e and a vertex of T1 have to be the same, which is impossible because
T1 has no points in common with T2.
Now, if we identify a Steiner triple system STS(n) with the complete geometric graph
of n points and we color each triangle with a different color, by Lemma 7.2, we have the
following.
Lemma 7.3. If n ≡ 1, 3 mod 6 and V is a set of n points in general position, then
n2
6
− n
6
=
1
3
(
n
2
)
≤ α(DV (n))
Therefore, we have the following theorem.
Theorem 7.4. For any natural number n and any set of n points V in general position,
n2
6
−Θ(n) ≤ α(DV (n)).
Further, if Kn has an even number of vertices, then there is a set F ⊆ E(Kn) such that
E(Kn) \ F can be partitioned into triangles. More precisely, if n ≡ 0, 2 mod 6,then F is a
perfect matching in Kn, and if n ≡ 4 mod 6, then F induces a spanning forest of n/2 + 1
edges in Kn with all vertices having an odd degree, see [11, 14].
A set V of n points in convex position is a set of n points in general position such
that they are the vertices of a convex polygon (each internal angle is strictly less than 180
degrees).
Theorem 7.5. For any even natural number n and any set of n points V in convex position,
n2
6
+ Θ(n) ≤ α(DV (n))
G. Araujo-Pardo et al.: Achromatic numbers of Kneser graphs 67
Figure 8: Configurations of the color classes of size one arising from F and a dashed
triangle of Kn − F in the proof of Theorem 7.5.
Proof. Take the edges of F in the convex hull of V , except for one in the case of n ≡ 4
mod 6, see Figure 8. Each component of F is a color class. Each triangle of Kn − F is a
color class. Essentially we use
(
n+1
2
)
/3 triangles, and the result follows.
Finally, the geometric type Kneser graph DV (n, k) for k ≥ 2 whose vertex set consists
of all subsets of k points in V . Two such sets X and Y are adjacent if and only if their
convex hulls are disjoint. Given a point set V , for a line dividing V into two sets V1 and V2
of n/2 points, having a coloring such that each color class has sets X ⊆ V1 and Y ⊆ V2,
we have that
ψ(DV (n, k)) ≥
(
n/2
k
)
= Ω
(
nk
2kkk
)
8 On odd graphs
It is obvious to prove that the achromatic and the pseudoachromatic number as well of (the
graph induced by) a matching of size
(
k
2
)
is equal to k. Therefore, a matching of sizem has
achromatic and pseudoachromatic number equal to
⌊
1
2 +
√
1
4 + 2m
⌋
, which means that in
the case n = 2k the upper bound of (1) for ψ(K(2k, k)) is equal to the lower bound for
α(K(2k, k)); in other words,
α(K(2k, k)) = ψ(K(2k, k)) =
⌊
1
2
+
√
1
4
+
(
2k
k
)⌋
However, the situation is different in the case of K(2k + 1, k), the Kneser graphs that
are called odd graphs. The better lower bound we have is
Ω
(
2k/2
)
=
⌊
1
2
+
√
1
4
+
(
2k
k
)⌋
≤ ψ(K(2k + 1, k)),
due to the fact that K(2k, k) is a subgraph of the odd graph K(2k + 1, k).
68 Ars Math. Contemp. 21 (2021) #P1.05 / 57–69
ORCID iDs
Gabriela Araujo-Pardo https://orcid.org/0000-0003-3804-1855
Christian Rubio-Montiel https://orcid.org/0000-0003-1474-8362
References
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.06 / 71–88
https://doi.org/10.26493/1855-3974.2354.616
(Also available at http://amc-journal.eu)
Coarse distinguishability of graphs with
symmetric growth*
Jesús Antonio Álvarez López
Departamento e Instituto de Matemáticas, Facultade de Matemáticas,
Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
Ramón Barral Lijó †
Research Organization of Science and Technology, Ritsumeikan University,
Nojihigashi 1-1-1, Kusatsu-Shiga, 525-8577, Japan
Hiraku Nozawa ‡
Department of Mathematical Sciences, Colleges of Science and Engineering,
Ritsumeikan Univesity, Nojihigashi 1-1-1, Kusatsu, Shiga, 525-8577, Japan
Received 5 June 2020, accepted 25 March 2021, published online 19 August 2021
Abstract
Let X be a connected, locally finite graph with symmetric growth. We prove that there
is a vertex coloring ϕ : X → {0, 1} and some R ∈ N such that every automorphism f
preserving ϕ is R-close to the identity map; this can be seen as a coarse geometric version
of symmetry breaking. We also prove that the infinite motion conjecture is true for graphs
where at least one vertex stabilizer Sx satisfies the following condition: for every non-
identity automorphism f ∈ Sx, there is a sequence xn such that lim d(xn, f(xn)) = ∞.
Keywords: Graph, coloring, distinguishing, coarse, growth, symmetry.
Math. Subj. Class. (2020): 05C15, 51F30
*The authors are partially supported by the Program for the Promotion of International Research by Rit-
sumeikan University and grants FEDER/Ministerio de Ciencia, Innovación y Universidades/AEI/MTM2017-
89686-P; and Xunta de Galicia/ED431C 2019/10. We would also like to thank the anonymous referee for a
careful reading of the paper.
†Corresponding author. Part of this work was carried out during the tenure of a Canon Foundation in Europe
Research Fellowship by B.L.
‡H.N. is partly supported by JSPS KAKENHI Grant Number 17K14195 and 20K03620.
E-mail addresses: jesus.alvarez@usc.es (Jesús Antonio Álvarez López), ramonbarrallijo@gmail.com
(Ramón Barral Lijó), hnozawa@fc.ritsumei.ac.jp (Hiraku Nozawa)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
72 Ars Math. Contemp. 21 (2021) #P1.06 / 71–88
1 Introduction
A (not necessarily proper) vertex coloring ϕ of a graph is distinguishing if the only au-
tomorphism that preserves ϕ is the identity. This notion was first introduced in [4] un-
der the name asymmetric coloring, where it was proved that 2 colors suffice to produce a
distinguishing coloring of a regular tree. Later, Albertson and Collins [1] defined the dis-
tinguishing number D(X) of a graph X as the least number of colors needed to produce
a distinguishing coloring. The problem of calculating D(X) and variants thereof has ac-
cumulated an extensive literature in the last 20 years, see e.g. [2, 14, 16, 17, 18, 22] and
references therein.
One of most important open problems in graph distinguishability is the Infinite Motion
Conjecture of T. Tucker. Let us introduce some preliminaries: The motion m(f) of a graph
automorphism f is the cardinality of the set of points that are not fixed by f . For a graphX
and a subset A ⊂ Aut(X), the motion of A is m(A) = inf{m(f) | f ∈ A, f ̸= id}, and
the motion of X is m(X) = m(Aut(X)). A probabilistic argument yields the following
result for finite graphs.
Lemma 1.1 (Motion Lemma, [20]). If X is a finite graph and 2m(X) ≥ |Aut(X)|2, then
D(X) ≤ 2.
We always have |Aut(X)|2 ≤ 2ℵ0 whenX is countable, which motivates the following
generalization.
Conjecture 1.2 (Infinite motion conjecture, [22]). If X is a connected, locally finite graph
with infinite motion, then D(X) ≤ 2.
The condition of local finiteness cannot be omitted [17]; note also that every connected,
locally finite graph is countable. This conjecture has been confirmed for special classes of
graphs: F. Lehner proved it in [16] for graphs with growth at most O(2(1−ϵ)
√
n
2 ) for some
ϵ > 0,1 and later, together with M. Pilśniak and M. Stawiski [18], for graphs with degree
less or equal to five.
The aim of this paper is to introduce a large-scale-geometric version of distinguisha-
bility for colorings, and to prove the existence of such colorings in graphs whose growth
functions are large-scale symmetric. This will result in a proof of Conjecture 1.2 for graphs
with a vertex stabilizer Sx satisfying that, for every automorphism f ∈ Sx \ {id}, there is
a sequence xn such that d(xn, f(xn)) → ∞; we can regard this condition as a geometric
refinement of having infinite motion.
Let X and Y be connected graphs, endowed with their canonical N-valued2 metric. In
the context of coarse geometry (see [19] for a nice exposition on the subject), two func-
tions f, g : X → Y are R-close (R ≥ 0) if d(f(x), g(x)) ≤ R for all x ∈ X , and we say
that f and g are close if they are R-close for some R ≥ 0. Let QI(X) denote the group
of closeness classes of quasi-isometries (in the sense of Gromov) f : X → X , and let
ι : Aut(X) → QI(X) denote the natural map that sends every automorphism to its close-
ness class. We can adapt the notion of distinguishing coloring to this setting as follows:
Definition 1.3. A coloring ϕ : X → N is coarsely distinguishing if every f ∈ Aut(X,ϕ)
is close to the identity; that is, ι(Aut(X,ϕ)) = {[idX ]}.
1The notation f = O(g) is used if there are C,N such that f(x) ≤ Cg(x) for all x > N .
2We will use the convention that 0 ∈ N.
J. A. Álvarez López et al.: Coarse distinguishability of graphs with symmetric growth 73
This new definition begs the following question: which connected, locally finite graphs
admit a coarsely distinguishing coloring by two colors? In Section 5.1 we present a sim-
ple example of a graph that does not admit such a coloring. The first main result of this
paper shows that graphs with symmetric growth admit coarsely distinguishing colorings
by two colors; this condition is satisfied by vertex-transitive graphs and, more generally,
coarsely quasi-symmetric graphs [3, Corollary 4.17]. The intuitive ideas behind these no-
tions are as follows: A connected, locally finite graph has the same growth type at all
vertices (see Section 2). If all of those growth types can be compared using the same con-
stants, then the graph is said to have symmetric growth (see Definition 2.3). Similarly,
given any pair of vertices, there is a quasi-isometry mapping one of them to the other one.
If all of those quasi-isometries can be obtained with the same distortion bounds, then the
graph is called coarsely quasi-symmetric [3, Definition 3.16]. This can be thought of as the
coarse-geometric analogue of being vertex-transitive.
Theorem 1.4. Let X be a connected, locally finite graph of symmetric growth. Then there
are R ∈ N and ϕ : X → {0, 1} such that every f ∈ Aut(X,ϕ) satisfies d(x, f(x)) ≤ R
for all x ∈ X .
Note that we obtain a uniform closeness parameter R for all f ∈ Aut(X,ϕ); fur-
thermore, we make no assumption on the motion of the graph. A slight modification of the
proof of Theorem 1.4 proves the infinite motion conjecture for graphsX containing a vertex
x ∈ X such that the restriction ι : Sx → QI(X) is injective. Let us rephrase this condition
in a language closer to the statement of Conjecture 1.2. Let X be a connected graph and
let f ∈ Aut(X). The geometric motion of f is then gm(f) = sup{d(x, f(x)) | x ∈ X};
for a subset A ⊂ Aut(X), the geometric motion of A is gm(A) = sup{gm(f) | f ∈
A, f ̸= id}. The definition of the “closeness” relation for functions yields that the restric-
tion ι : A → QI(X) is injective if and only if gm(A) = ∞. The second main result of the
paper therefore reads as follows.
Theorem 1.5. Let X be a connected, locally finite graph with symmetric growth. If
m(X) = ∞ and there exists x ∈ X such that gm(Sx) = ∞, then D(X) ≤ 2.
In Sections 5.3 and 5.4 we present two families of graphs satisfying the hypothesis of
Theorem 1.5: the Diestel-Leader graphs DL(p, q), p, q ≥ 2, and graphs with bounded
cycle length. The origin of Diestel-Leader graphs goes back to the following question,
posed in [21, 23] by W. Woess:
Question 1.6. Is there a locally finite vertex-transitive graph that is not quasi-isometric to
the Cayley graph of some finitely generated group?
R. Diestel and I. Leader introduced in [10] the graph DL(2, 3) and conjectured that it
satisfies the conditions of Question 1.6. A. Eskin, D. Fisher, and K. Whyte proved in [11,
12, 13] that in fact all graphs DL(p, q) with p ̸= q answer Question 1.6 positively. On the
other hand, graphs with bounded cycle length are hyperbolic (in the sense of Gromov) and
contain as examples free products of finite graphs.
A preliminary version of this paper stated that the authors did not know of any proof
in the literature for the existence of distinguishing colorings by 2 colors for these families
of graphs. An anonymous referee has pointed to us that, in the case of Diestel-Leader
graphs, this actually follows from the fact that they satisfy the Distinct Spheres Condition
74 Ars Math. Contemp. 21 (2021) #P1.06 / 71–88
(DSC) [15, Theorem 4]. A connected graph X satisfies the DSC if there is a vertex v ∈ X
such that, for all distinct u,w ∈ X ,
d(v, u) = d(v, w) =⇒ S(u, n) ̸= S(w, n) for infinitely many n. (1.1)
Since both symmetric growth and the DSC prove the existence of distinguishing colorings
by 2 colors for the same family of graphs, it is natural to ask if there is any relation between
these two notions; in Section 5 we present simple examples showing that all four possible
Boolean combinations of these two conditions can be realized. This shows to some extent
that our results and those in [15] are independent.
We can sketch the idea behind the proofs of Theorems 1.4 and 1.5 as follows: Choose
a suitable R > 0 and a subset Y ⊂ X such that d(x, Y ) ≤ R for all x ∈ X . Suppose that
there is a partial coloring ψ by two colors such that, if ϕ : X → {0, 1} is an extension of ψ
and f is an automorphism of X preserving ϕ, then f(Y ) = Y . Thus we can regard every
extension ϕ of ψ as a coloring ϕ̄ : Y → N by more than two colors. The hypothesis of sym-
metric growth ensures that, for R large enough, we have sufficiently many local extensions
of ψ around every point y ∈ Y so that, gluing them, we can find a global extension ϕwith ϕ̄
distinguishing. Theorems 1.4 and 1.5 then follow from a simple geometrical argument. In
general, we cannot find a partial coloring ψ as above, but the same idea works with minor
modifications; this technique is similar to that used in [2].
The outline of the paper is as follows: In the next section we introduce some prelim-
inaries to be used in the proof of the main theorems, which comprises Sections 3 and 4.
Finally, Section 5 contains several examples illustrating some of the concepts that appear
in the paper.
2 Preliminaries
In what follows we only consider undirected, simple graphs, so there are no loops and no
multiple edges. We identify a graph with its vertex set, and by abuse of notation we write
X = (X,EX). The degree of a vertex x ∈ X , deg x, is the number of edges incident to
x, and the degree of X is degX = sup{deg x | x ∈ X}. A graph X is locally finite if
deg x <∞ for all x ∈ X . A path γ inX of length l ∈ N is a finite sequence x0, x1, . . . , xl
of vertices such that xi−1EXxi for all i = 1, . . . , l; when the sequence of vertices is
infinite, we call γ a ray. We may also think of a path (respectively, a ray) as a function
σ : {0, . . . , n} → X (respectively, σ : N → X). A graph is connected if every two vertices
can be joined by a path. All graphs in this paper are assumed to be connected and locally
finite, hence countable. We consider every graph to be endowed with its canonical N-valued
metric, where d(x, y) is the length of the shortest path joining x and y; a length-minimizing
path is termed a geodesic path.
A partial coloring of a graph X is a map ψ : Y → N, where Y ⊂ X; if Y = X ,
we simply call ψ a coloring. We use the term (partial) 2-coloring when ψ takes values in
{0, 1}. For every graph X and coloring ϕ : X → N, let Aut(X,ϕ) denote the group of
automorphisms f of X satisfying ϕ = ϕ ◦ f . A coloring ϕ : X → N is distinguishing if
Aut(X,ϕ) = {id}.
For a graph X , x ∈ X , and r ∈ N, let
D(x, r) = { y ∈ X | d(y, x) ≤ r }, S(x, r) = { y ∈ X | d(y, x) = r }
denote the disk and the sphere of center x and radius r, respectively. We may write
DX(x, r) for D(x, r) when the ambient space X is not clear from context. A subset Y
J. A. Álvarez López et al.: Coarse distinguishability of graphs with symmetric growth 75
of X is R-separated (R > 0) if d(y, y′) ≥ R for all y, y′ ∈ Y with y ̸= y′; it is R-coarsely
dense if, for every x ∈ X , there is some y ∈ Y with d(x, y) ≤ R.
Lemma 2.1 (E.g. [2, Corollary 2.2.]). Let X be a graph and let R > 0. For every x ∈ X ,
there is a (2R+ 1)-separated, 2R-coarsely dense subset Y ⊂ X containing x.
Remark 2.2. The proof in [2, Corollary 2.2.] makes use of Zorn’s Lemma, but the result
can be proved for countable graphs without assuming the Axiom of Choice: First, note that
the proof in [2, Corollary 2.2.] does not require the Axiom of Choice for finite graphs.
Let X be a countable graph, and let An be an increasing and exhausting sequence of finite
subsets of X . Since we can use Lemma 2.1 with finite subsets, there is a sequence of
(2R + 1)-separated, 2R-coarsely dense subsets Sn ⊂ An. The space 2X is sequentially
compact with the topology of pointwise convergence3, so there is a convergent subsequence
Sni → S. It is now elementary to check that S is a (2R+1)-separated, 2R-coarsely dense
subset of X .
Let βx : N → N and σx : N → N be the functions defined by
βx(r) = |D(x, r)|, σx(r) = |S(x, r)|.
Given two non-decreasing functions f, g : N → R+, f is dominated by g if there are
integers k, l,m such that f(r) ≤ kg(lr) for all r ≥ m. Two functions have the same
growth type if they dominate one another. The growth type of βx does not depend on the
choice of point x ∈ X , so every graph has a well-defined growth type. The functions βx,
x ∈ X , however, may not dominate one another with a uniform choice of constants, which
motivates the following definition.
Definition 2.3 ([3, Definition 4.13]). A graphX has symmetric growth if there are k, l,m ∈
N such that βx(r) ≤ kβy(lr) for all r ≥ m and x, y ∈ X .
Lemma 2.4. If X has symmetric growth, then degX <∞.
Proof. Let x ∈ X , then we have deg y < βy(1) ≤ kβx(lm) <∞ for every y ∈ X .
Let X be a graph with ∆ := degX < ∞, then the following holds for all x ∈ X and
r ≥ 1 [2, Lemma 2.12]:
σx(1) ≤ ∆, (2.1)
σx(r + 1) ≤ σx(r)(∆− 1), (2.2)
σx(r + 1) ≤ ∆(∆− 1)r. (2.3)
We will later fix a graph with ∆ > 2; note that in this case ∆/(∆− 2) ≤ 3, so
βx(r) ≤ 1 + ∆
r−1∑
s=0
(∆− 1)s = 1 + ∆((∆− 1)
r − 1)
∆− 2
≤ 1 + 3(∆− 1)r − 1
= 3(∆− 1)r. (2.4)
We say that X has exponential growth if lim inf log βx(r)r > 0 for some, and hence all
x ∈ X , else it has subexponential growth. The following lemmas have elementary proofs.
3It is well-known that, for a countable product of compact subsets of the real line, the Tychonoff theorem can
be proved without using the Axiom of Choice.
76 Ars Math. Contemp. 21 (2021) #P1.06 / 71–88
Lemma 2.5. LetX be a graph with symmetric exponential growth. Then there are k, l,m ∈
N such that er ≤ kβx(lr) for all x ∈ X and r ≥ m.
Lemma 2.6. If X has symmetric subexponential growth, then, for every a, b > 0, there is
some m ∈ N such that βx(r) ≤ aebr for all x ∈ X and r ≥ m.
3 Construction of the coloring
LetR be a large enough odd number, to be determined later. Let Y be a (2R+1)-separated,
2R-coarsely dense subset of X; we define a graph structure EY on Y as follows:
yEY y
′ if and only if 0 < d(y, y′) ≤ 4R+ 1. (3.1)
Lemma 3.1. The graph (Y,EY ) is connected with degY y ≤ |DX(y, 4R+ 1)| − 1 for all
y ∈ Y .
Proof. The inequality follows trivially from (3.1), so let us prove that Y is connected. Let
y, y′ ∈ Y , and let (y, x1, . . . , xn−1, y′) be a path in X . Since Y is 2R-coarsely dense, for
every i = 1, . . . , n there is some yi ∈ Y with dX(xi, yi) ≤ 2R. The triangle inequality
and (3.1) then yield that (y, y1, . . . , yn−1, y′) is a path on (Y,EY ).
Recall that R is a large enough odd number, so assume R ≥ 5. Let
A = { 2n | 2 ≤ n ≤ R− 1
2
}, B = { 2n+ 1 | 1 ≤ n ≤ R− 1
2
}, (3.2)
and, for r ≤ R, let
D(Y, r) =
⋃
y∈Y
D(y, r), S(Y, r) = D(Y, r) \D(Y, r − 1) =
⋃
y∈Y
S(y, r),
where the last equality holds because Y is (2R + 1)-separated. Let us define a partial
coloring
ψ : X \
⋃
r∈B
S(Y, r) → {0, 1}
as follows (Cf. [9, Lemma 3.2], see Figure 1 for an illustration):
ψ(x) =

0, x ∈
⋃
r=0,1 S(Y, r),
1, x ∈ S(Y, 2),
1, x ∈
⋃
r∈A S(Y, r),
1, x /∈ D(Y,R).
(3.3)
Note that the vertices that are not colored by this formula are precisely those in S(y, r) for
r ∈ B.
Lemma 3.2 (Cf. [9, Lemma 3.2.]). Let ϕ : X → {0, 1} be an extension of ψ, and let
f ∈ Aut(X,ϕ). For each y ∈ Y , there is some ȳ ∈ Y such that d(ȳ, f(y)) ≤ 1 and
d(z, ȳ) = d(z, f(y)) for all z ∈ X \ {ȳ, f(y)}.
J. A. Álvarez López et al.: Coarse distinguishability of graphs with symmetric growth 77
Figure 1: An illustration of the coloring ψ, where y1, y2 ∈ Y , black represents the color 0,
and white represents 1. The grey vertices are those where ψ is not defined.
Proof. Let
Y ′ = { z ∈ X | ϕ(z′) = 0 for all z′ ∈ D(z, 1) },
then (3.3) yields Y ′ ⊂ D(Y, 1), and clearly f(Y ′) = Y ′ for all f ∈ Aut(X,ϕ). For
y ∈ Y , let ȳ be the unique vertex in Y which is adjacent to f(y). We have ϕ(z) = 0
for every vertex z ∈ D(f(y), 1) and D(f(y), 1) ⊂ D(ȳ, 2), so D(f(y), 1) ⊂ D(ȳ, 1)
by (3.3). Since D(ȳ, 1) ⊂ D(f(y), 2), we also get D(ȳ, 1) ⊂ D(f(y), 1), and the result
follows.
Corollary 3.3. If X has infinite motion, then f(Y ) = Y .
Proof. Let f ∈ Aut(X,ϕ) and suppose f(y) ̸= ȳ. By the previous lemma we have
D(f(y), 1) = D(ȳ, 1), so there is a non-trivial automorphism exchanging f(y) and ȳ and
leaving all other vertices in X fixed. This contradicts the assumption that X has infinite
motion.
Remark 3.4. Note that there might be automorphisms f ∈ Aut(X,ϕ) with f(Y ) ̸= Y
when m(X) < ∞. The graph in Figure 1 provides such an example: the map f that
interchanges y1 and z and leaves the rest of vertices fixed is an automorphism preserving
ψ, but f(Y ) ̸= Y .
Since domψ = X \
⋃
r∈B S(Y, r), an extension of ψ to X is the same thing as a
coloring of
⋃
r∈B S(Y, r); for such an extension ϕ, let ϕ̄ denote the induced coloring Y →∏
B N defined by
ϕ̄(y) = (ϕ̄r(y))r∈B , where ϕ̄r(y) = |S(y, r) ∩ ϕ−1(1)|. (3.4)
Lemma 3.5. If ξ := (ξr)r∈B : Y →
∏
B N is such that ξr(y) ≤ σy(r) for every y ∈ Y ,
then there is at least one extension ϕ satisfying ϕ̄ = ξ.
78 Ars Math. Contemp. 21 (2021) #P1.06 / 71–88
Proof. Since Y is (2R + 1)-separated, the spheres S(y, r), y ∈ Y , r ∈ B, are pairwise
disjoint. Thus we can define ϕ independently over each sphere S(y, r) by coloring ξr(y)
vertices with the color 1 and the rest with the color 0.
Lemma 3.6. For each extension ϕ : X → {0, 1} of ψ and every automorphism f ∈
Aut(X,ϕ), there is a unique automorphism f̄ ∈ Aut(Y, ϕ̄) such that d(f̄(y), f(y)) ≤ 1
for all y ∈ Y .
Proof. Let f̄ be defined by the formula f̄(y) = ȳ, where ȳ ∈ Y denotes the point given by
Lemma 3.2. This point satisfies d(f̄(y), z) = d(f(y), z) for all z ∈ X \ {f(y), f̄(y)}, so
d(y, y′) = d(f(y), f(y′)) = d(f̄(y), f̄(y′))
for every y, y′ ∈ Y , y ̸= y′. This equation and (3.1) yield that f̄ is an automorphism of Y ;
moreover,
f(S(y, r)) = S(f(y), r) = S(f̄(y), r)
for r ≥ 1 by Lemma 3.2, so f̄ preserves ξ by (3.4).
Proposition 3.7. If X has symmetric growth, then we can choose R large enough so that∏
r∈B(σx(r) + 1) > βx(4R+ 1) for all x ∈ X .
In order to keep with the flow of the argument, we defer the proof of Proposition 3.7 to
Section 4. Assume for the remainder of this section that X has symmetric growth and that
R has been chosen satisfying the statement of Proposition 3.7.
Proposition 3.8. There is a distinguishing coloring ξ := (ξr)r∈B : Y →
∏
B N such that
ξr(y) ≤ σy(r) + 1.
Proof. Choose a spanning tree T for (Y,EY ) and a root y0 ∈ Y . In order to define ξ, first
let ξ(y0) = (0, . . . , 0). Every y ∈ Y with y ̸= y0 has at most |DX(y, 4R+1)|− 1 siblings
in T by Lemma 3.1. Using Proposition 3.7, we can define ξ so that ξ(y) ̸= (0, . . . , 0) for
all y ̸= y0, and every vertex is colored differently from its siblings in T . It can be easily
checked that such a coloring is distinguishing [8, Lemma 4.1].
Proof of Theorem 1.4. Lemma 3.5 and Proposition 3.8 prove the existence of some ϕ : X →
{0, 1} extending ψ and such that ϕ̄ : Y → N is distinguishing. By Lemma 3.6, every
f ∈ Aut(X,ϕ) satisfies d(f(y), y) ≤ 1 for all y ∈ Y . Since Y is 2R-coarsely dense, the
triangle inequality yields d(x, f(x)) ≤ 4R+ 1 for all x ∈ X .
Proof of Theorem 1.5. Let X have infinite motion and pick x ∈ X so that Sx has infi-
nite geometric motion; Lemma 2.1 ensures that we can choose Y so that x ∈ Y . Us-
ing Lemma 3.5 and Proposition 3.8, we construct a coloring ϕ : X → {0, 1} extending
ψ and such that ϕ̄ is distinguishing. Since X has infinite motion, Corollary 3.3 yields
f(Y ) = Y for every f ∈ Aut(X,ψ). Moreover, Lemma 3.6 and the fact that ϕ̄ is distin-
guishing show that f |Y = idY , so Aut(X,ϕ) ⊂ Sx. Since gm(Sx) = ∞ by hypothesis,
gm(Aut(X,ϕ)) = ∞. But Y is a 2R-coarsely dense subset and is fixed pointwise by
every automorphism f , so the triangle inequality yields d(x, f(x)) ≤ 4R for all x ∈ X , a
contradiction.
J. A. Álvarez López et al.: Coarse distinguishability of graphs with symmetric growth 79
4 Growth estimates
In this section we assume that X is a graph with symmetric growth. We will derive Propo-
sition 3.7 from the following result:
Proposition 4.1. For R large enough, we have
∏R
r=3(σx(r)+1) > (∆−1)[βx(4R+1)]2
for all x ∈ X .
Proof. First, note that this result is trivial in the case where X is a graph of symmetric
subexponential growth. Indeed, since X is infinite, we have σx(r) ≥ 1 for all x ∈ X ,
r ≥ 0, so
R∏
r=3
(σx(r) + 1) ≥ 2R−2 =
1
4
eR log 2. (4.1)
Using Lemma 2.6, we have that, for R large enough,
βx(4R+ 1) ≤
1
8(∆− 1)
e[(4R+1) log 2]/10 ≤ 1
8(∆− 1)
e(R log 2)/2 (4.2)
for every x ∈ X . Combining now (4.1) and (4.2), we get
(∆− 1)[βx(4R+ 1)]2 ≤
1
8
eR log 2 ≤
R∏
r=3
(σx(r) + 1),
as desired. So, for the purposes of this proof, we will assume from now on that X is a
graph with symmetric exponential growth.
In order to obtain lower bounds for the function
∏R
r=3(σx(r) + 1), let us consider the
following optimization problem: given ∆, Q,R ∈ N with
∆ > 2, R > 3, Q > ∆2 +R− 1, (4.3)
minimize the function
f(a1, . . . , aR) =
R∏
i=3
(ai + 1) (4.4)
for a = (a1, . . . , aR) ∈ (Z+)R satisfying
a1 ≤ ∆, (C1)
ai ≤ ai−1(∆− 1), (C2)
R∑
i=1
ai = Q− 1 (C3)
for i = 1, . . . , R.
Claim 4.2. The above problem has a minimizer (a1, . . . , aR) satisfying:
(i) a1 = ∆, and a2 = ∆(∆− 1).
(ii) There is 0 ≤ I ≤ R − 2 such that the sequence a2, . . . , a2+I is increasing and
ai < ∆(∆− 1) for i > 2 + I .
80 Ars Math. Contemp. 21 (2021) #P1.06 / 71–88
(iii) For 3 ≤ i ≤ 2 + I , we have ai + 1 > (ai−1 − 1)(∆− 1).
Suppose that (a1, . . . , aR) is a minimizer that does not satisfy (i), let n ∈ {1, 2} be the
first index such that an < ∆(∆− 1)n−1, and let m ≥ 3 be such that am = max{ai | i ≥
3}. Conditions (C1) and (C2) yield
a1 + a2 ≤ ∆+∆(∆− 1) = ∆2. (4.5)
If ai = 1 for all i ≥ 3, then
R∑
i=1
ai = a1 + a2 +
R∑
i=3
ai ≤ ∆2 +R− 2 < Q− 1
by (4.3), contradicting (C3); this shows that am > 1. The sequence (a′1, . . . , a
′
R) given by
a′i =

ai + 1 for i = n,
ai − 1 for i = m,
ai otherwise.
still satifies (C1)–(C3), and clearly f(a′1, . . . , a
′
R) < f(a1, . . . , aR) since the index n does
not appear in (4.4). It follows that every minimizer has to satisfy (i).
Let us prove that we can obtain a minimizer satisfying both (i) and (ii). Let (a1, . . . , aR)
be a minimizer, and let s be a permutation of {1, . . . , R} so that s(1) = 1, s(2) = 2, and
(a′1, . . . , a
′
R) = (as(1), . . . , as(R))
satisfies (ii); it is obvious that such a permutation always exists. Since s leaves the subset
{3, . . . , R} invariant and the function f is symmetric in those indices, (a′1, . . . , a′R) is also
a minimizer if it satisfies (C1)–(C3).
Let us prove that (a′1, . . . , a
′
R) satisfies (C1)–(C3): Condition (C1) holds because s(1) =
1. In order to prove (C2), we begin by showing the following claim.
Claim 4.3. For every i ∈ {3, . . . , R} with ai > a2, there is some j ∈ {2, . . . R} such that
j ̸= i and a2 ≤ aj < ai ≤ (∆− 1)aj .
Let l be an integer to be determined later, we are going to define a sequence of indices
m1, . . . ,ml in {2, . . . , R}. Let
m1 = inf{ i ∈ {2, . . . , R} | ai ≥ aj for all 2 ≤ j ≤ R },
and assume am1 > a2, since otherwise the claim is vacuously true. Suppose now that, for
i > 1, we have defined mj for 1 ≤ j < i. If ami−1 = a2, then let l = i− 1, so that mi−1
is the last element in the sequence. If ami−1 > a2, then let
mi = inf{ i ∈ {2, . . . ,mi−1} | ai ≥ aj for all 2 ≤ j ≤ mi−1 }.
The claim is again vacuously true if l = 1, so assume l ≥ 2. It follows easily from the
definition of mi that ami−1 = ami+1 for all 1 ≤ i < l, and thus (C2) yields
ami ≤ (∆− 1)ami−1 = (∆− 1)ami+1 . (4.6)
J. A. Álvarez López et al.: Coarse distinguishability of graphs with symmetric growth 81
Observe that, for every i ∈ {3, . . . , R} such that a2 < ai, there is some j ∈ {1, . . . , l− 1}
such that amj+1 ≤ ai ≤ amj , which combined with (4.6) gives
amj+1 ≤ ai ≤ amj ≤ (∆− 1)amj+1 .
This concludes the proof of Claim 4.3.
We resume the proof of (C2), so let I be the largest non-negative integer so that
a′2, . . . a
′
2+I is increasing. Recall that a
′
2 = a2, and let 3 ≤ i ≤ 2 + I . If a′i = a′2,
then a′i−1 = a
′
2 = a
′
i, so (C2) is satisfied. If a
′
i > a
′
2, then by Claim 4.3 there is some
j ∈ {2, . . . , R} such that a2 ≤ aj < as(i) ≤ (∆ − 1)aj . Since aj > a2, we have
2 ≤ s−1(j) ≤ 2 + I by (ii). Also, the sequence a′2, . . . , a′2+I is increasing, so aj ≤ a′i−1
and therefore a′i ≤ (∆−1)a′i−1. Thus Condition (C3) is satisfied because the sum
∑R
i=1 ai
is invariant by permutations, and we have obtained a minimizer (a′1, . . . , a
′
R) that satis-
fies (i) and (ii).
Finally, suppose that (a1, . . . , aR) is a minimizer satisfying (i) and (ii), but not (iii). Let
n be an index such that 3 ≤ n ≤ R − 1 and an + 1 ≤ (an−1 − 1)(∆ − 1), then one can
easily check that the solution (a′1, . . . , a
′
R) given by
a′i =

ai − 1 for i = n− 1,
ai + 1 for i = n,
ai otherwise.
still satifies (C1)–(C3). Furthermore, an+1 ≥ an implies (an+1 + 1)(an − 1) < an+1an,
so f(a′1, . . . , a
′
R) < f(a1, . . . , aR), contradicting the assumption that (a1, . . . , aR) was a
minimizer. This completes the proof of Claim 4.2.
One can easily check that, for every graph X of bounded degree ∆, every x ∈ X , and
every R > 3, the sequence (σx(1), . . . , σx(R)) satisfies (C1)–(C3) for Q = βx(R). Then
Claim 4.2 shows that, for every x ∈ X , there is a sequence (ax,1, . . . , ax,R) satisfying
Claim 4.2(i)–(iii) for Q = βx(R) and such that
R∏
r=3
(σx(r) + 1) ≥
R∏
r=3
(ax,r + 1) (4.7)
Fix such a sequence ax,r for every point x ∈ X . Now (4.5) and Claim 4.2(ii) yield
2+I∑
r=3
ax,r =
R∑
r=1
ax,r −
R∑
r=3+I
ax,r −
2∑
r=1
ax,r
≥ βx(R)− (R− 2− I)∆(∆− 1)−∆2
≥ βx(R)−R∆(∆− 1)− (∆− 1)2. (4.8)
By (C2), we have ax,2+r ≤ ax,2(∆− 1)r for r = 1, . . . , I , so
2+I∑
r=3
ax,r ≤
I∑
r=1
ax,2(∆− 1)r = ax,2(∆− 1)
(∆− 1)I − 1
∆− 2
≤ ax,2∆(∆− 1)I
≤ ∆3(∆− 1)I . (4.9)
82 Ars Math. Contemp. 21 (2021) #P1.06 / 71–88
Since X has symmetric exponential growth, by Lemma 2.5 we have
R∆(∆− 1) + (∆− 1)2 < βx(R)/2
for R large enough and all x ∈ X , so
2+I∑
r=3
ax,r ≥ βx(R)/2 (4.10)
by (4.8), and now (4.9) and (4.10) yield
(∆− 1)I ≥ βx(R)/2∆3. (4.11)
From Claim 4.2(iii) we obtain by induction the following inequality for r = 1, . . . , I .
ax,2+r ≥ ax,2(∆− 1)r − 1− 2
r−1∑
i=1
(∆− 1)i
≥ ax,2(∆− 1)r − 1− 2(∆− 1)
(∆− 1)r−1 − 1
∆− 2
≥ (∆− 1)r(ax,2 −
2
∆− 2
)− 1.
Since ax,2 = ∆(∆− 1) > 2/(∆− 2) + 1, we have
ax,2+r ≥ (∆− 1)r.
Letting C = 1/2∆3, (4.11) yields
R∏
r=3
(ax,r + 1) ≥
2+I∏
r=3
(ax,r + 1) ≥
I∏
r=1
(∆− 1)r = ((∆− 1)I+1)I/2
≥ [Cβx(R)](log∆−1 Cβx(R))/2. (4.12)
Since X has symmetric exponential growth, by Lemma 2.5 there are k, l,m ∈ N such
that kβx(ln) ≥ en for all x ∈ X and n ≥ m. So, if R ≥ lm, then (4.12) yields
R∏
r=3
(ax,r + 1) ≥ (Ck−1e⌊R/l⌋)(⌊R/l⌋+logCk
−1)/2.
Since (Ck−1e⌊R/l⌋)(⌊R/l⌋+logCk
−1)/2 grows faster than ∆8R+7, we can assume that R is
large enough so that
R∏
r=3
(ax,r + 1) > ∆
8R+7
for all x ∈ X . Noting that (∆− 1)2 > 3, equations (2.4) and (4.7) yield
R∏
r=3
(σx(r) + 1) ≥
R∏
r=3
(ax,r + 1)∆[(∆− 1)4R+3]2 ≥ (∆− 1)[βx(4R+ 1)]2.
J. A. Álvarez López et al.: Coarse distinguishability of graphs with symmetric growth 83
Proof of Proposition 3.7. The definitions of A and B in (3.2) yield
R∏
r=3
(σx(r) + 1) =
[∏
r∈A
(σx(r) + 1)
][∏
r∈B
(σx(r) + 1)
]
. (4.13)
We have r − 1 ∈ B for every r ∈ A, so∏
r∈A
(σx(r) + 1) ≤ (∆− 1)
∏
r∈B
(σx(r) + 1) (4.14)
because σx(r) ≤ (∆ − 1)σx(r − 1) by (2.2). The combination of (4.13) and (4.14) then
yields ∏
r∈B
(σx(r) + 1) ≥
√∏R
r=3(σx(r) + 1)
∆− 1
,
and the result follows from Proposition 4.1.
5 Examples
5.1 A connected, locally finite graph with no coarsely distinguishing 2-coloring
For n ∈ Z+, let In = {v0, . . . , vn} be a graph with edges {vm, vm+1} for m = 0, . . . , n−
1, and let X = {um}∞m=1 be a graph with edges {um, um+1} for m ∈ Z+. For every
n ∈ Z+, take 2n + 1 copies of In and denote them by
Iin = { vim | i = 0, . . . , n }, i = 1, . . . , 2n + 1.
For every n and i, glue the graph Iin to X by identifying the points un and v
i
0; denote the
resulting graph by Y (see Figure 2), and let Yn be the full subgraph whose vertex set is the
image of
⋃
i I
i
n by the quotient map.
Figure 2: A graph without coarsely distinguishing 2-colorings
Let ϕ be an arbitrary 2-coloring of Y . Since we have 2n + 1 copies of In glued to un
(n ∈ Z+), by the pigeonhole principle there are at least two indices i(n) ̸= j(n) such that
the restrictions of ϕ to Ii(n)n and I
j(n)
n are equal. So there exists an isomorphism fn of Yn
that preserves ϕ and maps Ii(n)n to I
j(n)
n , and therefore d(f(v
i(n)
n ), v
i(n)
n ) = 2n. Choose
such an isomorphism fn for every n ∈ Z+, and combine them into an isomorphism f of Y
preserving ϕ. Since d(f(vi(n)n ), v
i(n)
n ) = 2n for all n ∈ Z+, the map f is not close to the
identity. Note that the vertex un has degree 4 + 2n, so deg Y = ∞ and hence Y does not
have symmetric growth.
84 Ars Math. Contemp. 21 (2021) #P1.06 / 71–88
5.2 Graphs with infinite motion but finite geometric motion
Perhaps the simplest example of a connected locally finite graph X with m(X) = ∞ and
gm(X) < ∞ is shown in Figure 3. This graph has symmetric linear growth. The only
non-trivial automorphism f is the obvious one interchanging the horizontal rays starting at
y and z, and it is easy to check that d(x, f(x)) ≤ 1 for all x ∈ X .
x
y
z
Figure 3: Example of a graph X with m(X) = ∞ and gm(X) <∞
We can modify this example to obtain graphs with infinite motion, finite geometric
motion, and faster growth. For example, let T3 be the regular tree of degree 4, and let
ϕ : T3 → {0, 1} be an distinguishing coloring. Substitute each edge in T3 by a “gadget”
depending on the colors of the incident vertices (see Figure 4). In this way we obtain a
graph Y with Aut(Y ) = {idY } and symmetric exponential growth. Moreover, we can
identify T3 with the subset Y of Y consisting of vertices of degree 4. Gluing one copy of
X to each vertex y ∈ Y by identifying it with x, we obtain a graph with infinite motion,
finite geometric motion, and exponential (but not symmetric) growth.
0 0 0 1 1 1
Figure 4: Substituting each edge in T4 by a graph
5.3 Diestel-Leader graphs
The Diestel-Leader graphs DL(p1, . . . , pn) are defined for n, p1, . . . , pn ≥ 2. For the sake
of simplicity, however, we will restrict our attention to the case n = 2; at any rate, the
following discussion can be easily adapted to include the case n > 2. In order to define
DL(p, q), let Tp and Tq be the regular trees of degree p + 1 and q + 1, respectively. For
i = p, q, choose a root oi ∈ Ti and fix an end ωi of Ti. These choices induce height or
Busemann functions hi : Ti → Z, and then
DL(p, q) := { (x, y) ∈ Tp × Tq | hp(x) + hq(y) = 0 }.
Let us write (x, y) ∈ DL(p, q) as xy for the sake of clarity, and let xEiy denote that x and
y are adjacent in Ti, then the graph structure E in DL(p, q) is defined by
xyEx′y′ if and only if xEpx′ and yEqy′.
J. A. Álvarez López et al.: Coarse distinguishability of graphs with symmetric growth 85
This yields
dDL(p,q)(xy, x
′y′) ≥ max{dTp(x, x′), dTq (y, y′)}
≥ max{| h(x)− h(x′)|, | h(y)− h(y′)|}. (5.1)
For i = p, q, let Aff(Ti) be the subgroup of automorphisms of Ti that fix ωi. For every
f ∈ Aff(Ti), the quantity h(f(x)) − h(x) is independent of x ∈ Ti, and we will denote it
by h(f). Let
Ap,q = { (f, f ′) ∈ Aff(Tp)×Aff(Tq) | hp(f) + hq(f ′) = 0 }.
Lemma 5.1 ([5, Theorem 2.7.], [6, Prop. 3.3]). If p ̸= q, then Aut(DL(p, q)) ∼= Ap,q . For
p = q, the group Aut(DL(p, p)) is generated by Ap,p and the map σ : xy 7→ yx.
Let us prove that DL(p, q) satisfies the hypothesis of Theorem 1.5.
Lemma 5.2. The group Aut(DL(p, q)) has infinite motion, and the stabilizer Sopoq has
infinite geometric motion.
Proof. Let a = (f, f ′) ∈ Ap,q . If a ̸= id, then at least one of f , f ′ is non-trivial, say f .
Therefore f is a non-trivial automorphism of a regular tree, hence m(f) = m(a) = ∞.
If moreover a ∈ Sopoq , then f(op) = op, and therefore gm(f) = ∞ when considered
as an automorphism of Tp (it is elementary to check that stabilizers in regular tres have
infinite geometric motion). Now (5.1) yields gm(a) = ∞, proving the result when p ̸= q
by Lemma 5.1.
If p = q, then every automorphism which is not in Ap,q can be written as σa, where
a = (f, f ′) ∈ Ap,p and σ is the map xy 7→ yx. Since f(op) = f ′(op) = op, we have
h(f) = h(f ′) = 0. Let xnyn be a sequence in DL(p, p) with hp(xn) = − hp(yn) = n.
Then
d(xnyn, σa(xnyn)) = d(xnyn, f
′(yn)f(xn)) ≥ | hp(xn)− hp(f ′(yn))|
= | hp(xn)− hp(yn)− hp(f)|
≥ 2n− hp(f),
so gm(a) = m(a) = ∞.
5.4 Graphs with bounded cycle length
A cycle of length n ∈ N in a graph is a path σ of length n with σ(0) = σ(n) and σ(i) ̸=
σ(j) for 0 ≤ i < j < n. A graph X has bounded cycle length if there is L ∈ N such
that every cycle in X has length ≤ L. It is not difficult to prove that all graphs of bounded
cycle length are hyperbolic in the sense of Gromov. There are in the literature several
non-equivalent definitions of the free product of graphs, see e.g. [7]; one can easily check,
however, that the following result holds for any of the definitions: The free product of a
finite family of graphs of bounded cycle length has bounded cycle length. In particular, the
free product of a finite family of finite graphs has bounded cycle length.
Lemma 5.3 (Cf. [16, Lemma 3.6]). Let X be a connected locally finite graph with infinite
motion, let x ∈ X , and let f ∈ Sx. Then there is a ray γ : N → X such that γ(0) =
f(γ(0)) and im(γ) ∩ im(f ◦ γ) = {γ(0)}.
86 Ars Math. Contemp. 21 (2021) #P1.06 / 71–88
Proof. See the proof of [16, Lemma 3.6].
Proposition 5.4. If X has infinite motion and bounded cycle length, then every vertex
stabilizer has infinite geometric motion.
Proof. Let x ∈ X and let f ∈ Sx. By Lemma 5.3, there is a ray γ such that, if we let
γ′ = f(γ), then γ(0) = γ′(0) and im(γ)∩im(γ′) = {γ(0)}. For n ∈ Z+, choose geodesic
paths σn from γ(n) to γ′(n). Let mn be the largest integer such that σn(mn) ∈ im γ, and
let m′n be the least integer such that σn(m
′
n) ∈ im γ′; clearly mn,m′n ≤ d(γ(n), γ′(n)).
The triangle Zn with sides
(γ(0), . . . , γ(i) = σ(mn)),
(σ(mn), σ(mn + 1), . . . , σ(m
′
n)), and
(γ′(j) = σ(m′n), γ
′(j − 1), . . . , γ′(0))
determines a cycle of length ≥ 2n − 2d(γ(n), γ′(n)). Now the assumption that X has
bounded cycle length yields lim d(γ(n), γ′(n)) = d(γ(n), f(γ(n)) = ∞, and the result
follows.
5.5 Symmetric growth and the distinct spheres condition
In this section we show, using examples and a short argument, that all four possible Boolean
combinations of the conditions “having symmetric growth” and “satisfying the DSC” can
be realized in very simple graphs. Recall thatX satisfies the DSC if there is a vertex v ∈ X
such that, for all distinct u,w ∈ X ,
d(v, u) = d(v, w) =⇒ S(u, n) ̸= S(w, n) for infinitely many n. (5.2)
Figure 5: We substitute a vertex x by two copies x1, x2 with the same sphere of radius one
We will begin by showing how to modify a graph X to obtain a similar graph X ′ that
does not satisfy the DSC. Let X be any connected graph, and take two different points
x, y ∈ X . Using the substitution shown in Figure 5 on x and y, we can obtain a graph X ′
that has two pairs of vertices xi, and yi (i = 1, 2), instead of x and y, and so that, for any
points u, v ∈ X with u, v ̸= x, y and i ∈ {1, 2},
dX′(xi, u) = dX(x, u), dX′(yi, u) = dX(y, u), dX′(u, v) = dX(u, v), (5.3)
where by abuse of notation we are identifying the points of X \ {x, y} with those of
X ′ \ {x1, x2, y1, y2}. It follows immediately from (5.3) that X ′ shares the same coarse-
geometric properties of X; in particular, X ′ has symmetric growth if and only if X does.
J. A. Álvarez López et al.: Coarse distinguishability of graphs with symmetric growth 87
Let us show that X ′ never satisfies the DSC: Let v ∈ X ′ be arbitrary, then at least one
pair of the new vertices does not contain v, assume v /∈ {x1, x2}. Now (5.3) yields that
d(v, x1) = d(v, x2), but S(x1, n) = S(x2, n) for every n > 0, so X ′ does not satisfy
the DSC. This procedure can be used to obtain examples of graphs of symmetric and non-
symmetric growth that do not satisfy the DSC.
Regarding graphs with symmetric growth that satisfy the DSC, as stated in the intro-
duction, the Diestel-Leader graphs constitute a family of such examples, but even simpler
examples like the Cayley graph of the integers satisfy this conditions.
Finally, as for graphs with non-symmetric growth that satisfy the DSC, let X de-
note the (unmarked, undirected) Cayley graph of Z2 with respect to the generating set
{(0, 1), (1, 0)}, and let Y be a semi-infinite ray; that is, the vertex set of Y is {yi}∞i=0 and
there is an edge yi ∼ yi+1 for every i ≥ 0. It is elementary to check that X satisfies the
DSC. Let Z be the graph obtained by gluing Y to X by identifying y0 and (0, 0), and let us
see that Z still satisfies the DSC: Let v = (0, 0), and let u,w be distinct vertices in Z with
d(v, u) = d(v, u). If u,w ∈ X ⊂ Z (we can obviously identify X and Y with subsets of
Z), then
S(u, n) ∩X ̸= S(w, n) ∩ S for infinitely many n
because X satisfies the DSC. If u ∈ X and w = yi ∈ Y for some i > 0, then, for every
n > 0, we have
yi+n ∈ S(w, n) but yi+n /∈ S(u, n)
because d(u, Y ) > 0, so Z also satisfies the DSC. Moreover, since Y has linear growth
and X has quadratic growth, it is easy to check that Z has non-symmetric growth.
ORCID iDs
Jesús Antonio Álvarez López https://orcid.org/0000-0001-6056-2847
Ramón Barral Lijó https://orcid.org/0000-0002-5597-1184
Hiraku Nozawa https://orcid.org/0000-0002-3658-5966
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.07 / 89–103
https://doi.org/10.26493/1855-3974.2554.856
(Also available at http://amc-journal.eu)
On complete multipartite derangement graphs*
Andriaherimanana Sarobidy Razafimahatratra
Department of Mathematics and Statistics, University of Regina
Regina, Saskatchewan S4S 0A2, Canada
Received 12 February 2021, accepted 31 March 2021, published online 19 August 2021
Abstract
Given a finite transitive permutation group G ≤ Sym(Ω), with |Ω| ≥ 2, the derange-
ment graph ΓG of G is the Cayley graph Cay(G,Der(G)), where Der(G) is the set of
all derangements of G. Meagher et al. [On triangles in derangement graphs, J. Combin.
Theory Ser. A, 180:105390, 2021] recently proved that Sym(2) acting on {1, 2} is the only
transitive group whose derangement graph is bipartite and any transitive group of degree
at least three has a triangle in its derangement graph. They also showed that there exist
transitive groups whose derangement graphs are complete multipartite.
This paper gives two new families of transitive groups with complete multipartite de-
rangement graphs. In addition, we prove that if p is an odd prime and G is a transitive
group of degree 2p, then the independence number of ΓG is at most twice the size of a
point-stabilizer of G.
Keywords: Derangement graph, cocliques, Erdős-Ko-Rado theorem, Cayley graphs.
Math. Subj. Class. (2020): 05C35, 05C69, 20B05
1 Introduction
This paper is concerned with Erdős-Ko-Rado (EKR) type theorems for finite transitive
groups. The classical EKR Theorem is stated as follows.
Theorem 1.1 (Erdős-Ko-Rado [9]). Suppose that n, k ∈ N such that 2k ≤ n. If F is a
family of k-subsets of [n] := {1, 2, . . . , n} such that A ∩ B ̸= ∅ for all A,B ∈ F , then
|F| ≤
(
n−1
k−1
)
. Moreover, if 2k < n, then equality holds if and only if F consists of all the
k-subsets which contain a fixed element of [n].
*The author would like to thank Roghayeh Maleki, Karen Meagher and Shaun Fallat for proofreading and
helping improve the presentation of this paper. The author is also grateful to the two anonymous referees for their
valuable comments and suggestions.
E-mail address: sarobidy@phystech.edu (Andriaherimanana Sarobidy Razafimahatratra)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
90 Ars Math. Contemp. 21 (2021) #P1.07 / 89–103
The EKR theorem has been well studied and generalized for numerous combinatorial
objects in the past 50 years [6, 7, 10, 11, 13, 19, 20, 22, 25]. Of interest to us is the
generalization of Theorem 1.1 for the symmetric group by Deza and Frankl in [10].
Given a finite transitive permutation group G ≤ Sym(Ω), we say that the permutations
σ, π ∈ G are intersecting if ωσ = ωπ , for some ω ∈ Ω. A subset or family F of G is
intersecting if any two permutations of F are intersecting.
Theorem 1.2 (Deza-Frankl, [10]). Let Ω be a set of size n ≥ 2. If F ⊂ Sym(Ω) is an
intersecting family, then |F| ≤ (n− 1)!.
The characterization of the maximum intersecting families of Sym(Ω) was solved al-
most three decades later by Cameron and Ku [6], and independently by Larose and Mal-
venuto [15].
Theorem 1.3 ([6, 15]). Let Ω be a set of size n ≥ 2. If F ⊂ Sym(Ω) is an intersecting
family of maximum size, that is |F| = (n− 1)!, then F is a coset of a stabilizer of a point
of Sym(Ω). In particular, there exist i, j ∈ Ω such that
F = {σ ∈ Sym(Ω) | iσ = j} .
The natural question that arises is whether analogues of Theorem 1.2 and Theorem 1.3
hold for different subgroups of Sym(Ω), i.e., permutation groups of degree n. All groups
considered in this paper are finite. We are interested in the following extremal problem.
Problem 1.4. Let G ≤ Sym(Ω) be transitive.
(1) What is the largest size of an intersecting family of G?
(2) If F is an intersecting family of G of maximum size, then describe the structure of
F .
Not surprisingly, the answer to this problem depends on the structure of the subgroup
of Sym(Ω). For instance, if σ1 = (1 2)(3 4), σ2 = (3 4)(5 6) and τ = (1 3 5)(2 4 6) are
permutations of Ω = {1, 2, 3, 4, 5, 6}, then ⟨σ1, σ2, τ⟩ has its point-stabilizers of size 2 but
F = {id, (1 2)(3 4), (3 4)(5 6), (1 2)(5 6)}
is a larger intersecting family. More examples of transitive permutation groups having
larger intersecting families than point-stabilizers are given in [3, 16, 18]. Due to this, we
consider the following definitions. We say that the group G has the EKR property if any
intersecting family of G has size at most |G||Ω| and G has the strict-EKR property if it has the
EKR property and an intersecting family of size |G||Ω| is a coset of a stabilizer of a point.
A typical approach in solving EKR-type problems is reducing it into a problem on a
graph theoretical invariant. The derangement graph ΓG of G ≤ Sym(Ω) is the graph
whose vertex set is G and two permutations σ, π are adjacent if and only if they are not
intersecting; that is, ωσ ̸= ωπ , for every ω ∈ Ω. In other words, ΓG is the Cayley graph
Cay(G,Der(G)), where Der(G) is the set of all derangements of G. Then, a family F ⊂ G
is intersecting if and only if F is an independent set or a coclique of the derangement graph
ΓG. Therefore, Problem 1.4 is equivalent to finding the size of the maximum cocliques
α(ΓG) and the structures of the cocliques of size α(ΓG).
A. S. Razafimahatratra: On complete multipartite derangement graphs 91
Our long term objective is to classify the transitive permutation groups that have the
EKR property and strict-EKR property. A big step toward this classification is the result
of Meagher, Spiga and Tiep [20], which says that every finite 2-transitive group has the
EKR property. More examples of primitive groups having the EKR property are given in
[1, 2, 5, 8, 17, 19, 22].
We are motivated to find more transitive groups that do not have the EKR property.
The group ⟨σ1, σ2, τ⟩ given above is special in the sense that its derangement graph is a
complete tri-partite graph. A recent result by Meagher, Spiga and the author [18] brought
to light the existence of many transitive groups that do not have the EKR property. The
most important of these are the transitive groups whose derangement graphs are complete
multipartite graphs. If G ≤ Sym(Ω) is transitive and ΓG is a complete multipartite graph,
then it is easy to see that the part H of ΓG, which contains the identity element id, consists
of the elements with at least one fixed point. Moreover, every element of G \ H is a
derangement. Therefore, H is a maximum coclique of ΓG and H is the union of all the
point-stabilizers of G. Thus, G does not have the EKR property unless H = {id}. An
important result on the structure of derangement graphs of transitive groups is given in the
next theorem.
Theorem 1.5 ([18]). Let G ≤ Sym(Ω) be transitive. Then, ΓG is bipartite if and only if
|Ω| ≤ 2. Further, if |Ω| ≥ 3, then ΓG contains a triangle.
Our motivation for this work is to find more transitive groups having complete multi-
partite derangement graphs. In this paper, we give two infinite families of transitive groups
whose derangement graphs are complete multipartite. Our main results are stated as fol-
lows.
Theorem 1.6. Let p be a prime and let q = pk, for some k ≥ 1. Then, there exists a
transitive group Gq , of degree q(q+1), such that ΓGq is a complete (q+1)-partite graph.
The following was conjectured in [18] on the existence of complete multipartite de-
rangement graphs.
Conjecture 1.7. If n is even but not a power of 2, then there is a transitive group G of
degree n such that ΓG is a complete multipartite graph with n/2 parts.
A transitive group of degree n = 2ℓ, where ℓ is odd, with a complete ℓ-partite de-
rangement graph was given in [18, Lemma 5.3]. We generalize this construction to find
another family of transitive groups with complete multipartite derangement graphs. This
result further reinforces Conjecture 1.7.
Theorem 1.8. For any odd ℓ, there exists a transitive permutation group of degree 4ℓ whose
derangement graph is a complete 2ℓ-partite graph.
The intersection density ρ(G) of a permutation group G was introduced in [16, 18] as
the ratio between the size of the largest intersecting families of G and the size of the largest
point-stabilizer of G. That is, if G ≤ Sym(Ω), then
ρ(G) :=
max{|F| : F ⊂ G is intersecting}
maxω∈Ω |Gω|
. (1.1)
For any n ∈ N, we define In := {ρ(G) | G is transitive of degree n} and I(n) := max In.
The following was conjectured in [18].
92 Ars Math. Contemp. 21 (2021) #P1.07 / 89–103
Conjecture 1.9 ([18]). (1) If n = pq where p and q are distinct odd primes, then I(n) =
1.
(2) If n = 2p where p is prime, then I(n) = 2.
In this paper, we also prove that Conjecture 1.9(2) holds.
Theorem 1.10. If p is an odd prime, then I(2p) = 2.
This paper is organized as follows. In Section 2, we give some background results on
complete multipartite derangement graphs and some properties of the intersection density
of transitive groups. In Section 3, Section 4, and Section 5, we give the proof of Theo-
rem 1.6, Theorem 1.8, and Theorem 1.10, respectively.
2 Background
Throughout this section, we let G ≤ Sym(Ω) be a transitive group and |Ω| = n.
2.1 Bound on maximum cocliques
We recall that the problem of finding the size of the maximum intersecting families of G is
equivalent to finding the size of the maximum cocliques of ΓG. We give a classical upper
bound on the size of the largest cocliques in vertex-transitive graphs (i.e., graphs whose
automorphism groups act transitively on their vertex sets). As the derangement graph of an
arbitrary finite permutation group is a Cayley graph, the right-regular representation of G
acts regularly on V (ΓG). In other words, ΓG is vertex transitive.
Lemma 2.1 ([13]). If X = (V,E) is a vertex-transitive graph, then α(X) ≤ |V (X)|ω(X) .
Moreover, equality holds if and only if a maximum coclique of X intersects each maximum
clique at exactly one vertex.
Lemma 2.1 can be used to prove the EKR property of groups. For instance, one can
prove that Sym(n), for n ≥ 3, has the EKR property [6, 10, 12] by showing first that
ω(ΓSym(n)) = n (a clique of ΓSym(n) is induced by a Latin square of size n) and applying
Lemma 2.1. A subset S ⊂ G with |S| = n that forms a clique in ΓG is called a sharply
1-transitive set. It is well-known that a transitive group need not have a sharply 1-transitive
set. Therefore, Lemma 2.1 does not hold with equality for the derangement graphs of many
transitive groups.
2.2 Intersection density
By (1.1), the intersection density of the transitive group G is the rational number
ρ(G) :=
max |{F ⊆ G | F is intersecting}|
|Gω|
,
where ω ∈ Ω.
The major result in [18] (see also Theorem 1.5) asserts that the intersection density of
the transitive group G cannot be equal to n2 . This is equivalent to saying that the derange-
ment graph of transitive groups cannot be bipartite if n ≥ 3 (see [18]). It is also proved in
[18] that for any transitive group K of degree n, ρ(K) is in the interval
[
1, n3
]
. We note
A. S. Razafimahatratra: On complete multipartite derangement graphs 93
that ρ(K) = 1 if and only if K has the EKR property. Moreover, the upper bound n3 is
sharp since there are transitive groups whose derangement graphs are complete tri-partite
graphs [18, Theorem 5.1]. It is conjectured that the only transitive groups that attain the
upper bound are those with complete tri-partite derangements graphs.
The study of the intersection density (see [16, 18]) of a transitive group was mainly
motivated by studying how far from having the EKR property a transitive group can be.
The intersection density, therefore, is a measure of the EKR property for transitive groups.
We make the following conjecture based on computer search using Sagemath [23].
Conjecture 2.2. For any n ≥ 3, almost all elements of the set In are integers. That is,
|{ρ(G) | G is transitive of degree n} ∩ N|
|In|
−−−−→
n→∞
1.
Note that the intersection density of a transitive group can be non-integer. For example,
the transitive groups of degree n and number k in the TransitiveGroup function of
Sagemath, with (n, k) ∈ {(12, 122), (12, 93)}, have non-integer intersection densities.
TransitiveGroup(12,122) and TransitiveGroup(12,93) have intersection density
equal to 32 and
17
16 , respectively.
Proposition 2.3. If the derangement ΓG has a clique of size k, then ρ(G) ≤ nk .
Proof. The proof follows by applying Lemma 2.1.
2.3 Complete multipartite derangement graphs
The transitive groups with complete multipartite derangement graphs are the most natural
examples of groups that do not have the EKR property. In this subsection, we give some
properties of transitive groups whose derangement graphs are complete multipartite.
The following lemma is a straightforward observation on the intersecting subgroups of
G.
Lemma 2.4 ([16, 18]). Let G ≤ Sym(Ω) and let H ≤ G. Then, H is intersecting if and
only if H does not have any derangement.
The next lemma illustrates that transitive groups with complete multipartite derange-
ment graphs have a very distinct algebraic structure.
Lemma 2.5 ([18]). If G ≤ Sym(Ω) is transitive such that ΓG is a complete multipartite
graph, then G is imprimitive.
A transitive group whose derangement graph is a complete multipartite graph is uniquely
determined by a specific subgroup of G. We define F(G) to be the subgroup of G generated
by all the permutations of G with at least one fixed point. That is,
F(G) :=
〈 ⋃
ω∈Ω
Gω
〉
.
Proposition 2.6. The subgroup F(G) is a normal subgroup of G.
Proof. The proof follows from the fact that F(G) is generated by all point-stabilizers.
94 Ars Math. Contemp. 21 (2021) #P1.07 / 89–103
Note that Lemma 2.5 follows from the normality of F(G) as its orbits form a non-trivial
system of imprimitivity of G acting on Ω.
A characterization of transitive groups with complete multipartite derangement graphs
is given in the next lemma.
Lemma 2.7 ([18]). Let G ≤ Sym(Ω) be transitive. The graph ΓG is complete multipartite
if and only if F(G) is intersecting. Moreover, if ΓG is a complete multipartite graph, then
the number of parts of ΓG is [G : F(G)].
Suppose that ΓG is a complete multipartite graph. When the subgroup F(G) is the
trivial group {id}, then ΓG is the complete multipartite graph that has |G| parts of size 1.
In other words, ΓG is the complete graph K|G|. When F(G) = G, then F(G) cannot be
intersecting since by Lemma 2.4, this would contradict the celebrated theorem of Jordan
[14, 21] on the existence of derangements in finite transitive groups. Hence, we say that
ΓG is a non-trivial complete multipartite graph if 1 < |F(G)| < |G|. In this paper, we
are only interested in transitive groups with non-trivial complete multiplartite derangement
graphs.
Next, we study the structure of F(G). If F(G) is intersecting, then by Lemma 2.4,
F(G) is derangement-free. Thus,
F(G) =
⋃
ω∈Ω
Gω.
Recall that if K ≤ Sym(Ω) and ω ∈ Ω, then the orbit of K containing ω is denoted by
ωK . Moreover, if S ⊂ Ω, then the setwise stabilizer of S in K is denoted by K{S}.
The following lemma is a standard result in the theory of permutation groups.
Lemma 2.8. Let G ≤ Sym(Ω) and ω ∈ Ω. If H is a non-trivial subgroup of G containing
Gω , then G{ωH} = H .
Corollary 2.9. Let G ≤ Sym(Ω) be transitive and let K be the subgroup of G fixing the
system of imprimitivity
{
ωF(G) | ω ∈ Ω
}
. Then K = F(G).
Proof. Since F(G) is generated by the point-stabilizers, by the previous lemma, we have
K =
⋂
ω∈Ω
G{ωF(G)} =
⋂
ω∈Ω
F(G) = F(G).
Remark 2.10. A representation of the derangement graph of the transitive group G as a
complete mutlipartite graph is unique. This is due to the fact that the part of ΓG, which
contains the identity element, must be equal to F (G).
3 Proof of Theorem 1.6
In this section, we describe the action of AGL(2, q) on the lines and give some basic results.
Then, we prove Theorem 1.6.
A. S. Razafimahatratra: On complete multipartite derangement graphs 95
3.1 An action of AGL(2, q) on the lines
Let q = pk be a prime power, where k ≥ 1. For b ∈ F2q and A ∈ GL(2, q), we let
(b, A) : F2q → F2q be the affine transformation such that (b, A)(v) := Av + b. The affine
group AGL(2, q) is the permutation group{
(b, A) | A ∈ GL(2, q), b ∈ F2q
}
,
with the multiplication (a,A)(b, B) := (a+Ab,AB).
Hence, AGL(2, q) acts naturally on the vectors of F2q . This action induces an action of
AGL(2, q) on the set Ω of all lines of F2q (i.e., the collection of all sets of the form Lu,v :=
{u+ tv | t ∈ Fq}, where u, v ∈ F2q and v ̸= 0). Recall that PG(1,Fq) := PG(1, q) is the
set of all 1-dimensional subspaces of the Fq-vector space F2q . The elements of PG(1, q)
are exactly the lines containing 0 ∈ F2q . By a simple counting argument, each vector of
F2q \ {0} determines a line, and each line passing through 0 has q − 1 points (excluding 0).
So there are q
2−1
q−1 = q + 1 subspaces in PG(1, q). For any line ℓ ∈ PG(1, q), we define
Ωℓ :=
{
ℓ+ b | b ∈ F2q
}
. The set Ωℓ consists of F2q-shifts of the 1-dimensional subspace ℓ,
thus its elements are affine lines of F2q that are parallel to ℓ. Therefore, Ω :=
⋃
ℓ∈PG(1,q) Ωℓ
is exactly the set of lines of F2q . Note that we can also view Ω as the lines of the incidence
structure
(
F2q, L,∼
)
, where L = {Lu,v | u, v ∈ F2q, v ̸= 0} and v ∼ ℓ, for v ∈ F2q and
ℓ ∈ L, if and only if v ∈ ℓ. This incidence structure is the affine plane AG(2, q).
As GL(2, q) acts transitively on PG(1, q), it is easy to see that AGL(2, q) acts transi-
tively on Ω. Since the elements of GL(2, q) ≤ AGL(2, q) leave PG(1, q) invariant, for any
ℓ ∈ PG(1, q), the set Ωℓ is either invariant by the action of an element of AGL(2, q) or is
mapped to some other Ωℓ′ , where ℓ′ ∈ PG(1, q) \ {ℓ}. That is, Ωℓ is a block for the action
of AGL(2, q) on Ω. Therefore, AGL(2, q) acts imprimitively on Ω.
As the elements of AGL(2, q) are affine transformations, the pair of parallel lines
(l, l′) ∈ Ωℓ×Ωℓ can be mapped by AGL(2, q) to any other pair of parallel lines. However,
if (l, l′) ∈ Ωℓ × Ωℓ′ , for distinct ℓ, ℓ′ ∈ PG(1, q), then no element of AGL(2, q) can map
(ℓ, ℓ′) to a pair of parallel lines. In addition, one can prove that any pair of non-parallel
lines can be mapped to any other pair of non-parallel lines. In other words, AGL(2, q)
acting on Ω2 has exactly 3 orbits. We formulate this result as the following lemma.
Lemma 3.1. The group AGL(2, q) acting on Ω is a rank 3 imprimitive group.
3.2 Action of Singer subgroups of GL(2, q) as subgroups of AGL(2, q)
We recall that for n ≥ 1, GL(n, q) admits elements of order qn − 1. These elements are
called Singer cycles, and a subgroup of order qn − 1 generated by a Singer cycle is called
a Singer subgroup. We recall the following observation about Singer cycles.
Proposition 3.2. If A is a Singer cycle of GL(2, q), then the subgroup ⟨A⟩ acts regularly
on F2q \ {0}.
For any matrix C ∈ GL(2, q), we define
Gq(C) :=
{
(b, B) | B ∈ ⟨C⟩, b ∈ F2q
}
.
Now, let A be an arbitrary Singer cycle of GL(2, q). By Proposition 3.2, it is easy to
see that the action of
Hq := {(0, B) ∈ AGL(2, q) | B ∈ ⟨A⟩}
96 Ars Math. Contemp. 21 (2021) #P1.07 / 89–103
on PG(1, q) is transitive. The latter implies that the action of the subgroup Gq(A) on Ω is
transitive. To see this, let ℓ = ℓ0 + b and ℓ′ = ℓ′0 + b
′ be two lines in Ω such that ℓ0 and ℓ′0
are 1-dimensional subspaces and b, b′ ∈ F2q . By transitivity of Hq on PG(1, q), there exists
(0, B) ∈ Hq such that (0, B)(ℓ0) = ℓ′0. Hence,
(b′ −Bb,B)(ℓ) = (b′ −Bb,B)(ℓ0 + b) = Bℓ0 +Bb+ b′ −Bb = ℓ′0 + b′ = ℓ′.
Thus, Gq(A) is transitive. It is straightforward to verify that for any ℓ ∈ PG(1, q), Ωℓ is a
block of Gq(A). Therefore, we have the following.
Proposition 3.3. The group Gq(A) acts imprimitively on Ω and Ωℓ is a block of Gq(A),
for any ℓ ∈ PG(1, q).
3.3 Kernel of the action of Gq(A)
In this subsection, we study the kernel of the action of Gq(A) on the system of imprimitivity
{Ωℓ | ℓ ∈ PG(1, q)}.
To avoid any confusion, we use the notation StabGq(A)(l) in the remainder of Sec-
tion 3 to denote the point-stabilizer of ℓ ∈ Ω in Gq(A), instead of the standard notation
used in the theory of permutation groups. Similarly, for any S ⊂ Ω, we use the notation
Stab(Gq(A), S) for the setwise stabilizer of S in Gq(A).
By Lemma 3.1, the action of AGL(2, q) on Ω has a unique system of imprimitivity,
namely the set {Ωℓ | ℓ ∈ PG(1, q)}. Define
Mq :=
⋂
ℓ∈PG(1,q)
Stab(Gq(A),Ωℓ).
We prove the following lemma.
Lemma 3.4. The affine transformation (b, B) ∈ Mq if and only if there exists k ∈ F∗q such
that B = kI , where I is the 2× 2 identity matrix.
Proof. It is easy to see that if B = kI , for some k ∈ F∗q , then (0, B) fixes every element of
PG(1, q). Therefore, (b, B) leaves Ωℓ invariant for any ℓ ∈ PG(1, q).
If (b, B) ∈ Mq , then (0, B) fixes every element of PG(1, q). In particular, there exists
k1, k2 ∈ F∗q such that
(0, B)
[
1
0
]
= B
[
1
0
]
= k1
[
1
0
]
, and (0, B)
[
0
1
]
= B
[
0
1
]
= k2
[
0
1
]
.
Therefore, the matrix B = diag(k1, k2). The 1-dimensional subspace generated by the
vector u =
[
1
1
]
forces k1 = k2, since Bu = ku for some k ∈ F∗q . Hence B = kI .
We present an immediate corollary of this.
Corollary 3.5. The subgroup Mq of Gq(A) is intersecting.
Proof. It suffices to prove that any element of Mq has a fixed point. Let (b, kI2) ∈ Mq . If
k = 1, then it is obvious that (b, I) fixes every line in the block Ωℓ, where ℓ ∈ PG(1, q)
such that b ∈ ℓ.
A. S. Razafimahatratra: On complete multipartite derangement graphs 97
If k ̸= 1, then we prove that there exist β ∈ F2q such that for any ℓ ∈ PG(1, q), (b, kI)
fixes the line ℓ+ β. If (b, kI) fixes this line, then we must have
(b, kI)(ℓ+ β) = kℓ+ kβ + b
= ℓ+ kβ + b = ℓ+ β.
In other words, we should find β such that (1 − k)β − b ∈ ℓ, for any ℓ ∈ PG(1, q). For
β = (1− k)−1b, we have (1− k)β − b = 0 ∈ ℓ. Moreover, the solution β = (1− k)−1b
does not depend on ℓ since every element of PG(1, q) contains 0.
We conclude that when k = 1, then (b, kI) fixes every line of the block Ωℓ, with b ∈ ℓ
and if k ̸= 1, then (b, kI) fixes every line of the form ℓ + (1 − k)−1b ∈ Ω, for any
ℓ ∈ PG(1, q).
We prove the following lemma about the relation between the kernel of the action
of Gq(A) on {Ωℓ | ℓ ∈ PG(1, q)} and the subgroup F(Gq(A)) generated by the non-
derangements of Gq(A).
Lemma 3.6. The subgroup F(Gq(A)) is equal to Mq .
Proof. Let (b, kI) ∈ Mq . In the proof of Corollary 3.5, we showed that a transformation
of (b, kI) either fixes every element of Ωℓ, for some ℓ ∈ PG(1, q), or it fixes exactly one
line in each Ωℓ. Therefore, Mq ≤ F(Gq(A)).
Next, we will prove that the point-stabilizer StabGq(A)(ℓ) of ℓ in Gq(A) is a subgroup
of Mq , for any ℓ ∈ Ω. First, let ℓ ∈ PG(1, q) be the line that contains the point
[
1
0
]
∈ F2q .
Observe that for b ∈ F2q , ℓ+ b = ℓ if and only if b ∈ ℓ. Therefore, the affine transformation
(b, kI) ∈ StabGq(A)(ℓ), for any b ∈ ℓ and k ∈ F∗q . There are q(q−1) affine transformations
of this form in StabGq(A)(ℓ). Arguing by the size of the stabilizer of ℓ in Gq(A), we have
|StabGq(A)(ℓ)| =
q2(q2 − 1)
q(q + 1)
= q(q − 1).
We conclude that the point-stabilizer of ℓ in Gq(A) is
StabGq(A)(ℓ) =
{
(b, B) ∈ Gq | B = kI, b ∈ ℓ, k ∈ F∗q
}
≤ Mq.
Since Mq ◁ Gq(A) and Gq(A) is transitive, we have StabGq(A)(ℓ) ≤ Mq for any ℓ ∈ Ω.
Therefore, F(Gq(A)) ≤ Mq . This completes the proof.
3.4 Proof of Theorem 1.6
We prove that the derangement graph ΓGq(A) of Gq(A) is a complete (q+1)-partite graph.
By Corollary 3.5, Mq is intersecting, and by Lemma 3.6, we have Mq = F(Gq(A)).
Therefore, ΓGq(A) is a complete k-partite graph, where k = [Gq(A) : Mq] =
q2(q2−1)
q2(q−1) =
q + 1.
Note that Lemma 3.6 is crucial to our proof. Indeed, the subgroup generated by the
permutations with fixed points in AGL(2, q) acting on Ω, i.e., F(AGL(2, q)), is the whole
of AGL(2, q); whereas the stabilizer of its unique system of imprimitivity is the proper
subgroup Mq .
98 Ars Math. Contemp. 21 (2021) #P1.07 / 89–103
4 Proof of Theorem 1.8
We will construct a transitive permutation group G of degree n = 4ℓ acting on [n], where
ℓ is an odd natural number. The derangement ΓG of this group G will be a complete
multipartite graph with n2 parts. The group G that we will construct is isomorphic to
(C2 × C2 × . . .× C2︸ ︷︷ ︸
ℓ−1
)⋊Dℓ,
where Dℓ is the dihedral group of order 2ℓ.
4.1 Kernel of the action
We would like to construct G so that it will have a system of imprimitivity
B = {{i, i+ 1} | for i ∈ [n] ∩ (2Z+ 1)} .
For any i, j ∈ (4Z+ 1) ∩ [n], define σi := (i i + 1)(i + 2 i + 3) and πj := σjσ4ℓ−3.
Let S = {πj | j ∈ (4Z+ 1) ∩ [n]}. Notice that |S| = ℓ, however, π4ℓ−3 = id ∈ S. We
consider the permutation group H = ⟨S⟩. It is easy to see that
H ∼= C2 × C2 × . . .× C2︸ ︷︷ ︸
ℓ−1
.
Moreover, for any fixed k ∈ [n]∩(4Z+1), any subset of the form {σiσk | i ∈ [n] ∩ (4Z+ 1)}
generates H .
A permutation of H either fixes, pointwise, an element of B or interchanges the pair of
elements in a set of B. Therefore, H leaves B invariant. Any g ∈ H can be written in the
form
g =
∏
j∈[n]∩(4Z+1)
π
kj
j , (4.1)
for some kj ∈ {0, 1}. Since π4ℓ−3 = id, there are at most ℓ − 1 (which is even) permu-
tations of the form πj in the expression of g in (4.1). If the number of non-identity terms
in (4.1) is even, then g fixes the points 4ℓ − 3, 4ℓ − 2, 4ℓ − 1, and 4ℓ. If the number of
non-identity terms in (4.1) is odd, then there exists j ∈ [n]∩(4Z+1), j ̸= 4ℓ−3, such that
kj = 0 (because ℓ− 1 is even). Therefore, g fixes the elements j, j + 1, j + 2, and j + 4.
We conclude that
H is an intersecting subgroup of degree n = 4ℓ. (4.2)
The group G will be defined so that H = F(G).
4.2 Action of a dihedral group on H
First, we give a permutation c, which is a product of four disjoint ℓ-cycles. Then, we
construct a transposition τ so that τcτ−1 = c−1. In other words, ⟨c, τ⟩ = Dℓ. This
subgroup will act on H so that ⟨H, c, τ⟩ is transitive.
For any i ∈ Z, define Ai := (i i+4 . . . i+4k . . . i+4(ℓ−1)) to be the permutation
of order ℓ, whose entries in the cycle notation are those of an arithmetic progression of step
4, and with initial value i. Let
c := A1A2A3A4.
A. S. Razafimahatratra: On complete multipartite derangement graphs 99
We note that A1, A2, A3, and A4 are pairwise disjoint ℓ-cycles. Consider the permutation
τ := (1 3)(2 4)
∏
i∈{1 2 ... ℓ−1}
(1 + 4i 3 + 4(ℓ− i)) (2 + 4i 4 + 4(ℓ− i)) .
The transpositions in the expression of τ are also pairwise disjoint. Moreover, τ is a de-
rangement of Sym(n). The following conditions are satisfied by τ
τA1τ
−1 = A−13 ,
τA2τ
−1 = A−14 ,
τA3τ
−1 = A−11 ,
τA4τ
−1 = A−12 .
(4.3)
From (4.3), we deduce that τcτ−1 = c−1. We conclude that ⟨τ, c⟩ ∼= Dℓ.
Next, we see how the subgroup ⟨c, τ⟩ acts on H . For i ∈ [n]∩(4Z+1) with i ̸= 4ℓ−3,
we have
νi := cπic
−1 = cσiσ4ℓ−3c
−1 = σi+4σ1.
Since {νi | i ∈ [n] ∩ (4Z + 1)} also generates H , we conclude that cHc−1 = H . In
addition, for any i ∈ [n] ∩ (4Z+ 1), we have
µi := τπiτ
−1 = τσiσ4ℓ−3τ
−1 = στ(i+2)σ5.
Since {µi | i ∈ [n] ∩ (4Z+ 1)} also generates H , we have τHτ−1 = H .
We conclude that G := H⟨τ, c⟩ is a permutation group of degree 4ℓ. In addition, it is
easy to see that H∩⟨τ, c⟩ = {id}, so we have G = H⋊⟨τ, c⟩. Furthermore, G is transitive
because
(1) the orbits of H⟨c⟩ are {1+4i | i ∈ {0, 1, 2, . . . , ℓ−1}}∪{2+4i | i ∈ {0, 1, 2, . . . , ℓ−
1}} and {3 + 4i | i ∈ {0, 1, 2, . . . , ℓ− 1}} ∪ {4 + 4i | i ∈ {0, 1, 2, . . . , ℓ− 1}}, and
(2) the orbits of ⟨τ⟩ are the sets of the form {1+4i, 3+4(ℓ− i)}, {2+4i, 4+4(ℓ− i)}
where i ∈ {0, 1, . . . , ℓ− 1} , {2, 4}, and {1, 3}.
4.3 Derangement graph of G
The derangement graph of G is a complete multipartite graph with 2ℓ parts. To prove this,
we need to show that H is intersecting and F(G) = H . We only need to prove the latter
since H is an intersecting subgroup (see (4.2)).
On one hand, as the elements of S all have fixed points, it is easy to see that ⟨S⟩ =
H ≤ F(G). On the other hand, the subgroup K = ⟨π5, π9, . . . , π4i+1, . . . , π4ℓ−7⟩ ≤ H
fixes 1; that is, K ≤ G1. Since |K| = 2ℓ−2 and |G1| = |G|4ℓ = 2
ℓ−2, we conclude that
G1 = K ≤ H . As G is transitive, the point-stabilizers of G are conjugate. Moreover, since
H ◁ G (because G = H ⋊ ⟨τ, c⟩) and G1 ≤ H , we can conclude that Gi ≤ H , for any
i ∈ [n]. Therefore, F(G) ≤ H .
In conclusion, we know that F(G) = H is intersecting. This is equivalent to ΓG being
a complete multipartite graph, with [G : H] = 2ℓ parts.
100 Ars Math. Contemp. 21 (2021) #P1.07 / 89–103
5 Proof of Theorem 1.10
We will prove that every transitive group of degree 2p, for any odd prime p, has intersec-
tion density at most 2 (Theorem 1.10) by showing that there is a clique of size p in the
derangement graph of G. In this case, we have ρ(G) ≤ |Ω|p = 2. Therefore, 1 ≤ ρ(G) ≤ 2
for any transitive group G of degree 2p. It is proved in [18, Lemma 5.3] that for any odd ℓ,
there is a transitive group of degree 2ℓ, whose intersection density is 2. Therefore, we will
have I(2p) = 2, for any odd prime p.
As p | |G|, by Cauchy’s theorem, there exists σ ∈ G whose order is p. Therefore, σ
is either a p-cycle or the product of two disjoint p-cycles. If the latter holds, then σ is a
derangement of G and ⟨σ⟩ is then a clique of size p in ΓG. So, we may suppose that σ is a
p-cycle.
5.1 Imprimitive case
Since G ≤ Sym(Ω) is imprimitive of degree 2p, a non-trivial block of imprimitivity of G
has size 2 or p. Assume that
σ = (x1 x2 x3 . . . xp).
As p is an odd prime and σ ∈ G, it is easy to see that G cannot have a system of imprimi-
tivity consisting of sets of size 2. We suppose that G has a set of imprimitivity Q consisting
of two subsets of size p of Ω. It is easy to see that σ cannot interchange the two blocks
of Q since the support of σ only has p elements. Thus, σ is in the setwise stabilizer of Q.
Suppose that Q = {B,B′}, where B = {x1, x2, . . . , xp} and B′ = {y1, y2, . . . , yp}. As
Gy1 and Gx1 are conjugate, there exists an element σ
′ ∈ Gx1 , which is a p-cycle. As σ′ is
a p-cycle, it must fix B pointwise and act as a p-cycle on B′.
We conclude that the permutation σσ′ ∈ G is a product of two disjoint p-cycles. The
subgroup ⟨σσ′⟩ is a clique of size p of ΓG.
5.2 Primitive case
Suppose that G ≤ Sym(Ω) is primitive of degree 2p. We derive the result of Theorem 1.10
from the following lemma.
Lemma 5.1 ([24]). Suppose that p is an odd prime. A primitive group of degree 2p is either
2-transitive or every non-identity element of a Sylow p-subgroup of G is a product of two
disjoint p-cycles.
By Lemma 5.1, we conclude that G is 2-transitive or G contains a derangement of order
p. Hence, either G has the EKR property [20] (in which case ρ(G) = 1) or ρ(G) ≤ 2.
This completes the proof of Theorem 1.10.
6 Further work
We finish this paper by posing some open questions. In Section 5, we proved that for any
odd prime p, a transitive group G of degree 2p has intersection density 1 ≤ ρ(G) ≤ 2. It
follows from the classification of finite simple groups that the only simply primitive groups
(i.e., primitive groups that are not 2-transitive) of degree 2p are Alt(5) and Sym(5), both
of degree 10. Using Sagemath [23], the largest intersecting family of Alt(5) is of size
A. S. Razafimahatratra: On complete multipartite derangement graphs 101
12, whereas its stabilizer of a point has size 6. The largest intersecting family of Sym(5)
is 12, which equals the size of its point-stabilizers. We conclude that the group Alt(5) of
degree 10 has the largest intersection density among all primitive groups of degree 2p, for
every odd prime p.
For the imprimitive case, there are infinitely many examples of transitive groups with
intersection density equal to 2. In [18, Lemma 5.3], the authors gave a family of transitive
groups of degree 2ℓ, for any odd ℓ, whose derangement graphs are ℓ-partite and whose in-
tersection densities are equal to 2. Based on a non-exhaustive search on the small transitive
groups of degree 2p (where p is an odd prime) available on Sagemath, we are inclined to
believe that the intersection density of a transitive group of degree 2p, where p is an odd
prime, is an integer. We ask the following question.
Question 6.1. Does there exist an odd prime p and a transitive group G of degree 2p such
that ρ(G) is not an integer?
In Theorem 1.8, we proved that there exists a family of transitive groups of degree
4ℓ, for any odd ℓ, with complete 2ℓ-partite derangement graphs. This further confirms
the validity of [18, Conjecture 6.6 (1)] (see also Conjecture 1.7) about the existence of
transitive groups of any degree n which is even but not a power of 2, with a complete
n
2 -partite derangement graph.
Problem 6.2. For any odd ℓ and an integer i ≥ 3, find a transitive group of degree 2iℓ
whose derangement graph is a complete 2i−1ℓ-partite graph.
In Section 3, we gave an example of a transitive group of degree q(q + 1), where q is
a prime power, whose intersection density is equal q. A non-exhaustive search on small
transitive groups of degree q(q + 1), which are available on Sagemath, shows that the
largest intersection density for these groups is q. We ask the following question.
Question 6.3. Does there exist a transitive group G of degree q(q+ 1), where q is a prime
power, such that ρ(G) > q?
Our motivation to work on the EKR property for the transitive group in Section 3 comes
from studying the EKR property for AGL(2, q) acting on the lines of AG(2, q) (see Sec-
tion 3), where q is a prime power. Observe that if H and G are transitive permutation
groups acting on Ω and H ≤ G, then ΓH is an induced subgraph of ΓG. Using the No-
Homomorphism Lemma [4], one can prove that α(ΓG) ≤ α(ΓH) |G||H| . We deduce from this
inequality that if H has the EKR property, then so does G. Moreover, ρ(G) ≤ ρ(H).
Recall that the subgroup Gq(A) defined in Section 3 is a subgroup of AGL(2, q) acting
on the lines of AG(2, q). Using the result from the previous paragraph, we know that
ρ(AGL(2, q)) ≤ ρ(Gq(A)) =
q2(q − 1)
q(q − 1)
= q,
where q is a prime power and A is a Singer cycle of GL(2, q). However, we believe that
this bound is not sharp. Indeed, from the observation of the behavior of the intersection
density of AGL(2, q) (q ∈ {3, 4, 5, 7, 8}) acting on the lines of AG(2, q), we make the
following conjecture.
Conjecture 6.4. For any ε > 0, there exists a prime power q0, such that for any prime
power q ≥ q0, 0 ≤ ρ(AGL(2, q)) − 1 ≤ ε. In particular, ρ(AGL(2, q)) ∈ Q \ N, for any
prime power q.
102 Ars Math. Contemp. 21 (2021) #P1.07 / 89–103
ORCID iDs
Andriaherimanana Sarobidy Razafimahatratra https://orcid.org/0000-0002-3542-9160
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.08 / 105–124
https://doi.org/10.26493/1855-3974.2450.1dc
(Also available at http://amc-journal.eu)
On 2-closures of rank 3 groups
Saveliy V. Skresanov *
Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia
Novosibirsk State University, 1 Pirogova street, 630090 Novosibirsk, Russia
Received 29 September 2020, accepted 2 April 2021, published online 23 August 2021
Abstract
A permutation group G on Ω is called a rank 3 group if it has precisely three orbits in
its induced action on Ω × Ω. The largest permutation group on Ω having the same orbits
as G on Ω × Ω is called the 2-closure of G. A description of 2-closures of rank 3 groups
is given. As a special case, it is proved that the 2-closure of a primitive one-dimensional
affine rank 3 group of sufficiently large degree is also affine and one-dimensional.
Keywords: 2-closure, rank 3 group, rank 3 graph, permutation group.
Math. Subj. Class. (2020): 20B25, 20B15, 05E18
1 Introduction
Let G be a permutation group on a finite set Ω. Recall that the rank of G is the number of
orbits in the induced action ofG on Ω×Ω; these orbits are called 2-orbits. If a rank 3 group
has even order, then its non-diagonal 2-orbit induces a strongly regular graph on Ω, which
is called a rank 3 graph. It is readily seen that a rank 3 group acts on the corresponding
rank 3 graph as an automorphism group. Notice that an arc-transitive strongly regular graph
need not be a rank 3 graph, since its automorphism group might be intransitive on non-arcs.
Related to this is the notion of a 2-closure of a permutation group [31]. The 2-closure
G(2) of a permutation group G is the largest permutation group having the same 2-orbits
as G. Clearly G ≤ G(2), the 2-closure of G(2) is equal to G(2), and G(2) has the same rank
as G. Note also that given a rank 3 graph Γ corresponding to the rank 3 group G, we have
Aut(Γ) = G(2).
The rank 3 groups are completely classified. A primitive rank 3 group either stabilizes
a nontrivial product decomposition, or is almost simple or is an affine group. The rank 3
groups stabilizing a nontrivial product decomposition are given by the classification of the
*The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613
with the Ministry of Science and Higher Education of the Russian Federation.
E-mail address: skresan@math.nsc.ru (Saveliy V. Skresanov)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
106 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
2-transitive almost simple groups, see Theorem 4.1 (ii)(a) and Section 5 in [6]. Almost
simple rank 3 groups were determined in [1] when the socle is an alternating group, in [16]
when the socle is a classical group and in [21] when the socle is an exceptional or sporadic
group. The classification of affine rank 3 groups was completed in [19].
In order to describe the 2-closures of rank 3 groups (or, equivalently, the automorphism
groups of rank 3 graphs), it is essential to know which rank 3 groups give rise to isomorphic
graphs. Despite all the rank 3 groups being known, it is not a trivial task (considerable
progress in describing rank 3 graphs was made in [4]). In the present paper we give a
detailed description of the 2-closures of rank 3 groups.
Theorem 1.1. Let G be a rank 3 permutation group on a set Ω. Then either G is one of the
groups from Table 8, or exactly one of the following is true.
(i) G is imprimitive, i.e. it preserves a nontrivial decomposition Ω = ∆ × X . Then
G(2) = Sym(∆) ≀ Sym(X).
(ii) G is primitive and preserves a product decomposition Ω = ∆2. Then G(2) =
Sym(∆) ↑ Sym(2).
(iii) G is primitive almost simple with socle L, i.e. L ⊴ G ≤ Aut(L). Then G(2) =
NSym(Ω)(L), and G(2) is almost simple with socle L.
(iv) G is a primitive affine group which does not stabilize a product decomposition. Then
G(2) is also an affine group. More precisely, there exist an integer a ≥ 1 and a prime
power q such that G ≤ AΓLa(q), and exactly one of the following holds (setting
F = GF(q)).
(a) G ≤ AΓL1(q). Then G(2) ≤ AΓL1(q).
(b) G ≤ AΓL2m(q) preserves the bilinear forms graph Hq(2,m), m ≥ 3. Then
G(2) = F 2m ⋊ ((GL2(q) ◦GLm(q))⋊Aut(F )).
(c) G ≤ AΓL2m(q) preserves the affine polar graph VOϵ2m(q), m ≥ 2, ϵ = ±.
Then
G(2) = F 2m ⋊ ΓOϵ2m(q).
(d) G ≤ AΓL10(q) preserves the alternating forms graph A(5, q). Then
G(2) = F 10 ⋊ ((ΓL5(q)/{±1})× (F×/(F×)2)).
(e) G ≤ AΓL16(q) preserves the affine half spin graph VD5,5(q). Then G(2) ≤
AΓL16(q) and
G(2) = F 16 ⋊ ((F× ◦ Inndiag(D5(q)))⋊Aut(F )).
(f) G ≤ AΓL4(q) preserves the Suzuki-Tits ovoid graph VSz(q), q = 22e+1,
e ≥ 1. Then
G(2) = F 4 ⋊ ((F× × Sz(q))⋊Aut(F )).
S. V. Skresanov: On 2-closures of rank 3 groups 107
Up arrow symbol in (ii) denotes the primitive wreath product (see Section 2), notation
for graphs in the affine case is explained in Section 3. Table 8 contains finitely many
permutation groups and can be found in Appendix. We note that the largest degree of a
permutation group from Table 8 is 312.
We also remark that the value of a in (iv) of Theorem 1.1 is not necessarily minimal
subject to G ≤ AΓLa(q), since it is not completely defined by the corresponding rank 3
graph and may depend on the group-theoretical structure of G. Minimal values of a can be
found in Table 1.
The proof of Theorem 1.1 can be divided into three parts. First we reduce the study to
the case when G(2) has the same socle as G, and deal with cases (i)–(iii) (Proposition 2.8).
In the affine case (iv) we apply the classification of affine rank 3 groups [19], and compare
subdegrees of groups from various classes (Lemma 3.5 and Proposition 3.6); that allows
us to deal with case (a). Finally, we invoke known results on automorphisms of some
families of strongly regular graphs to cover cases (b)–(d), while cases (e) and (f) are treated
separately.
The case (iv), (a) of Theorem 1.1 can be formulated as a standalone result that may be
of the independent interest.
Theorem 1.2. Let G be a primitive affine permutation group of rank 3 and suppose that
G ≤ AΓL1(q) for some prime power q. Then G(2) ≤ AΓL1(q), unless the degree and the
smallest subdegree of G are as in Table 7.
It is important to stress that the group G in Theorem 1.2 can have a nonsolvable 2-
closure; such an example of degree 26 has been found in [28].
The main motivation for the present study is the application of Theorem 1.1 to the
computational 2-closure problem. Namely, the problem asks if given generators of a rank 3
group one can find generators of its 2-closure in polynomial time. This task influenced the
scope of the present paper considerably, for instance, while one can determine the structure
of the normalizer in Theorem 1.1 (iii) explicitly depending on the type of the corresponding
rank 3 graph, this is not required for the computational problem as this normalizer can be
computed in polynomial time [22]. In other cases it is possible to compute automorphism
groups of associated rank 3 graphs directly (for example, of Hamming graphs), but in many
situations a more detailed study of relevant groups is required. The author plans to turn to
the computational problem in his future work.
Finally, the author would like to express his gratitude to professors M. Grechkoseeva,
I. Ponomarenko, A. Vasil’ev and to the anonymous referees for numerous helpful com-
ments and suggestions.
2 Reduction to affine case
We will prove Theorem 1.1 by dealing with rank 3 groups on a case by case basis. Recall
the following well-known general classification of rank 3 groups.
Proposition 2.1. Let G be a rank 3 group with socle L. Then G is transitive and one of the
following holds:
(i) G is imprimitive,
(ii) L is a direct product of two isomorphic simple groups, and G preserves a nontrivial
product decomposition,
108 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
(iii) L is nonabelian simple,
(iv) L is elementary abelian.
Proof. Transitivity part is clear. If G is primitive, Theorem 4.1 and Proposition 5.1 from
[6] imply that G belongs to one of the last three cases from the statement.
Suppose that G ≤ Sym(Ω). Observe that G acts imprimitively on Ω if and only if it
preserves a nontrivial decomposition Ω = ∆×X , i.e. the action domain Ω can be identified
with a nontrivial Cartesian product ∆×X , |∆| > 1, |X| > 1, where G permutes blocks of
the form ∆ × {x}, x ∈ X . Denote by Sym(∆) ≀ Sym(X) ≤ Sym(Ω) the wreath product
of Sym(∆) and Sym(X) in the imprimitive action, so G ≤ Sym(∆) ≀ Sym(X).
Proposition 2.2. Let G be an imprimitive rank 3 permutation group on Ω. Let ∆ be a
nontrivial block of imprimitivity of G, so Ω can be identified with ∆ ×X for some set X .
Then G(2) = Sym(∆) ≀ Sym(X).
Proof. Set H = Sym(∆) ≀ Sym(X). Then G ≤ H and since G and H are both groups of
rank 3, we have G(2) = H(2). By [15, Lemma 2.5] (see also [8, Proposition 3.1]), we have
(Sym(∆) ≀ Sym(X))(2) = Sym(∆)(2) ≀ Sym(X)(2) = Sym(∆) ≀ Sym(X),
so H is 2-closed. Hence G(2) = H(2) = H , as claimed.
Suppose that the action domain is a Cartesian power of some set: Ω = ∆m, m ≥ 2 and
|∆| > 1. Denote by Sym(∆) ↑ Sym(m) the wreath product of Sym(∆) and Sym(m) in
the product action, i.e. the base group acts on ∆m coordinatewise, while Sym(m) permutes
the coordinates. We say that G ≤ Sym(Ω) preserves a nontrivial product decomposition
Ω = ∆m if G ≤ Sym(∆) ↑ Sym(m).
If G preserves a nontrivial product decomposition Ω = ∆m, then G induces a per-
mutation group G0 ≤ Sym(∆). Recall that we can identify G with a subgroup of G0 ↑
Sym(m). We need the following well-known formula for the rank of a primitive wreath
product; the proof is provided for completeness (see also [18]).
Lemma 2.3. Let G be a transitive group of rank r. Then G ↑ Sym(m) has rank
(
r+m−1
m
)
.
Proof. Let G ≤ Sym(∆), and recall that Γ = G ↑ Sym(m) acts on ∆m. Choose α1 ∈ ∆
and set α1 = (α1, . . . , α1) ∈ ∆m. Let α1, . . . , αr be representatives of orbits of Gα1 on
∆. Since the point stabilizer Γα1 is equal to Gα1 ↑ Sym(m), the points (αi1 , . . . , αim),
1 ≤ i1 ≤ · · · ≤ im ≤ r, form a set of representatives of orbits of Γα1 on ∆m. The
number of indices i1, . . . , im satisfying 1 ≤ i1 ≤ · · · ≤ im ≤ r is equal to the number of
weak compositions of m into r parts, hence the claim is proved.
Observe that in the particular case when Ω = ∆2, the wreath product Sym(∆) ↑
Sym(2) has rank 3 and its 2-orbit of size |∆|2(|∆| − 1) is the edge set of the Hamming
graph H(2, |∆|).
Proposition 2.4. Let G be a primitive rank 3 permutation group on Ω, preserving a non-
trivial product decomposition Ω = ∆m, m ≥ 2. Then m = 2, a 2-orbit of G induces a
Hamming graph and G(2) = Sym(∆) ↑ Sym(2).
S. V. Skresanov: On 2-closures of rank 3 groups 109
Proof. SetH = Sym(∆) ↑ Sym(m), and recall that by Lemma 2.3,H has rank
(
2+m−1
m
)
=
m+ 1 as a permutation group. Since G ≤ H , we have m+ 1 ≤ 3. Therefore m = 2 and
H is a rank 3 group. Then G(2) = H(2) and it suffices to show that H is 2-closed.
A 2-orbit of H induces the Hamming graph H(2, q) on Ω, where q = |∆|. By [3,
Theorem 9.2.1], Aut(H(2, q)) = Sym(q) ↑ Sym(2). It readily follows that H(2) =
Aut(H(2, q)) = H , completing the proof.
In order to find 2-closures in the last two cases of Proposition 2.1, we need to show that
2-closure almost always preserves the socle of a rank 3 group.
Lemma 2.5. LetG be a primitive rank 3 group and suppose thatG andG(2) have different
socles. Then either G preserves a nontrivial product decomposition, or G is an almost
simple group with socle and degree as in Table 8.
Proof. From [25, Theorem 2] it follows that eitherG preserves a nontrivial product decom-
position, or G and G(2) are almost simple groups. By [20, Theorem 1], the latter situation
applies only to a finite number of rank 3 groups, namely, either G is one of exceptional
examples from [20, Table 1], or the socle of G is G2(q), q ≥ 3, or the socle is Ω7(q). Since
rank 3 graphs are distance-transitive, [20, Proposition 1] implies q ∈ {3, 4, 8} in the case
of G2(q), while [20, Proposition 2] yields q ∈ {2, 3} in the case of Ω7(q).
Lemma 2.6. LetG be a primitive rank 3 group with nonabelian simple socle. ThenG does
not preserve a nontrivial product decomposition.
Proof. LetG ≤ Sym(Ω) andL be the socle ofG. Suppose on the contrary thatG preserves
a nontrivial product decomposition. Since G is primitive, and L is a nonabelian simple
minimal normal subgroup of G, [26, Theorem 8.21] implies that either L is A6 and |Ω| =
36, or L = M12 and |Ω| = 144, or L = Sp4(q), q ≥ 4, q even and |Ω| = q4(q2 − 1)2.
One can easily check that neither of these situations occurs in rank 3 by inspecting the
classification of almost simple rank 3 groups. The reader is referred to [5, Table 5] for
alternating socles, [5, Table 9] for sporadic socles and [5, Tables 6 and 7] for classical
socles.
It should be noted that an almost simple group with rank larger than 3 might preserve a
nontrivial product decomposition, see [26, Section 1.3].
Proposition 2.7. Let G be a primitive rank 3 permutation group on Ω with nonabelian
simple socle L. Then either G appears in Table 8, or G(2) has socle L and G(2) =
NSym(Ω)(L).
Proof. By Lemma 2.6, G does not preserve a nontrivial product decomposition, hence by
Lemma 2.5, either G belongs to Table 8, or 2-closure G(2) has the same socle as G. Set
N = NSym(Ω)(L). Clearly G(2) ≤ N , and to establish equality it suffices to show that N
is a rank 3 group.
Suppose that this not the case and N is 2-transitive. By [6, Proposition 5.2], N has
a unique minimal normal subgroup, and since L is a minimal normal subgroup of N , the
socle of N must be equal to L. Hence N is an almost simple 2-transitive group with
socle L.
The possibilities for a socle of a 2-transitive almost simple group are all known and
moreover, such a socle is a 2-transitive group itself, unless G acts on 28 points and L =
110 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
PSL2(8) (see Theorem 5.3 (S) and the following notes in [6]). By [16, Theorem 1.2], there
is no rank 3 group of degree 28 with socle PSL2(8), hence L and thus G are 2-transitive,
which is a contradiction. Therefore N is a rank 3 group and G(2) = N .
We summarize the results of this section in the following.
Proposition 2.8. Let G be a rank 3 permutation group on Ω. Then either G appears in
Table 8, or exactly one of the following holds.
(i) G is imprimitive, i.e. it preserves a nontrivial decomposition Ω = ∆ × X . Then
G(2) = Sym(∆) ≀ Sym(X).
(ii) G is primitive and preserves a product decomposition Ω = ∆2. Then G(2) =
Sym(∆) ↑ Sym(2).
(iii) G is a primitive almost simple group with socle L, i.e. L ⊴ G ≤ Aut(L). Then
G(2) = NSym(Ω)(L), and G(2) is almost simple with socle L.
(iv) G is a primitive affine group which does not stabilize a product decomposition. Then
G(2) is also an affine group.
3 Affine case
In the previous section we reduced the task of describing the 2-closures of rank 3 groups
to the case when the group in question is affine. Recall that a primitive permutation group
G ≤ Sym(Ω) is called affine, if it has a unique minimal normal subgroup V equal to its
socle, such that V is an elementary abelian p-group for some prime p and G = V ⋊ G0
for some G0 < G. The permutation domain Ω can be identified with V in such a way that
V acts on it by translations, and G0 acts on it as a subgroup of GL(V ). Clearly G0 is the
stabilizer of the zero vector in V under such identification.
If G0 acts semilinearly on V as a GF(q)-vector space, where q is a power of p, then we
write G0 ≤ ΓLm(q), where ΓLm(q) is the full semilinear group and V ≃ GF(q)m. If the
field is clear from the context, we may use ΓL(V ) = ΓLm(q) instead. We write AΓLm(q)
for the full affine semilinear group.
Now we are ready to state the classification of affine rank 3 groups.
Theorem 3.1 ([19]). Let G be a finite primitive affine permutation group of rank 3 and
degree n = pd, with socle V ≃ GF(p)d for some prime p, and let G0 be the stabilizer of
the zero vector in V . Then G0 belongs to one of the following classes.
(A) Infinite classes. These are:
(1) G0 ≤ ΓL1(pd);
(2) G0 is imprimitive as a linear group;
(3) G0 stabilizes the decomposition of V ≃ GF(q)2m into V = V1 ⊗ V2, where
pd = q2m, dimV1 = 2 and dimV2 = m;
(4) G0 ⊵ SLm(
√
q) and pd = qm, where 2 divides dm ;
(5) G0 ⊵ SL2( 3
√
q) and pd = q2, where 3 divides d2 ;
(6) G0 ⊵ SUm(q) and pd = q2m;
S. V. Skresanov: On 2-closures of rank 3 groups 111
(7) G0 ⊵ Ω±2m(q) and p
d = q2m;
(8) G0 ⊵ SL5(q) and pd = q10;
(9) G0 ⊵B3(q) and pd = q8;
(10) G0 ⊵D5(q) and pd = q16;
(11) G0 ⊵ Sz(q) and pd = q4.
(B) ‘Extraspecial’ classes.
(C) ‘Exceptional’ classes.
Moreover, classes (B) and (C) consist of finitely many groups.
Observe that the only case when a primitive affine rank 3 group can lie in some other
class from the statement of Proposition 2.8 is when it preserves a nontrivial product de-
composition. This is precisely case (A2) of the classification, and this situation does occur.
Recall that each rank 3 group gives rise to a rank 3 graph. By [4, Table 11.4], the groups
from case (A) of Theorem 3.1 correspond to the following series of graphs:
• One-dimensional affine graphs (i.e. those arising from case (A1)). These graphs are
either Van Lint–Schrijver, Paley or Peisert graphs [23];
• Hamming graphs. These graphs correspond to linearly imprimitive groups;
• Bilinear forms graph Hq(2,m), where m ≥ 2 and q is a prime power. These graphs
correspond to groups fixing a nontrivial tensor decomposition;
• Affine polar graph VOϵ2m(q), where m ≥ 2, ϵ = ±, and q is a prime power;
• Alternating forms graph A(5, q), where q is a prime power;
• Affine half spin graph VD5,5(q), where q is a prime power;
• Suzuki-Tits ovoid graph VSz(q), where q = 22e+1, e ≥ 1.
The reader is referred to [4] for the construction and basic properties of the mentioned
graphs.
It should be noted that different cases of Theorem 3.1 may lead to isomorphic graphs.
Table 3 lists affine rank 3 groups from case (A) and indicates the corresponding rank 3
graphs. In Tables 1 and 2 we provide degrees and subdegrees of affine rank 3 groups in
case (A). These and some other relevant tables and comments on sources of data used are
collected in Appendix.
Our first goal is to show that almost all pairs of affine rank 3 graphs can be distinguished
based on their subdegrees. We start with the class (A1). The following lemma summarizes
some of the arithmetical conditions for the subdegrees of the corresponding groups.
Lemma 3.2. Let G be a primitive affine rank 3 group from class (A1), having degree
n = pd, where p is a prime. Denote by m1,m2 the subdegrees of G and suppose that
m1 < m2. Then m1 divides m2 and m2m1 divides d.
Proof. See [10, Proposition 3.3] for the first claim and [10, Theorem 3.7, (4)] for the sec-
ond.
112 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
The following lemmas apply conditions from Lemma 3.2 to groups from classes (B),
(C) and (A).
Lemma 3.3. Let G be a primitive affine rank 3 group from class (B). Suppose that G
has the same subdegrees as a group from class (A1). Then the degree and subdegrees of
G are one of the following: (72, 24, 24), (172, 96, 192), (232, 264, 264), (36, 104, 624),
(472, 1104, 1104), (34, 16, 64), (74, 480, 1920).
Proof. Let n denote the degree of G, and let m1 ≤ m2 be the subdegrees. In Table 5 all
possible subdegrees of groups from class (B) are listed. We apply Lemma 3.2. For instance,
if n = 292 then m1 = 168, m2 = 672. The quotient m2m1 = 4 does not divide 2, hence this
case cannot happen. The other cases are treated in the same manner.
Lemma 3.4. Let G be a primitive affine rank 3 group from class (C). Suppose that G has
the same subdegrees as a group from class (A1). Then the degree and subdegrees of G are
(34, 40, 40) or (892, 2640, 5280).
Proof. Follows from Lemma 3.2 and Table 6.
Lemma 3.5. LetG be a primitive affine rank 3 group from class (A) and suppose thatG has
the same subdegrees as a group from class (A1). Then either G lies in (A1) or degree and
subdegrees ofG are one of the following: (32, 4, 4), (34, 16, 64), (36, 104, 624), (24, 5, 10),
(26, 21, 42), (28, 51, 204), (210, 93, 930), (212, 315, 3780), (216, 3855, 61680), (52, 8, 16).
Proof. Suppose that G does not lie in class (A1), but shares subdegrees with some group
from (A1). Notice that in cases (A3) through (A11), exactly one of the subdegrees is
divisible by p, so the subdegrees are not equal (see Table 1). In case (A2) subdegrees are
the same if and only if pm = 3, and consequentially n = 9. This situation is the first
example in our list of parameters, hence from now on we may assume that the subdegrees
of G are not equal.
Let m1 and m2 denote the subdegrees of G, where, as shown earlier, we may assume
m1 < m2. Sincem1 andm2 are subdegrees of some group from the class (A1), Lemma 3.2
yields that m1 divides m2 and the number u = m2m1 divides d, where n = p
d.
Now, since G belongs to one of the classes (A2)–(A11), we apply the above arithmeti-
cal conditions in each case. We consider some classes together, since they give rise to
isomorphic rank 3 graphs and hence have the same formulae for subdegrees. The reader is
referred to Table 1 for the list of subdegrees in question.
(A2) Subdegrees in this case are 2(pm−1) and (pm−1)2. If 2(pm−1) > (pm−1)2,
then pm = 2 and n = 4. It can be easily seen that G is not primitive in this
case, contrary to our hypothesis. Therefore we can assume that 2(pm − 1) <
(pm − 1)2.
Then u = p
m−1
2 and since u divides d = 2m, we have p
m − 1 ≤ 4m. It
follows that (n,m1,m2) is one of (32, 4, 4), (34, 16, 64) or (52, 8, 16).
(A3)–(A5) We write r for the highest power of p dividing m2, so the second subdegree is
equal to r(rm − 1)(rm−1 − 1) for some m ≥ 2.
We have u = r r
m−1−1
r+1 and hence u ≥
rm−1−1
2 . Now r
2m = pd ≥ p r
m−1−1
2 .
Using inequalitiesm ≥ 2 and p ≥ 2, we obtain 2r8(m−1) ≥ 2rm−1 . Therefore
S. V. Skresanov: On 2-closures of rank 3 groups 113
rm−1 ≤ 44 and there are finitely many choices for r and m. Checking these
values of r and m against original divisibility conditions we yield the follow-
ing possibilities for (n,m1,m2): (26, 21, 42), (210, 93, 930), (212, 315, 3780),
(36, 104, 624).
(A6), (A7) u = qm−1 q−1qm−1±1 . Numbers q
m−1 and qm−1 ± 1 are coprime, so qm−1 ± 1
divides q − 1. That is possible only when m = 2, so we have u = q. Now
2q ≤ pq ≤ pd = q4, so q ≤ 16. Hence we have the following possibilities for
n, m1, m2 in this case: (24, 5, 10), (28, 51, 204), (216, 3855, 61680).
(A8) u = q3 − q2 q+1q2+1 . Since q
2 and q2 +1 are coprime, q2 +1 must divide q+1.
This can not happen, so this case does not occur.
(A9) u = q3 q−1q3+1 . Since q
3 + 1 does not divide q − 1, this case does not occur.
(A10) u = q5 − q3 q
2+1
q3+1 . Since q
3 + 1 does not divide q2 + 1, this case does not
occur.
(A11) u = q and pd = q4. Hence we obtain the same possible parameters as in cases
(A6), (A7).
In all cases considered we either got a contradiction or got one of the possible excep-
tions recorded in the statement. The claim is proved.
As an immediate corollary we derive that 2-closures of primitive rank 3 subgroups of
AΓL1(q) also lie in AΓL1(q) (Theorem 1.2), apart from a finite number of exceptions.
Proof of Theorem 1.2. Suppose that G and G(2) have different socles. Since G is not al-
most simple, Lemma 2.5 implies that G(2) and thus G must preserve a nontrivial product
decomposition. In that situation G has subdegrees of the form 2(
√
n − 1), (
√
n − 1)2, in
particular, G has subdegrees as a group from class (A2) and hence parameters of G are
listed in Lemma 3.5. We may assume that G does not preserve a nontrivial product decom-
position and so G and G(2) have equal socles. The claim now follows from Theorem 3.1
and Lemmas 3.3–3.5.
Note that Lemmas 3.3–3.5 list degrees and subdegrees of possible exceptions to Theo-
rem 1.2; in Table 7 of Appendix we collect these data in one place.
Now we move on to establish a partial analogue of Lemma 3.5 for classes (A2)–(A11).
First we need to recall some notions related to quadratic and bilinear forms.
Let V be a vector space over a field F . Given a symmetric bilinear form f : V × V →
F , the radical of f is rad(f) = {x ∈ V | f(x, y) = 0 for all y}; we say that f is non-
singular, if rad(f) = 0. If κ : V × V → F is a quadratic form with an associated bilinear
form f , then the radical of κ is rad(κ) = rad(f) ∩ {x ∈ V | κ(x) = 0}. We say that κ is
non-singular, if rad(κ) = 0, and we say that κ is non-degenerate, if rad(f) = 0.
If F has odd characteristic, then rad(κ) = rad(f). If F has even characteristic and
κ is non-singular, then the dimension of rad(f) is at most one, f induces a non-singular
alternating form on V/ rad(f) and, hence, the dimension of V/ rad(f) is even (see [32,
Section 3.4.7]). Therefore if the dimension of V is even, then the notions of non-singular
and non-degenerate quadratic forms coincide regardless of the characteristic.
114 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
Now we can describe the affine polar graph VOϵ2m(q), m ≥ 2. Let V be a 2m-
dimensional vector space over GF(q), and let κ : V → GF(q) be a non-singular quadratic
form of type ϵ. Vertices of the graph VOϵ2m(q) are identified with vectors from V , and two
distinct vertices u, v ∈ V are joined by an edge if κ(u − v) = 0. Up to isomorphism,
VOϵ2m(q) does not depend on the form κ.
Allowing some abuse of terminology, we say that subdegrees of a rank 3 graph are
simply subdegrees of the respective rank 3 group.
Proposition 3.6. If two affine rank 3 graphs have the same subdegrees, then they are iso-
morphic apart from the following exceptions:
• graphs arising from affine groups from Table 8,
• VSz(q) and VO−4 (q) for q = 2
2e+1, e ≥ 1,
• Paley and Peisert graphs.
In particular, graphs Hq(2, 2) and VO+4 (q) are isomorphic.
Proof. Since classes (B) and (C) of Theorem 3.1 and all exceptional parameter sets of
Lemma 3.5 are included in Table 8, we may assume that our graphs come from the case
(A) and their subdegrees are not among the exceptions from Lemma 3.5.
By Lemma 3.5, if one of the graphs in question arises from the case (A1), then the
second graph also comes from (A1). By Table 2, Van Lint-Schrijver graph has unequal
subdegrees, while Paley and Peisert graphs have equal subdegrees, hence in this case graphs
are either isomorphic or it is a Paley graph and a Peisert graph. We may now assume that
our graphs do not come from (A1).
Notice that given n = pd for p prime, the largest subdegree of graphs from classes
(A3)–(A11) is divisible by p. This is not the case in class (A2), unless n = 4 with subde-
grees 2 and 1. The corresponding rank 3 group is imprimitive in that situation, contrary to
our assumptions. Thus we may assume that none of the two graphs comes from (A2).
We compare subdegrees of classes (A3)–(A11) and collect the relevant information in
Table 4. Let us explain the procedure in the case Hq(2,m) vs. VO±2m(q) only, since other
cases are treated similarly.
Consider the graph Hq(2,m). The number of its vertices is equal to n = q2m and the
second subdegree is equal to q(qm − 1)(qm−1 − 1). Recall that n = pd for some prime p,
and the largest power of p dividing the second subdegree is q. In the case of the graph
VOϵ2m(q), we have n = q
2m and the largest power of p dividing the second subdegree
is qm−1. We obtain a system of equations
q2m = q2m, q = qm−1,
which is written in the relevant cell of Table 4. We derive that m = mm−1 , and hence
m = m = 2, q = q. Now, the second subdegree for VOϵ4(q) is q(q − 1)(q2 + (−1)ϵ).
Therefore ϵ = +, which gives us the first example of affine rank 3 graphs with same
subdegrees. Other cases are dealt with in the same way.
Now, Table 4 lists two cases when graphs from different classes have the same subde-
grees, namely, Hq(2, 2), VO+4 (q) and VSz(q), VO
−
4 (q). To finish the proof of the propo-
sition, we show that graphs Hq(2, 2) and VO+4 (q) are in fact isomorphic.
S. V. Skresanov: On 2-closures of rank 3 groups 115
Identify vertices ofHq(2, 2) with 2×2 matrices over GF(q), and recall that two vertices
are connected by an edge if the rank of their difference is 1. A nonzero 2 × 2 matrix has
rank 1 precisely when its determinant is zero:
rk
(
u1 u3
u4 u2
)
= 1 ⇐⇒ u1u2 − u3u4 = 0.
It can be easily seen that u1u2 − u3u4 is a non-degenerate quadratic form on GF(q)4, so
Hq(2, 2) is isomorphic to the affine polar graph VOϵ4(q). By comparing subdegrees we
derive that ϵ = +, and we are done.
It should be noted that VSz(q) and VO−4 (q) in fact have the same parameters as strongly
regular graphs (see [5, Table 24]). In Lemma 3.13 we will see that these graphs are actually
not isomorphic since they have non-isomorphic automorphism groups.
Paley and Peisert graphs are generally not isomorphic (see [24]), but have the same
parameters since they are strongly regular and self-complementary (i.e. isomorphic to their
complements).
Recall that in order to describe 2-closures of rank 3 groups it suffices to find full au-
tomorphism groups of corresponding rank 3 graphs. Hamming graphs were dealt with in
Proposition 2.4, and graphs arising in the case (A1) were covered in Theorem 1.2. We are
left with five cases: bilinear forms graph, affine polar graph, alternating forms graph, affine
half spin graph and the Suzuki-Tits ovoid graph. In most of these cases the full automor-
phism group was described earlier in some form, and we state relevant results here.
For two groups G1 and G2 let G1 ◦ G2 denote their central product. Note that the
central product GL(U) ◦GL(W ) has a natural action on the tensor product U ⊗W .
Proposition 3.7 ([3, Theorem 9.5.1]). Let q be a prime power and m ≥ 2. Set G =
Aut(Hq(2,m)) and F = GF(q). If m > 2, then
G = F 2m ⋊ ((GL2(q) ◦GLm(q))⋊Aut(F )).
If m = 2, then
G = F 4 ⋊ (((GL2(q) ◦GL2(q))⋊Aut(F ))⋊ C2),
where the additional automorphism of order 2 exchanges components of simple tensors.
Let V be a vector space endowed with a quadratic form κ. We say that a nonzero vector
v ∈ V is isotropic if κ(v) = 0.
Lemma 3.8 ([27]). Let V be a vector space over some (possibly finite) field F , and suppose
that dimV ≥ 3. Let κ : V → F be a non-singular quadratic form, possessing an isotropic
vector. If f is a permutation of V with the property that
κ(x− y) = 0 ⇔ κ(xf − yf ) = 0,
then f ∈ AΓL(V ) and f : x 7→ xϕ + v, v ∈ V , where ϕ ∈ ΓL(V ) is a semisimilarity of κ,
i.e. there exist λ ∈ F× and α ∈ Aut(F ) such that κ(xϕ) = λκ(x)α for all x ∈ V .
116 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
Denote by ΓOϵ2m(q) the group of all semisimilarities of a non-degenerate quadratic
form of type ϵ on the vector space of dimension 2m over the finite field of order q.
The reader is referred to [17, Sections 2.7 and 2.8] for the structure and properties of
groups ΓOϵ2m(q).
Proposition 3.9. Let q be a prime power and m ≥ 2. Set F = GF(q). Then
Aut(VOϵ2m(q)) = F
2m ⋊ ΓOϵ2m(q), ϵ = ±.
Proof. Recall that the graph VOϵ2m(q) is defined by a vector space V = F
2m over F and a
non-singular (or, equivalently, non-degenerate) quadratic form κ : V → F . Since m ≥ 2,
we have dimV ≥ 3 and κ possesses an isotropic vector. The claim now follows from
Lemma 3.8.
Proposition 3.10 ([3, Theorem 9.5.3]). Let q be a prime power and set F = GF(q). Then
Aut(A(5, q)) = F 10 ⋊ ((ΓL5(q)/{±1})× (F×/(F×)2)).
Denote byD5(q) an orthogonal group of universal type, in particular, recall the formula
|Z(D5(q))| = gcd(4, q5 − 1) (see [7, Table 5]).
Lemma 3.11. Let q be a prime power, q16 = pd, let F = GF(q), V = F 16 and set
G = Aut(VD5,5(q)). Then G = V ⋊G0, and
F× ◦D5(q) ≤ G0 = NGLd(p)(D5(q)),
where D5(q) acts on the spin module. Moreover, G0/F× is an almost simple group and
G0 ≤ ΓL16(q).
Proof. Set H = V ⋊ (F× ◦ D5(q)). By [19, Lemma 2.9], D5(q) has two orbits on the
set of lines P1(V ), so H is an affine rank 3 group of type (A10). Clearly G = H(2) so by
Lemma 2.5, G is an affine rank 3 group. By Proposition 3.6, G belongs to class (A10) and
the main result of [19] implies that G0 ≤ NGLd(p)(D5(q)). By [19, (1.4)], the generalized
Fitting subgroup of G0/F× is simple, hence this quotient group is almost simple. By
Hering’s theorem [12] (see also [19, Appendix 1]), the normalizer NGLd(p)(D5(q)) cannot
be transitive on the nonzero vectors of V , so G0 = NGLd(p)(D5(q)) as claimed.
Finally, let a be the minimal integer such that G0 ≤ ΓLa(pd/a). By Table 1, a = 16,
so the last inclusion follows.
We write Inndiag(D5(q)) for the overgroup of D5(q) in Aut(D5(q)), containing all
diagonal automorphisms.
Proposition 3.12. Let q be a prime power, and set F = GF(q). Then
Aut(VD5,5(q)) = F
16 ⋊ ((F× ◦ Inndiag(D5(q)))⋊Aut(F )).
Proof. We follow [19, Lemma 2.9]. Take K = E6(q) to be of universal type, so that
|Z(K)| = gcd(3, q − 1). The Dynkin diagram of K is:
α1◦ −α3◦ −α4◦
|
◦α2
−α5◦ −α6◦
S. V. Skresanov: On 2-closures of rank 3 groups 117
Let Σ be the set of roots and let xα(t), hα(t) be Chevalley generators of K. Write Xα =
{xα(t)|t ∈ F}. Let P be a parabolic subgroup of K corresponding to the set of roots
{α2, α3, α4, α5, α6}, and let P = UL be its Levi decomposition. Moreover, L = MH
and we may choose P such that
U = ⟨Xα | α ∈ Σ+, α involves α1⟩,
M = ⟨X±αi | 2 ≤ i ≤ 6⟩,
where M is of universal type and H = ⟨hαi(t)|t ∈ F, 1 ≤ i ≤ 6⟩ is the Cartan subgroup.
In [19, Lemma 2.9] it was shown that M ≃ D5(q), the group U is elementary abelian of
order q16 and in fact, it is a spin module for M . By [11, Theorem 2.6.5 (f)], H induces
diagonal automorphisms on M , and by [30, Section 1, B] it induces the full group of
diagonal automorphisms. Recall that for an element h of H we have xα(t)h = xα(k · t)
for some k ∈ F . In particular, diagonal automorphisms of D5(q) commute with the action
of the field F on U .
Let ϕ be a generator of the field automorphisms group of K, and note that one can
identify that group with Aut(F ); in particular, ϕ acts on F under such an identification.
By [11, Theorem 2.5.1 (c)], generators xα(t) and hα(t) are carried to xα(tϕ) and hα(tϕ)
by ϕ, so field automorphisms normalize U , M and H . Furthermore, ϕ induces the full
group of field automorphisms on M .
Set T = L⋊ ⟨ϕ⟩. We have M ⊴T and T induces all field and diagonal automorphisms
on M . Set T = T/Z(K) and M = MZ(K)/Z(K). By [11, Theorem 2.6.5 (e)], the
centralizer CAut(K)(U) is the image of Z(U) in Aut(K). Therefore T acts faithfully on
U , and since |Z(M)| is coprime to |Z(K)|, we derive that M ≃ M ≃ D5(q). Hence we
have an embedding T ≤ GLd(p), where |U | = pd, and, with some abuse of notation, T ≤
NGLd(p)(D5(q)). By Lemma 3.11, the latter normalizer is an almost simple group (modulo
scalars), and thus we have shown that it contains all field and diagonal automorphisms of
D5(q). It is left to show that it does not contain graph automorphisms.
Suppose that a graph automorphism ψ lies in G0 = NGLd(p)(D5(q)), and recall that
M ≃ D5(q). By [19, Lemma 2.9], there is an orbit ∆ of G0 on the nonzero vectors of
U , such that the point stabilizer Mδ , δ ∈ ∆ is a parabolic subgroup of type A4. Since ψ
preserves the orbit ∆ and normalizes M , it must take a point stabilizer Mδ to the point
stabilizer Mδ′ for some δ′ ∈ ∆, in particular, it takes M δ to a conjugate subgroup. That
is impossible, since by [11, Theorem 2.6.5 (c)], automorphism ψ interchanges conjugacy
classes of parabolic subgroups of type A4, so the final claim is proved.
Recall the construction of the graph VSz(q), q = 22e+1, e ≥ 1. Set F = GF(q),
V = F 4 and let σ be an automorphism of F acting as σ(x) = x2
e+1
. Define the subset O
of the projective space P1(V ) by
O = {(0, 0, 1, 0)} ∩ {(x, y, z, 1) | z = xy + x2xσ + yσ},
where vectors are written projectively. The vertex set of VSz(q) is V and two vectors are
connected by an edge, if a line connecting them has a direction in O.
Recall that Sz(q) ≤ GL4(q) is faithfully represented on P1(V ) and induces the group
of all collineations which preserve the Suzuki-Tits ovoid O (see [14, Chapter XI, Theo-
rem 3.3]). Clearly scalar transformations preserve the preimage of O in V , and it can be
118 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
easily seen that Oα = O for any α ∈ Aut(F ). Hence the following group
H = V ⋊ ((F× × Sz(q))⋊Aut(F ))
acts as a group of automorphisms of VSz(q). By [13, Lemma 16.4.6], Sz(q) acts transi-
tively on P1(V ) \O, hence H is a rank 3 group.
We will show that H is the full automorphism group of VSz(q), but first we need to
note the following basic fact.
Lemma 3.13. If q = 22e+1, e ≥ 1, then there is no subgroup of Aut(VO−4 (q)) isomorphic
to Sz(q). In particular, graphs VO−4 (q) and VSz(q) are not isomorphic.
Proof. Suppose the contrary, so that Sz(q) is a subgroup of Aut(VO−4 (q)). By Proposi-
tion 3.9, we have Aut(VO−4 (q)) ≃ V ⋊ ΓO
−
4 (q) for some elementary abelian group V .
Recall that the orthogonal group Ω−4 (q) is a normal subgroup of ΓO
−
4 (q), and the quotient
ΓO−4 (q)/Ω
−
4 (q) is solvable. Clearly V is also solvable, and since Sz(q) is simple, we ob-
tain an embedding of Sz(q) into Ω−4 (q). Yet that is impossible, as can be easily seen by
inspection of maximal subgroups of Ω−4 (q), see, for instance, [2, Table 8.17]. That is a
contradiction, so the first claim is proved.
The second claim follows from the fact that Sz(q) lies in Aut(VSz(q)).
Proposition 3.14. Let q = 22e+1, where e ≥ 1, and set F = GF(q). Then
Aut(VSz(q)) = F 4 ⋊ ((F× × Sz(q))⋊Aut(F )).
Proof. Let H = F 4 ⋊ ((F× × Sz(q))⋊ Aut(F )) be a rank 3 group acting on VSz(q) by
automorphisms. Set G = Aut(VSz(q)) and recall that G = H(2). By Lemma 2.5, G is an
affine group with the same socle asH , and by Proposition 3.6 and Table 3, it follows thatG
lies in class (A7) or (A11), or it is one of the groups from Table 8. It can be easily checked
that there is no group with degree q4 and subdegrees (q2 + 1)(q − 1), q(q2 + 1)(q − 1)
in Table 8, so the last possibility does not happen. By Lemma 3.13, G does not lie in
(A7), so it is a group from class (A11). Denote by H0 and G0 zero stabilizers in H and G
respectively. Notice that H0 ≤ G0.
By Theorem 3.1 and Table 1, we have G0 ≤ ΓL4(q) and Sz(q) ⊴ G0. By [19, (1.4)],
given Z = Z(GL4(q)) ≃ F×, the generalized Fitting subgroup ofG0/(G0∩Z) is a simple
group. Hence G0/(G0 ∩ Z) is an almost simple group with socle Sz(q).
The outer automorphisms group of Sz(q) consists of field automorphisms only (see [7,
Table 5]), so
|G0| ≤ |Z| · |Aut(Sz(q))| ≤ |F×||Sz(q)||Aut(F )|.
Since H0 ≃ (F× × Sz(q)) ⋊ Aut(F ), the order of H0 coincides with the value on right-
hand side of the inequality. Now H0 = G0 and the claim is proved.
Proof of Theorem 1.1. Let G be a rank 3 group, and suppose that G is not listed in
Table 8. By Proposition 2.8, we may assume that G is a primitive affine group which
does not stabilize a product decomposition and, moreover, G(2) is also an affine group.
By Theorem 3.1, G is either a one-dimensional affine group (class (A1)), or preserves a
bilinear forms graph Hq(2,m), m ≥ 2, an affine polar graph VOϵ2m(q), ϵ = ±, m ≥ 2,
alternating forms graph A(5, q), affine half-spin graph VD5,5(q), Suzuki-Tits ovoid graph
VSz(q) or lies in class (B) or (C).
S. V. Skresanov: On 2-closures of rank 3 groups 119
The full automorphism groups of these graphs (i.e. 2-closures of respective groups) are
described in Theorem 1.2 (one-dimensional affine groups), Proposition 3.7 (bilinear forms
graph), Proposition 3.9 (affine polar graph), Proposition 3.10 (alternating forms graph),
Proposition 3.12 (affine half-spin graph) and Proposition 3.14 (Suzuki-Tits ovoid graph).
Notice that we do not need to consider classes (B) and (C) as they are included in Table 8.
Since by Proposition 3.6, the graph Hq(2, 2) is isomorphic to VO+4 (q), we may ex-
clude it from the bilinear forms case. Now it is easy to see that cases considered in Theo-
rem 1.1 (iv) are mutually exclusive. Indeed, it suffices to prove that graphs from different
cases are not isomorphic. By Proposition 3.6, if two affine rank 3 graphs have the same
subdegrees, then they belong to the same case except for VSz(q) and VO−4 (q), q = 2
2e+1,
e ≥ 1 (note that we group one-dimensional affine graphs into one case). By Lemma 3.13,
graphs VSz(q) and VO−4 (q) are not isomorphic, which proves the claim.
Finally, inclusions of the form G ≤ AΓLa(q) can be read off Table 1. Notice that in
some cases we do not give the minimal value of a, for example, if SUm(q) ≤ G lies in class
(A6), then G ≤ AΓLm(q2), but we list the inclusion G ≤ AΓL2m(q). This completes the
proof of the theorem.
ORCID iDs
Saveliy V. Skresanov https://orcid.org/0000-0002-8397-5609
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S. V. Skresanov: On 2-closures of rank 3 groups 121
A Appendix
In this section we collect some relevant tabular data. Table 1 lists information on affine
rank 3 groups from class (A), namely, for each groupG it provides rough group-theoretical
structure (column “Type of G”), degree n and subdegrees. Column “a” gives the smallest
integer a such that the stabilizer of the zero vector G0 lies in ΓLa(pd/a). Most of the
information in Table 1 is taken from [19, Table 12], see also [5, Table 10] for the values
of a.
Table 2 lists the subdegrees of one-dimensional affine rank 3 groups. The first col-
umn specifies the type of graph associated to the group in question, next two columns
provide degree and subdegrees, and the last column lists additional constraints on param-
eters involved. By [23], these graphs turn out to be either Van Lint–Schrijver, Paley or
Peisert graphs. See [29, Section 2] for the parameters of the Van Lint–Schrijver graph;
parameters of Paley and Peisert graphs are computed using the fact that these graphs are
self-complementary.
Table 3 lists rank 3 graphs corresponding to rank 3 groups from classes (A1)–(A11),
cf. [4, Table 11.4]. Terminology and graph notation is mostly consistent with [4], see also
[5, Table 10].
Table 4 records information on when some families of affine rank 3 graphs can have
identical subdegrees, the procedure for building this table being described in Proposi-
tion 3.6. Trivial cases (when two graphs are the same) are not listed, also graphs from
cases (A1) and (A2) are omitted, since they are dealt with separately.
Tables 5 and 6 list degrees and subdegrees of affine rank 3 groups from classes (B)
and (C), without repetitions (i.e. parameter sets are listed only once, regardless of whether
several groups possess same parameters). If the smaller subdegree divides the largest, the
last column gives the respective quotient; otherwise a dash is placed. Information in Ta-
ble 5 is taken from [9, Theorem 1.1] and [19, Table 13], see also [5, Table 11]. Information
in Table 6 before the horizontal line is taken from [10, Theorem 5.3], but notice that we ex-
clude the case of 1192, since 119 is not a prime number (that error was observed by Liebeck
in [19]). Information in Table 6 after the horizontal line is taken from [19, Table 14], with
the correction for the case of Alt(9), where subdegrees should be 120, 135 instead of 105,
150, as noted in [5, Table 12].
Table 7 lists parameters of possible exceptions to Theorem 1.2. The table consists of
three subtables, corresponding to classes (A), (B) and (C) of Theorem 3.1, i.e. values for
the first subtable are taken from Lemma 3.5, for the second from Lemma 3.3, and for the
third from Lemma 3.4. Each subtable lists degrees and smallest subdegrees of possible
exceptions. Notice that parameters of one-dimensional affine rank 3 groups stabilizing a
nontrivial product decomposition are collected in the subtable for the class (A).
Table 8 lists possible exceptions to Theorem 1.1. The first column references the state-
ment where a possible exception first appears, the second column describes the structure of
the group, and the third column gives its degree, either explicitly or by referencing another
table. Notice that we include classes (B) and (C) of Theorem 3.1 in Table 8; corresponding
groups can be found in [19, Table 1 and 2].
Finally, we mention that Tables 7 and 8 list potential exceptions to Theorems 1.2 and 1.1
respectively, in particular, it might be possible to remove some parameter sets and groups
by a more careful analysis.
122 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
Table 1: Class (A) in the classification of affine rank 3 groups
Type of G n = pd a Subdegrees
(A1): G0 < ΓL1(pd) pd 1 See Table 2
(A2): G0 imprimitive p2m 2m 2(pm − 1), (pm − 1)2
(A3): tensor product q2m 2m (q + 1)(qm − 1), q(qm − 1)(qm−1 − 1)
(A4): G0 ⊵ SLm(
√
q) qm m (
√
q + 1)(
√
qm − 1), √q(√qm − 1)(√qm−1 − 1)
(A5): G0 ⊵ SL2( 3
√
q) q2 2 ( 3
√
q + 1)(q − 1), 3√q(q − 1)( 3√q2 − 1)
(A6): G0 ⊵ SUm(q) q2m m
{
(qm − 1)(qm−1 + 1), qm−1(q − 1)(qm − 1), m even
(qm + 1)(qm−1 − 1), qm−1(q − 1)(qm + 1), m odd
(A7): G0 ⊵ Ωϵ2m(q) q
2m 2m
{
(qm − 1)(qm−1 + 1), qm−1(q − 1)(qm − 1), ϵ = +
(qm + 1)(qm−1 − 1), qm−1(q − 1)(qm + 1), ϵ = −
(A8): G0 ⊵ SL5(q) q10 10 (q5 − 1)(q2 + 1), q2(q5 − 1)(q3 − 1)
(A9): G0 ⊵ B3(q) q8 8 (q4 − 1)(q3 + 1), q3(q4 − 1)(q − 1)
(A10): G0 ⊵ D5(q) q16 16 (q8 − 1)(q3 + 1), q3(q8 − 1)(q5 − 1)
(A11): G0 ⊵ Sz(q) q4 4 (q2 + 1)(q − 1), q(q2 + 1)(q − 1)
Table 2: Subdegrees of one-dimensional affine rank 3 groups
Graph Degree Subdegrees Comments
Van Lint–Schrijver q = p(e−1)t 1e (q − 1),
1
e (e − 1)(q − 1) e > 2 is prime, p is primitive (mod e)
Paley q 12 (q − 1),
1
2 (q − 1) q ≡ 1 (mod 4)
Peisert q = p2t 12 (q − 1),
1
2 (q − 1) p ≡ 3 (mod 4)
Table 3: Rank 3 graphs in class (A)
Type of G Graph Comments
(A1): G0 < ΓL1(pd) Van Lint–Schrijver, Paley or Peisert graph
(A2): G0 imprimitive Hamming graph
(A3): tensor product bilinear forms graph Hq(2,m)
(A4): G0 ⊵ SLm(
√
q) bilinear forms graph H√q(2,m) SLm(
√
q) stabilizes an m-dimensional
subspace over GF(
√
q)
(A5): G0 ⊵ SL2( 3
√
q) bilinear forms graph H 3√q(2, 3) SL2( 3
√
q) stabilizes a 2-dimensional
subspace over GF( 3
√
q)
(A6): G0 ⊵ SUm(q) affine polar graph VOϵ2m(q), ϵ = (−1)
m
(A7): G0 ⊵ Ωϵ2m(q) affine polar graph VO
ϵ
2m(q)
(A8): G0 ⊵ SL5(q) alternating forms graph A(5, q)
(A9): G0 ⊵ B3(q) affine polar graph VO
+
8 (q)
(A10): G0 ⊵ D5(q) affine half spin graph VD5,5(q)
(A11): G0 ⊵ Sz(q) Suzuki-Tits ovoid graph VSz(q)
Table 4: Intersections between classes based on subdegrees
VO±2m(q) A(5, q) VD5,5(q) VSz(q)
Hq(2,m)
q2m = q2m
q = qm−1
m = mm−1
m = m = 2, q = q
q2m = q10
q = q2
m = 104
Impossible
q2m = q16
q = q3
m = 83
Impossible
q2m = q4
q = q
m = 2
q(q2 − 1)(q − 1) = q(q2 + 1)(q − 1)
Impossible
VO±2m(q)
q2m = q10
qm−1 = q2
m = 53
Impossible
q2m = q16
qm−1 = q3
m = 85
Impossible
q2m = q4
qm−1 = q
m = 2, q = q
A(5, q)
q10 = q16
q2 = q3
Impossible
q10 = q4
q2 = q
Impossible
VD5,5(q)
q16 = q4
q3 = q
Impossible
S. V. Skresanov: On 2-closures of rank 3 groups 123
Table 5: Subdegrees of rank 3 groups in class (B)
n = pd Subdegrees m1, m2 m2m1 if it is an integer
26 27, 36 —
34 32, 48 —
72 24, 24 1
132 72, 96 —
172 96, 192 2
192 144, 216 —
232 264, 264 1
36 104, 624 6
292 168, 672 4
312 240, 720 3
472 1104, 1104 1
34 16, 64 4
54 240, 384 —
74 480, 1920 4
38 1440, 5120 —
Table 6: Subdegrees of rank 3 groups in class (C)
n = pd Subdegrees m1, m2 m2m1 if it is an integer
34 40, 40 1
312 (31− 1) · 12, (31− 1) · 20 —
412 (41− 1) · 12, (41− 1) · 30 —
74 (72 − 1) · 20, (72 − 1) · 30 —
712 (71− 1) · 12, (71− 1) · 60 5
792 (79− 1) · 20, (79− 1) · 60 3
892 (89− 1) · 30, (89− 1) · 60 2
26 18, 45 —
54 144, 480 —
28 45, 210 —
74 720, 1680 —
28 120, 135 —
28 102, 153 —
36 224, 504 —
74 240, 2160 9
35 22, 220 10
35 110, 132 —
211 276, 1771 —
211 759, 1288 —
312 65520, 465920 —
212 1575, 2520 —
56 7560, 8064 —
124 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
Table 7: Possible exceptions to Theorem 1.2
(A)
Degree 24 26 28 210 212 216 32 34 36 52
Subdegree 5 21 51 93 315 3855 4 16 104 8
(B)
Degree 34 36 72 74 172 232 472
Subdegree 16 104 24 480 96 264 1104
(C)
Degree 34 892
Subdegree 40 2640
Table 8: Possible exceptions to Theorem 1.1
Appearance Type of group Degree
Lemma 2.5 PΓL2(8) 36
M11 55
M12 66
M23 253
M24 276
Alt(9) 120
G2(q)⊴G q3(q3 − 1)/2, where q ∈ {3, 4, 8}
Ω7(q)⊴G q3(q4 − 1)/ gcd(2, q − 1), where q ∈ {2, 3}
Theorem 3.1 (B) and (C) Tables 5 and 6
Theorem 1.2 G ≤ AΓL1(q) Table 7
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.09 / 125–132
https://doi.org/10.26493/1855-3974.2488.f5c
(Also available at http://amc-journal.eu)
Nonlinear maps preserving the elementary
symmetric functions
Constantin Costara
Faculty of Mathematics and Informatics, Ovidius University of Constanţa,
Mamaia Boul. 124, Constanţa, Romania
Received 24 November 2020, accepted 5 April 2021, published online 1 September 2021
Abstract
Let Mn be the algebra of all n × n complex matrices, and for a natural number 2 ≤
k ≤ n denote by Ek (x) the kth elementary symmetric function on the eigenvalues of
x ∈ Mn. For two maps φ,ψ : Mn → Mn, one of them being surjective, we prove that if
Ek(λx+ y) = Ek(λφ (x) + ψ (y)) for each λ ∈ C and x, y ∈ Mn, then φ = ψ on Mn,
the common value being a linear map from Mn into itself. In particular, for 3 ≤ k ≤ n the
general form of φ and ψ can be computed explicitly.
Keywords: Elementary symmetric function, nonlinear, preserver.
Math. Subj. Class. (2020): 15A15, 15A86
1 Introduction and statement of the result
For a natural number n, let us denote by Mn the algebra of all n × n matrices over the
complex field C. By In ∈ Mn we shall denote the n × n identity matrix. For x ∈ Mn,
by tr (x) we shall denote its usual trace, and by det (x) its determinant. Also, by xt ∈ Mn
we shall denote the transpose of x.
For k ∈ {1, ..., n}, a k-by-k principal submatrix of x ∈ Mn is the submatrix of xwhich
lies in the rows and columns of x indexed by J ⊆ {1, ..., n} with |J | = k. Equivalently, we
eliminate from the matrix x the rows and the columns which are not in J . The determinant
of the k-by-k principal submatrix given by J ⊆ {1, ..., n} is called a k-by-k principal
minor, and shall be denoted by ∆J (x). There are
(
n
k
)
different k-by-k principal minors,
and put
Ek (x) =
∑
|J|=k
∆J (x) (x ∈ Mn, k = 1, n). (1.1)
E-mail address: cdcostara@univ-ovidius.ro (Constantin Costara)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
126 Ars Math. Contemp. 21 (2021) #P1.09 / 125–132
In particular, k = 1 in (1.1) gives E1 (x) = tr (x) and k = n gives En (x) = det (x) for
each x ∈ Mn.
For k ∈ {1, ..., n}, the kth elementary symmetric function on the complex numbers
λ1, ..., λn is
Sk (λ1, ..., λn) =
∑
1≤i1<···<ik≤n
k∏
j=1
λij . (1.2)
(We have a sum of
(
n
k
)
products in (1.2).) For x ∈ Mn, if the spectrum σ (x) of x (taking
into account multiplicities) is {α1, ..., αn}, the equality
det (λIn + x) = λ
n + λn−1E1 (x) + · · ·+ En (x) (x ∈ Mn, λ ∈ C)
implies that
Ek (x) = Sk(σ (x)) (x ∈ Mn, k = 1, n).
Thus, for each k ∈ 1, n and x ∈ Mn we have that Ek (x) is the kth elementary symmetric
function on the eigenvalues of x.
Frobenius studied in [5] linear maps on Mn which preserve the determinant. He proved
that if ϕ : Mn → Mn is a bijective linear map such that det (ϕ (x)) = det(x) for each
x ∈ Mn, there exist then invertible matrices a, b ∈ Mn satisfying det (ab) = 1 such
that either ϕ (x) = axb for each x ∈ Mn, or ϕ (x) = axtb for each x ∈ Mn. One
way to generalize this result is to relax the linearity assumption on the map ϕ. In [4],
Dolinar and Šemrl proved that we arrive at the same conclusion if we merely suppose that
ϕ : Mn → Mn is a surjective map such that
det (λx+ y) = det(λϕ(x) + ϕ(y)) (x, y ∈ Mn, λ ∈ C). (1.3)
(In this case, the linearity for the map ϕ is part of the conclusion.) Another way to gener-
alize the result of Frobenius is to consider the elementary symmetric functions Ek instead
of the determinant. The following theorem was proved by Marcus and Purves in [7] for
4 ≤ k < n and by Beasley in [1] for 3 = k < n.
Theorem 1.1 ([7, Theorem 3.1] and [1, Theorem 1.1]). Let 3 ≤ k < n and let ϕ : Mn →
Mn be a linear map such that
Ek (x) = Ek(ϕ(x)) (x ∈ Mn).
There exist then an invertible matrix a ∈ Mn and η ∈ C satisfying ηk = 1 such that either
ϕ (x) = ηaxa−1 (x ∈ Mn),
or
ϕ (x) = ηaxta−1 (x ∈ Mn).
The result of Frobenius shows that the conclusion of Theorem 1.1 does not hold if
k = n. The same happens if k = 1 (see, for example, [6, Section 2]) or k = 2 (see, for
example, [7, Section 4] or [6, Section 3].) However, if the linear map ϕ : Mn → Mn
preserves both the trace and the determinant, then there exists an invertible matrix a ∈ Mn
such that either
ϕ (x) = axa−1 (x ∈ Mn), (1.4)
C. Costara: Nonlinear maps preserving the elementary symmetric functions 127
or
ϕ (x) = axta−1 (x ∈ Mn). (1.5)
(See, for example, [8, Theorem 1] or [9, Theorem 3].) Thus, ifEk(x) = Ek(ϕ(x)) for each
k ∈ {1, n} and x ∈ Mn, then ϕ is of the form given by (1.4) or (1.5). Also, if the linear
map ϕ : Mn → Mn satisfies Ek (x) = Ek(ϕ(x)) for each k ∈ {2, n} and x ∈ Mn, there
exists then an invertible matrix a ∈ Mn such that either
ϕ (x) = ηaxa−1 (x ∈ Mn), (1.6)
or
ϕ (x) = ηaxta−1 (x ∈ Mn), (1.7)
where η = 1 if n is odd, and η = −1 or η = 1 if n is even. (See, for example, [9,
Theorem 4].)
The aim of this article is to improve the results of Theorem 1.1 in a way that is similar
to [4, Theorem 1.1]. Thus, we eliminate the linearity assumption on the map ϕ and we
impose a strengthened preservation condition which is suggested by (1.3). Incidentally, the
result holds with a preservation condition which is stated for two maps φ and ψ on Mn
instead of a single map ϕ.
Theorem 1.2. Let 2 ≤ k ≤ n and consider two maps φ,ψ : Mn → Mn, one of them
being surjective, such that
Ek (λx+ y) = Ek(λφ(x) + ψ (y)) (x, y ∈ Mn, λ ∈ C). (1.8)
Then φ = ψ on Mn, the common value being a linear map of Mn into itself.
As a corollary, we obtain the following generalization of Theorem 1.1.
Corollary 1.3. Let 3 ≤ k < n and consider two maps φ,ψ : Mn → Mn, one of them
being surjective, such that (1.8) holds. There exist then an invertible matrix a ∈ Mn and
η ∈ C satisfying ηk = 1 such that either
φ (x) = ψ (x) = ηaxa−1 (x ∈ Mn),
or
φ (x) = ψ (x) = ηaxta−1 (x ∈ Mn).
Of course, if we suppose that (1.8) holds for k ∈ {1, n}, then φ = ψ on Mn, the
common value being a linear map of the form (1.4) or (1.5), and if we suppose that (1.8)
holds for k ∈ {2, n}, then φ = ψ on Mn, the common value being a linear map of the
form (1.6) or (1.7).
Since Theorem 1.2 also holds for k = n, we obtain a different proof for the following
slight generalization of [4, Theorem 1.1]. (See also [2, Theorem 1] and [3, Theorem 1].)
Corollary 1.4. Consider two maps φ,ψ : Mn → Mn, one of them being surjective, such
that
det (λx+ y) = det(λφ(x) + ψ (y)) (x, y ∈ Mn, λ ∈ C).
There exist then invertible matrices a, b ∈ Mn satisfying det (ab) = 1 such that either
φ (x) = ψ (x) = axb (x ∈ Mn),
or
φ (x) = ψ (x) = axtb (x ∈ Mn).
128 Ars Math. Contemp. 21 (2021) #P1.09 / 125–132
2 Preliminary lemmas
Let 2 ≤ k ≤ n. For x, y ∈ Mn, consider the complex polynomial (with respect to λ) given
by
λ 7→ Ek(λx+ y).
This section is devoted to the study of these polynomials. As a general property, let us
observe that its degree is always at most k, the coefficient of λk being exactly Ek (x). In
particular, the degree of the polynomial is also bounded by the rank of the matrix x. Also,
if we fix x ∈ Mn, then the coefficient of λk−1 is linear with respect to y ∈ Mn. Indeed,
this comes from (1.1) and the fact that for J ⊆ {1, ..., n} with |J | = k we have
∆J(λx+ y) = λ
k∆J (x) + λ
k−1tr(adj(xJ)yJ) + · · ·+∆J (y) ,
where xJ (respectively, yJ ) is the principal submatrix of x (respectively, y) corresponding
to J , and for z ∈ Mk by adj (z) ∈ Mk we have denoted the (classical) adjoint of the
matrix z, obtained from its cofactors.
This coefficient will play an important role in our approach to prove Theorem 1.1, and
shall be studied thoroughly in this section.
Lemma 2.1. Let 2 ≤ k ≤ n, and let 1 ≤ i, j ≤ n, with i ̸= j. There exists then a matrix
x0 ∈ Mn such that, for each y ∈ Mn we have that the coefficient of λk for the polynomial
λ 7→ Ek(λx0 + y) is zero, and the coefficient of λk−1 for the same polynomial is yij .
Proof. Suppose, without lost of generality, that i = 1 and j = 2. Put then J0 = {1, ..., k},
and let
x0 =

0 0
−1 0 0 0
0 Ik−2 0
0 0 0n−k
 ∈ Mn.
For y ∈ Mn, let us observe that
∆J0(λx0 + y) = det

y11 y12
−λ+ y21 y22
· · · y1k
· · · y2k
...
...
yk1 yk2
λIk−2 + (yst)3≤s,t≤k

= 0 · λk + y12λk−1 + · · · .
Also, if |J | = k and J ̸= J0, then the degree of λ 7→ ∆J(λx0 + y) with respect to λ is
at most k − 2, since we have at most k − 2 appearances of λ in the principal submatrix of
λx0 + y corresponding to J . We use now (1.1) to finish the proof.
The remaining of this section is devoted to prove that the same type of statement as the
one of Lemma 2.1 also holds for i = j. This requires a little extra work.
Lemma 2.2. Let 2 ≤ k ≤ n, and let J0 ⊆ {1, ..., n} with |J0| = k − 1. There exists then
a matrix x0 ∈ Mn such that, for each y ∈ Mn we have that the coefficient of λk for the
polynomial λ 7→ Ek(λx0 + y) is zero, and the coefficient of λk−1 for the same polynomial
is
∑
j /∈J0 yjj .
C. Costara: Nonlinear maps preserving the elementary symmetric functions 129
Proof. Suppose, without lost of generality, that J0 = {1, ..., k − 1}. Let then
x0 =
[
Ik−1 0
0 0n−k+1
]
∈ Mn.
For y ∈ Mn, let us observe that for each j ∈ {1, ..., n}\J0 we have that
∆J0∪{j}(λx0 + y) = 0 · λ
k + yjjλ
k−1 + · · · .
Also, if |J | = k and |J ∩ J0| ≤ k−2, then the degree of λ 7→ ∆J(λx0+y) with respect to
λ is at most k− 2, since we have at most k− 2 appearances of λ in the principal submatrix
of λx0 + y corresponding to J . We use again (1.1) to finish the proof.
Corollary 2.3. Let 2 ≤ k ≤ n, and let 1 ≤ i, j ≤ n, with i ̸= j. There exist then
matrices x1, x2 ∈ Mn such that, for each y ∈ Mn we have that the coefficient of λk for
the polynomial λ 7→ Ek(λx1 + y) − Ek(λx2 + y) is zero, and the coefficient of λk−1 for
the same polynomial is yjj − yii.
Proof. Consider J ⊆ {1, ..., n}\{i, j} such that |J | = k− 2. We apply then Lemma 2.2 to
J ∪ {i} to find a matrix x1 ∈ Mn such that, for each y ∈ Mn we have that the coefficient
of λk for the polynomial λ 7→ Ek(λx1 + y) is zero, and the coefficient of λk−1 for the
same polynomial is
∑
t/∈(J∪{i}) ytt, and we apply the same lemma to J ∪ {j} to find a
matrix x2 ∈ Mn such that, for each y ∈ Mn we have that the coefficient of λk for the
polynomial λ 7→ Ek(λx2+ y) is zero, and the coefficient of λk−1 for the same polynomial
is
∑
t/∈(J∪{j}) ytt. To finish the proof, observe that (
∑
t/∈(J∪{i}) ytt)−(
∑
t/∈(J∪{j}) ytt) =
yjj − yii.
Lemma 2.4. Let 2 ≤ k ≤ n, and let j ∈ {1, ..., n}. There exist then q > 0, m, p ≥ 0
in Z and matrices x0, x1, .., xm+p ∈ Mn such that, for each y ∈ Mn we have that the
coefficient of λk for the polynomial
λ 7→
m∑
i=0
Ek(λxi + y)−
(
m+p∑
i=m+1
Ek(λxi + y)
)
is zero, and the coefficient of λk−1 for the same polynomial is q · yjj .
Proof. Consider J ⊆ {1, ..., n}\{j} such that |J | = k − 1. We apply Lemma 2.2 to J
to find a matrix x0 ∈ Mn such that, for each y ∈ Mn we have that the coefficient of λk
for the polynomial λ 7→ Ek(λx0 + y) is zero, and the coefficient of λk−1 for the same
polynomial is
∑
t/∈J ytt = yjj +
∑
t/∈(J∪{j}) ytt. For each t /∈ (J ∪ {j}) in {1, ..., n}, we
apply Corollary 2.3 to t ̸= j to find matrices x(1)t , x
(2)
t ∈ Mn such that, for each y ∈ Mn
we have that the coefficient of λk for the polynomial λ 7→ Ek(λx(1)t + y)−Ek(λx
(2)
t + y)
is zero, and the coefficient of λk−1 for the same polynomial is yjj−ytt. To finish the proof,
observe that ∑
t/∈J
ytt +
 ∑
t/∈(J∪{j})
(yjj − ytt)
 = q · yjj ,
for some strictly positive integer q.
130 Ars Math. Contemp. 21 (2021) #P1.09 / 125–132
3 Proof of the main result
Let 2 ≤ k ≤ n. Let us first observe that if φ,ψ : Mn → Mn satisfy (1.8), dividing
by λ ∈ C\{0} we obtain that Ek (x+ µy) = Ek(φ(x) + µψ (y)) for all x, y ∈ Mn
and µ ∈ C\{0}. By continuity, the same holds for µ = 0, too. Thus
Ek (x+ µy) = Ek(φ(x) + µψ (y)) (x, y ∈ Mn, µ ∈ C). (3.1)
That is, the same type of equalities as the ones in (1.8) hold, with the role of φ and ψ
interchanged. Thus, without lost of generality, we may suppose for the remaining of the
paper that the map φ is surjective. (If not, then ψ must be surjective, and we work with
(3.1) instead of (1.8).)
Another immediate observation is the fact that if φ and ψ satisfy (1.8), for λ = 0 in
(1.8) and µ = 0 in (3.1) we see that Ek (y) = Ek(ψ (y)) for all y ∈ Mn, respectively
Ek (x) = Ek(φ(x)) for all x ∈ Mn.
As a corollary of Lemma 2.1 and Lemma 2.4, the following result holds.
Theorem 3.1. Suppose 2 ≤ k ≤ n, and let i, j ∈ {1, ..., n}. There exist then a nonzero
scalar α, positive integers m and p and matrices x0, x1, .., xm+p ∈ Mn such that, for
each y ∈ Mn we have that the coefficient of λk for the polynomial
λ 7→
m∑
s=0
Ek(λxs + y)−
(
m+p∑
s=m+1
Ek(λxs + y)
)
is zero, and the coefficient of λk−1 for the same polynomial is α · yij .
As a direct corollary of Theorem 3.1, we obtain the following test for the equality to
0 ∈ Mn in terms of the functions Ek. (See also [7, Lemma 3.1].)
Corollary 3.2. Suppose 2 ≤ k ≤ n. Let y ∈ Mn such that
Ek(x+ y) = Ek (x) (x ∈ Mn). (3.2)
Then y = 0.
Proof. Observe that (3.2) gives
Ek(λx+ y) = λ
kEk (x) (x ∈ Mn, λ ∈ C). (3.3)
Let i, j ∈ {1, ..., n}. By Theorem 3.1, there exist α ̸= 0, positive integers m and p and
matrices x0, x1, .., xm+p ∈ Mn such that, for all λ ∈ C,
m∑
s=0
Ek(λxs + y)−
(
m+p∑
s=m+1
Ek(λxs + y)
)
= 0 · λk + (αyij) · λk−1 + · · · .
Using (3.3), for all λ ∈ C we have that
m∑
s=0
Ek(λxs + y)−
(
m+p∑
s=m+1
Ek(λxs + y)
)
= λk
(
m∑
s=0
Ek(xs)−
(
m+p∑
s=m+1
Ek(xs)
))
= 0.
Thus αyij = 0, and therefore yij = 0. Since this holds for any i and j, we obtain that
y = 0 ∈ Mn.
C. Costara: Nonlinear maps preserving the elementary symmetric functions 131
Theorem 3.1 gives us also linearity for the maps φ and ψ from the statement of Theo-
rem 1.1.
Proof of Theorem 1.1. Let us prove first that (1.8) and the surjectivity of φ implies that ψ
is linear on Mn. To see this, consider i, j ∈ {1, ..., n} and let us prove that ψij : Mn → C
is linear, where ψij is the (i, j) entry of the map ψ. By Theorem 3.1, there exist α ̸= 0
in C, natural numbers m and p and matrices x0, x1, .., xm+p ∈ Mn such that, for each
y ∈ Mn we have that the coefficient of λk for the polynomial
λ 7→
m∑
s=0
Ek(λxs + y)−
(
m+p∑
s=m+1
Ek(λxs + y)
)
is zero, and the coefficient of λk−1 for the same polynomial is αyij . Since φ is supposed
surjective, let w0, w1, .., wm+p ∈ Mn such that φ (wj) = xj for j = 0, ...,m + p. Then
for each z ∈ Mn, we have that the coefficient of λk for the polynomial
λ 7→
m∑
s=0
Ek(λφ (ws) + ψ (z))−
(
m+p∑
s=m+1
Ek(λφ (ws) + ψ (z))
)
is zero, and the coefficient of λk−1 for the same polynomial is αψij (z). Using (1.8), for
all λ ∈ C we have that
m∑
s=0
Ek(λφ (ws) + ψ (z))−
(
m+p∑
s=m+1
Ek(λφ (ws) + ψ (z))
)
equals
m∑
s=0
Ek(λws + z)−
(
m+p∑
s=m+1
Ek(λws + z)
)
.
The remark at the beginning of Section 2 shows that the coefficient of λk−1 for the polyno-
mial λ 7→
∑m
s=0Ek(λws+z)−(
∑m+p
s=m+1Ek(λws+z)) is linear with respect to z ∈ Mn.
Therefore, ψij is linear with respect to z ∈ Mn.
Thus ψ : Mn → Mn is linear and Ek (x) = Ek (ψ (x)) for each x ∈ Mn. If
ψ (y) = 0, then for each x ∈ Mn we have that
Ek (x) = Ek (ψ (x)) = Ek (ψ (x) + ψ (y)) = Ek (ψ (x+ y))
= Ek (x+ y) .
Then Corollary 3.2 gives y = 0. Thus the linear map ψ is injective on Mn, and therefore
bijective. Using (1.8), the linearity of ψ and the fact that Ek (z) = Ek(ψ−1 (z)) for each
z, then for each x, y ∈ Mn we have that
Ek (x+ y) = Ek(φ(x) + ψ (y)) = Ek(ψ
−1(φ(x) + ψ (y)))
= Ek((ψ
−1 ◦ φ)(x) + y).
Denoting z = x + y, we conclude that Ek (z) = Ek(((ψ−1 ◦ φ)(x) − x) + z) for each
x, z ∈ Mn. Then Corollary 3.2 gives (ψ−1 ◦ φ)(x) − x = 0, equality which holds for
every x ∈ Mn. Thus φ = ψ on Mn.
132 Ars Math. Contemp. 21 (2021) #P1.09 / 125–132
ORCID iDs
Constantin Costara https://orcid.org/0000-0001-7621-5591
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.10 / 133–150
https://doi.org/10.26493/1855-3974.2405.b43
(Also available at http://amc-journal.eu)
A characterization of exceptional pseudocyclic
association schemes by multidimensional
intersection numbers
Gang Chen * , Jiawei He †
School of Mathematics and Statistics, Central China Normal University, Wuhan, China
Ilia Ponomarenko ‡
Steklov Institute of Mathematics at St. Petersburg, Russia;
Sobolev Institute of Mathematics, Novosibirsk, Russia; and
School of Mathematics and Statistics of Central China Normal University, Wuhan, China
Andrey Vasil’ev §
Sobolev Institute of Mathematics, Novosibirsk, Russia; and
Novosibirsk State University, Novosibirsk, Russia
Received 10 August 2020, accepted 6 April 2021, published online 13 September 2021
Abstract
Recent classification of 32 -transitive permutation groups leaves us with three infinite
families of groups which are neither 2-transitive, nor Frobenius, nor one-dimensional affine.
The groups of the first two families correspond to special actions of PSL(2, q) and PΓL(2, q),
whereas those of the third family are the affine solvable subgroups of AGL(2, q) found by
D. Passman in 1967. The association schemes of the groups in each of these families are
known to be pseudocyclic. It is proved that apart from three particular cases, each of these
exceptional pseudocyclic schemes is characterized up to isomorphism by the tensor of its
3-dimensional intersection numbers.
Keywords: Association schemes, groups, coherent configurations.
Math. Subj. Class. (2020): 05E30, 05E18
*The author was supported by the NSFC grant No. 11971189.
†The author was supported by the NSFC grant No. 11971189.
‡Corresponding author. The author was supported by the Mathematical Center in Akademgorodok under
agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.
§The author was supported by the Mathematical Center in Akademgorodok under agreement No. 075-15-
2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.
E-mail addresses: chengangmath@mail.ccnu.edu.cn (Gang Chen), hjwywh@mails.ccnu.edu.cn (Jiawei He),
inp@pdmi.ras.ru (Ilia Ponomarenko), vasand@math.nsc.ru (Andrey Vasil’ev)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
134 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
1 Introduction
In the late 1960s, H. Wielandt proposed a method for studying permutation groups via
invariant relations. Later, D. Higman axiomatized a part of this method (connected with
binary relations) by introducing a new object called a coherent configuration [6]. The
coherent configuration of a permutation group G is formed by the orbits of the induced
action of G on the Cartesian square of the underlying set of points (for exact definitions,
see Section 2). Looking only at the parameters of this coherent configuration, the so-called
intersection numbers, one can easily determine whether the original group is transitive,
primitive, 2-transitive, etc. For example, the transitivity of a group G means exactly that
the coherent configuration of G is an association scheme.
The concept of pseudocyclic (association) scheme goes back to research of D. Mes-
ner [11], related with constructing of designs and strongly regular graphs; the defining
property of such a scheme is that the ratio of the multiplicity and degree of its nonprin-
cipal irreducible character does not depend on the choice of the character. It was proved
in [13, Theorem 3.2] that this is equivalent to a certain relation for intersection numbers,
see Subsection 2.6.
The class of pseudocyclic schemes contains all Frobenius schemes, i.e., the coherent
configurations of the Frobenius groups, and, moreover, every pseudocyclic scheme of rank
sufficiently large comparing with its degree is Frobenius [13, Theorem 1.1]. Thus the
pseudocyclic schemes can be considered as combinatorial analogs of the Frobenius groups.
It should be mentioned that the analogy is not complete, because there exist schurian (i.e.,
those associated with permutation groups) pseudocyclic schemes which are not Frobenius,
as well as non-schurian pseudocyclic schemes [2, Example 2.6.15].
It is well known that every Frobenius group is 32 -transitive, i.e., is transitive and all the
orbits of the stabilizer of a point α, other than {α}, have the same size greater than 1. It
immediately follows that so is the automorphism group of any Frobenius scheme. More-
over, the above mentioned relation for intersection numbers implies that the automorphism
group of a schurian pseudocyclic scheme is also 32 -transitive. A recent classification of
3
2 -transitive permutation groups shows that in most cases the coherent configuration of a
3
2 -transitive group is pseudocyclic, see Subsection 6.1. We cite a part of this classification
in the following theorem, see [10, Corollaries 2,3].
Theorem 1.1. Let G be a 32 -transitive permutation group of degree n. Assume that neither
G is 2-transitive or Frobenius nor G ≤ AΓL(1, q) for some prime power q. Then apart
from finitely many cases,
(1) n = q(q − 1)/2, q = 2d ≥ 8, and either G = PSL(2, q), or d is prime and
G = PΓL(2, q),
(2) n = q2, q is odd, and G ≤ AGL(2, q) is the affine group with point stabilizer of
order 4(q − 1), consisting of all monomial matrices of determinant ±1.
Remark 1.2. The finitely many cases mentioned in Theorem 1.1 include some affine
groups of degree at most 134 and the groups Alt(7) and Sym(7) both of degree 21.
The association schemes of the groups in statement (1) of Theorem 1.1 appeared in
Master Thesis of H. Hollmann (1982); such a scheme is called a large or small Hollmann
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 135
scheme depending on whether G = PSL(2, q) or G = PΓL(2, q).1 These schemes have
been studied in [8]; in particular, it was proved there that both are pseudocyclic. The group
in statement (2) of Theorem 1.1 appeared in D. Passman’s characterization of solvable 32 -
transitive groups [16]. The association scheme of this group, the Passman scheme, is also
pseudocyclic [13].
The goal of the present paper is to establish combinatorial characterizations of the Holl-
mann and Passman schemes; from the point of view of Theorem 1.1, they can naturally be
considered as exceptional. One of the best possible combinatorial characterization of an
association scheme is obtained when the scheme in question is determined up to (combina-
torial) isomorphism by its intersection numbers; in this case the scheme is called separable.
However, most of association schemes are not separable. In [4], multidimensional inter-
section numbers and separability number s(X ) of a coherent configuration X have been
introduced and studied (see also [2, Section 3.5 and 4.2]). According to the definition,
s(X ) ≤ m if and only if X is determined up to isomorphism by its m-dimensional inter-
section numbers; thus s(X ) = 1 if and only if X is separable. It was proved in [4] that
s(X ) = 1 or 2 if X is the scheme of a Hamming, Johnson, or Grassmann graph; later, the
estimate s(X ) ≤ 3 has been established in [5] for any cyclotomic scheme X over finite
field.
Theorem 1.3. Let X be a large Hollmann scheme. Then s(X ) ≤ 2.
The proof of Theorem 1.3 is given in Section 3. The difficult step in the proof is to
verify that the one point extension of the scheme X (which is a combinatorial analog of a
one point stabilizer of a permutation group) is a coherent configuration of the stabilizer of
this point in Aut(X ). In proving this fact we use the formulas for the intersection numbers
of X , which were calculated in [8].
The proofs of the following two theorems are based on Theorem 4.1 (see Section 4),
giving a sufficient condition for an arbitrary coherent configuration X to be partly regular,
i.e., to be the coherent configuration of a permutation group having a faithful regular orbit.
Using this sufficient condition we are able to show that if X is the small Hollmann scheme
(apart from several exceptions) or the Passman scheme, then a two point extension of X is
partly regular. Modulo known results, this immediately implies that s(X ) ≤ 3.
Theorem 1.4. Let X be a small Hollmann scheme of degree q(q − 1)/2, where q = 2d
with prime d ̸= 7, 11, 13. Then s(X ) ≤ 3.
In the three exceptional cases of Theorem 1.4, the sufficient condition given in Theo-
rem 4.1 does not work. It seems that the conclusion of Theorem 1.4 is also true for them.
However, the corresponding schemes are too large to check this statement by a direct cal-
culation.
Theorem 1.5. Let X be a Passman scheme. Then s(X ) ≤ 3.
Throughout the paper, we actively use the notation, concepts, and statements from the
theory of coherent configurations. All of them can be found in the monograph [2]. In
Section 2, we give a brief extract from the theory of coherent configurations that is relevant
for this paper.
1According to the Galois correspondence between permutation groups and coherent configurations [2, Sec-
tion 2.2], the smaller groups correspond to larger coherent configurations
136 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
The authors are grateful to the anonymous referees whose valuable comments helped
to improve the presentation and eliminate inaccuracies, and to Saveliy Skresanov for his
supplement to the GAP-package COCO2P, that was used for the computations related to
Theorem 5.5.
Notation.
• For a prime power q, Fq is a finite field of order q.
• Throughout the paper, Ω is a finite set.
• The diagonal of the Cartesian product Ω × Ω is denoted by 1Ω; for α ∈ Ω, we
set 1α := 1{α}.
• For r ⊆ Ω×Ω, we set r∗ = {(β, α) : (α, β) ∈ r} and αr = {β ∈ Ω : (α, β) ∈ r},
α ∈ Ω.
• For relations r, s ⊆ Ω× Ω, we set r · s = {(α, β) : (α, γ) ∈ r, (γ, β) ∈ s}.
• For a set S of relations on Ω, we define S∗ = {s∗ : s ∈ S} and put S∪ to be the set
of all unions of the relations of S.
2 Coherent configurations
2.1 Rainbows
Let Ω be a finite set and S a partition of Ω × Ω. A pair X = (Ω, S) is called a rainbow
on Ω if
1Ω ∈ S∪, and S∗ = S.
The elements of the sets Ω, S = S(X ), and S∪ are called, respectively, the points, basis
relations, and relations of X . The numbers |Ω| and |S| are called the degree and rank of X ,
respectively. The unique basis relation containing a pair (α, β) ∈ Ω × Ω is denoted by
rX (α, β); we omit the subscript X wherever it does not lead to misunderstanding.
A set ∆ ⊆ Ω is called a fiber of the rainbow X if 1∆ ∈ S; the set of all fibers is denoted
by F := F (X ). The point set Ω is the disjoint union of fibers. If ∆ is a union of fibers,
then the pair
X∆ = (∆, S∆)
is a rainbow, where S∆ consists of all s∆ = s ∩ (∆×∆), s ∈ S.
Let X = (Ω, S) and X ′ = (Ω′, S′) be rainbows. A bijection f : Ω → Ω′ is called
a combinatorial isomorphism (or simply isomorphism) from X to X ′ if Sf = S′, where
Sf = {sf : s ∈ S}. When X = X ′, the set of all these isomorphisms form a permutation
group on Ω. This group has a (normal) subgroup
Aut(X ) = {f ∈ Sym(Ω): sf = s for all s ∈ S},
called the automorphism group of X .
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 137
2.2 Coherent configurations
A rainbow X = (Ω, S) is called a coherent configuration if for any r, s, t ∈ S, the number
ctrs = |αr ∩ βs∗|
does not depend on the choice of (α, β) ∈ t; the numbers ctrs are called the intersection
numbers of X . If, in addition, 1Ω ∈ S, then the coherent configuration X is said to be
homogeneous, an association scheme, or just a scheme. A scheme X is called symmetric if
s = s∗ for all s ∈ S.
Let X be a coherent configuration. Then for any s ∈ S, there exist uniquely determined
∆,Γ ∈ F such that s ⊆ ∆× Γ. Denote by S∆,Γ the set of all s contained in ∆× Γ. Then
the union
S =
⋃
∆,Γ∈F
S∆,Γ
is disjoint. The positive integer |δs|, δ ∈ ∆, equals the intersection number c1∆ss∗ , and hence
does not depend on the choice of δ. It is called the valency of s and denoted by ns. In
homogeneous case, ns = ns∗ and also
ntc
t∗
rs = nrc
r∗
st = nsc
s∗
tr , r, s, t ∈ S. (2.1)
A basis relation s ∈ S is called a matching if ns = ns∗ = 1; note that s is not
necessarily symmetric like in theory of undirected graphs. The matching s ∈ S∆,Γ defines
a bijection from ∆ to Γ, taking δ ∈ ∆ to the unique point of the singleton δs. Furthermore,
one can see that if r ∈ S and t = s · r (respectively, t = r · s) is nonempty, then t ∈ S.
Let G be a permutation group on Ω. Denote by (α, β)G the orbit of the induced action
of G on Ω× Ω, that contains the pair (α, β). Then
Inv(G) = Inv(G,Ω) = (Ω, {(α, β)G : α, β ∈ Ω})
is a coherent configuration; we say that Inv(G) is the coherent configuration associated
with G. A coherent configuration X is said to be schurian if X = Inv(Aut(X )).
2.3 Separability
Let X = (Ω, S) and X ′ = (Ω′, S′) be coherent configurations. A bijection φ : S →
S′, s 7→ s′, is called an algebraic isomorphism from X to X ′ if
ctrs = c
t′
r′s′ , r, s, t ∈ S. (2.2)
When X = X ′, the set of all such φ forms a subgroup of Sym(S), denoted by Autalg(X ).
Each isomorphism f from X to X ′ induces an algebraic isomorphism from X to X ′,
which maps r ∈ S to rf ∈ S′. A coherent configuration X is said to be separable if every
algebraic isomorphism from X is induced by a suitable bijection (which is an isomorphism
of the coherent configurations in question).
The algebraic isomorphism φ induces a bijection from S∪ to (S′)∪: the union r∪s∪· · ·
of basis relations of X is mapped to r′ ∪ s′ ∪ · · · . This bijection is also denoted by φ. It
preserves the dot product, i.e., φ(r · s) = φ(r) · φ(s) for all r, s ∈ S.
138 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
2.4 Coherent closure
There is a natural partial order ≤ on the set of all rainbows on the same set Ω. Namely,
given two such rainbows X and X ′, we set
X ≤ X ′ ⇔ S(X )∪ ⊆ S(X ′)∪.
The minimal and maximal elements with respect to this order are the trivial and discrete
coherent configurations, respectively: the basis relations of the former are the reflexive
relation 1Ω and (if |Ω| > 1) its complement in Ω × Ω, whereas the basis relations of the
latter are singletons.
The functors X → Aut(X ) and G → Inv(G) form a Galois correspondence between
the posets of coherent configurations and permutation groups on the same set, i.e.,
Y ≤ X ⇒ Aut(Y) ≥ Aut(X ) and L ≤ K ⇒ Inv(L) ≥ Inv(K),
and
Aut(Inv(Aut(X ))) = Aut(X ) and Inv(Aut(Inv(G))) = Inv(G).
The coherent closure WL(T ) of a set T of relations on Ω, is defined to be the smallest
coherent configuration on Ω, for which T is a set of relations. The point extension Xα,β,...
of the rainbow X with respect to the points α, β, . . . ∈ Ω is defined to be WL(T ), where T
consists of S(X ) and the relations 1α, 1β , . . .. In other words, Xα,β,... is the smallest co-
herent configuration on Ω that is larger than or equal to X and has singletons {α}, {β}, . . .
as fibers.
2.5 Multidimensional intersection numbers
The theory of multidimensional extensions of coherent configurations has been developed
in [4], see also [2, Section 3.5].
Let m ≥ 1 be an integer. The m-extension of a coherent configuration X on Ω is
defined to be the smallest coherent configuration on Ωm, which contains the Cartesian m-
power of X and for which the set Diag(Ωm) is a union of fibers. The intersection numbers
of the m-extension are called the m-dimensional intersection numbers of the configura-
tion X . If m = 1, then the m-extension of X coincides with X and the m-dimensional
intersection numbers of X are the ordinary intersection numbers.
An algebraic isomorphism φ from X to X ′ is said to be m-dimensional if it can be
extended to an algebraic isomorphism from the m-extension of X to that of X ′, that takes
Diag(Ωm) to Diag(Ω′m). The separability number s(X ) of the coherent configuration X
is defined to be the smallest positive integer m for which every m-dimensional algebraic
isomorphism from X is induced by some isomorphism. Thus, the equality s(X ) = m ex-
presses the fact that X is determined up to isomorphism by its tensor of the m-dimensional
intersection numbers. The following statement was proved in [4, Theorem 4.6(1)].
Lemma 2.1. Let X be a coherent configuration. Then s(X ) ≤ s(Xα) + 1 for any point α
of X .
2.6 Pseudocyclic schemes
Let X = (Ω, S) be a coherent configuration. The indistinguishing number of a relation
s ∈ S(X ) is defined to be the sum c(s) of the intersection numbers csrr∗ , r ∈ S. For each
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 139
pair (α, β) ∈ s, we have c(s) = |c(α, β)|, where
c(α, β) = {γ ∈ Ω : r(γ, α) = r(γ, β)}. (2.3)
The maximum c(X ) of the numbers c(s), where s runs over the set of all irreflexive basis
relations of X , is called the indistinguishing number of X . It is easily seen that c = 0 if
and only if ns = 1 for each s ∈ S.
Assume that X is a scheme. In accordance with [13, Theorem 3.2], X is pseudocyclic
of valency k if the equalities
c(s) + 1 = k = ns
hold for all irreflexive s ∈ S. The class of pseudocyclic schemes includes all (homoge-
neous) coherent configurations associated with regular or Frobenius groups.
2.7 Partly regular coherent configurations
A coherent configuration X is said to be partly regular if there exists a point α ∈ Ω such
that |αs| ≤ 1 for all s ∈ S; the point α is said to be regular. When all the points of X are
regular, we say that X is semiregular, and regular if X is a scheme. Thus, X is semiregular
if and only if c(X ) = 0.
The following statement taken from [2, Theorem 3.3.19] shows, in particular, that the
partly regular (respectively, semiregular, regular) coherent configurations are in one-to-one
correspondence with those of the form Inv(G), where G is a permutation group having a
faithful orbit (respectively, G is semiregular, regular).
Theorem 2.2. Every partly regular coherent configuration X is schurian and separable.
In particular, s(X ) = 1.
The key point in the proof of Theorem 2.2 is the lemma below [2, Lemma 3.3.20]; it is
also used in the proof of Theorem 1.3.
Lemma 2.3. Let X be a coherent configuration and ∆ a union of fibers of X . Assume that
for every Γ ∈ F there exists s ∈ S∆,Γ such that ns = 1. Then
(1) the restriction mapping Aut(X ) → Aut(X∆) is a group isomorphism,
(2) X is schurian and separable whenever X∆ is schurian and separable.
3 Large Hollmann schemes
3.1 General properties
Throughout this section, d ≥ 3 is an integer, q = 2d, and G = PSL(2, q) the permutation
group of degree n = q(q − 1)/2 from Theorem 1.1(1). The lemma below immediately
follows from Theorem 1.2(iii) and Lemma 6.2 proved in [1].
Lemma 3.1. For any point α, we have Gα = D2(q+1). Moreover,
|∆| = q + 1, ∆ ∈ Orb(Gα), ∆ ̸= {α}. (3.1)
To study the large Hollmann scheme X = Inv(G), we make use of some results proved
in [8]. However, formally, the definition of the scheme given there is different from the
140 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
definition of X . Thus our first goal is to verify that X is exactly the symmetric pseudocyclic
scheme X ′ of degree n and valency q + 1 defined in [13] and studied in [8].
First, we note that Aut(X ) is a 32 -transitive group of the same degree as G; in particular,
Aut(X ) is not a subgroup of AΓL(1, r) for some r. Moreover, it is neither 2-transitive,
nor Frobenius (because G ≤ Aut(X )). By Theorem 1.1 and Remark 1.2, this implies that
Aut(X ) = PSL(2, q) or PΓL(2, q). The latter case is impossible by Lemma 3.1, because
G and Aut(X ) have the same subdegrees. Consequently, G = Aut(X ). On the other
hand, the scheme X ′ is also associated with G. Therefore a similar argument shows that
Aut(X ′) = G = Aut(X ). Thus,
X ′ = Inv(Aut(X ′)) = Inv(Aut(X )) = X . (3.2)
Proposition 3.2. The large Hollmann scheme X is symmetric and pseudocyclic of degree
q(q − 1)/2, rank q/2, and valency q + 1. Moreover,
Aut(X ) = G and Aut(Xα) = Gα = D2(q+1) for all α. (3.3)
Proof. By the remark before the proposition, we need to verify the second equality in (3.3)
only. By [2, Proposition 3.3.3(1)], we have Aut(Xα) = Aut(X )α. Thus the required
statement immediately follows from the first equality in (3.3) and Lemma 3.1.
Equality (3.2) allows us to use formulas for the intersection numbers of the scheme X ′,
given in [8, Theorem 2.2]. Namely, let S = S(X ) and
T0 = {x ∈ F2d : Tr(x) = 0},
where Tr(x) is the trace of x over the prime subfield of the field F2d . Then there is a
bijection T0 → S, x 7→ sx, such that s0 is reflexive and
cszsx,sy = 1 ⇔ Tr(xz) = 0 and x+ y + z = 0. (3.4)
As is easily seen, T0 is a linear space of dimension d− 1 over F2.
3.2 One point extension
Let us analyze the extension Xα of the large Hollmann scheme X with respect to a point α.
Since the scheme X is schurian, each fiber of the coherent configuration Xα is of the form
∆ = αs for some s ∈ S [2, Theorem 3.3.7]. When s = sx for some x ∈ T0, the fiber ∆
is denoted by ∆x. Thus,
F (Xα) = {∆x : x ∈ T0}.
Theorem 3.3. Let x and y be nonzero elements of T0. Then the set S(Xα)∆x,∆y contains
a matching.
Proof. For d = 3, the statement has been checked with the help of the computer pack-
age COCO2P [9]. Assume that d ≥ 4. We need auxiliary lemmas.
Lemma 3.4. Theorem 3.3 holds whenever Tr(xy) = 0.
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 141
Proof. Let z = x+ y. Then obviously z ∈ T0. Moreover
Tr(xz) = Tr(x2 + xy) = Tr(x2) + Tr(xy) = 0.
By formula (3.4), this implies that cszsx,sy = 1. If z = 0, then x = y and 1∆x is a desired
matching. Assume that z is nonzero. Then nsx = nsz , because X is a pseudocyclic scheme
(Proposition 3.2). Since X is also symmetric, we have
csxsy,sz =
nsz
nsx
cszsx,sy = 1,
see (2.1). Therefore if r = sz ∩ (αsx × αsy), then |βr| = 1 for all β ∈ ∆x, and |βr∗| = 1
for all β ∈ ∆y (here we use the fact that |∆x| = |∆y|). Since r is a relation of Xα [2,
Lemma 3.3.5], this implies that r belongs to S(Xα)∆x,∆y . Thus, r is a required matching.
Let us define a graph X with nonzero elements of T0 as the vertices and in which two
distinct vertices x and y are adjacent if and only if Tr(xy) = 0. One can see that X is an
undirected graph with exactly |T0| − 1 vertices.
Lemma 3.5. The graph X is connected.
Proof. Given x ∈ T0, we define the linear mapping
fx : T0 → F2, y 7→ Tr(xy).
Now let x ̸= y be two vertices of the graph X. If Tr(xy) = 0, then x and y are connected by
an edge. Let Tr(xy) ̸= 0. Then both fx and fy are nonzero linear mappings, and ker(fx)
and ker(fy) are subspaces of T0 of codimension one. Therefore, ker(fx) ∩ ker(fy) is a
subspace of T0 of codimension at most two. Consequently,
dim(ker(fx) ∩ ker(fy)) ≥ dim(T0)− 2 = d− 3 ≥ 1.
It follows that ker(fx)∩ker(fy) contains at least one nonzero vector, say z. Then Tr(xz) =
Tr(yz) = 0. Since Tr(xy) ̸= 0, this implies that z ̸= x, y. Thus the vertices x, y are at
distance two in the graph X, in particular, X is connected.
Let us return to the proof of Theorem 3.3. By Lemma 3.5, the vertices x and y of the
graph X are connected by a path
x = x0, x1, . . . , xk = y,
where k ≥ 1. For i = 0, . . . , k − 1, the vertices xi and xi+1 are adjacent and hence
Tr(xixi+1) = 0. Denote by si the matching in S(Xα)∆xi ,∆xi+1 the existence of which is
guaranteed by Lemma 3.4. Then the dot product
s = s0 · s1 · · · sk−1
is a desired matching belonging to S(Xα)∆x,∆y .
Corollary 3.6. Let ∆ = ∆x for nonzero x ∈ T0. Then the coherent configuration Y =
(Xα)∆ is schurian and separable. Moreover, the extension of Y with respect to at least one
point is partly regular.
142 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
Proof. By Proposition 3.2, we have Aut(Xα) = D2(q+1). Furthermore, the hypothesis of
Lemma 2.3 is satisfied for X = Xα by Theorem 3.3. Thus by statement (1) of that lemma,
we have
H := Aut(Y) ∼= Aut(Xα) = D2(q+1). (3.5)
On the other hand, |∆| = q + 1 by formula (3.1). Consequently, the group H contains a
normal regular cyclic subgroup C of order q + 1. In terms of [2, Section 4.4], this means
that Y is isomorphic to a normal circulant scheme. The radical of such a scheme, being a
subgroup of the group C, is of order at most 2; this follows from the implication (1)⇔(3)
in [5, Theorem 6.1]. Since the number |C| = q + 1 = 2d + 1 is odd, the radical is trivial.
Thus, the scheme Y is schurian by [2, Corollary 4.4.3], and every its extension with respect
to at least one point is partly regular by [2, Theorem 4.4.7].
It remains to verify that Y is separable. Since Y is schurian by above, we have Y =
Inv(Aut(Y)) = Inv(H). By virtue of (3.5), this means that Y is the coherent configuration
associated with D2(q+1). Thus, the required statement follows from [2, Exercise 2.7.33].
3.3 Proof of Theorem 1.3
By Lemma 2.1, it suffices to verify that a one point extension of a large Hollmann scheme
is separable. But this immediately follows from Theorem 3.7 below.
Theorem 3.7. Let q = 2d where d ≥ 3. Then the extension of the large Hollmann
scheme of degree q(q − 1)/2 with respect to at least one point is schurian and separa-
ble.
Proof. Let X ′ be the extension of the large Hollmann scheme X with respect to m ≥ 1
points α = α1, α2, . . . , αm. Let x ∈ T0 be nonzero and ∆ = ∆x. Then the hypothesis of
Lemma 2.3 is satisfied for X = X ′. Indeed, each Γ ∈ F (X ′) other than {α} is contained
in ∆y for some nonzero y ∈ T0. By Theorem 3.3, there is a matching s′ ∈ S(Xα)∆x,∆y ,
and as the required relation s one can take s′ ∩ (∆× Γ).
By Lemma 2.3(2), the coherent configuration X ′ is schurian and separable whenever
so is X ′∆. If m = 1, then X ′ = Xα and we are done by Corollary 3.6. Let m > 1. We
claim that there exist β2, . . . , βm ∈ ∆ such that
X ′ = Xα,β2,...,βm . (3.6)
Indeed, without loss of generality we may assume that none of the αi, i > 1, equals α. By
Theorem 3.3, there is a matching si ∈ S(Xα)∆i,∆, where ∆i is the fiber of Xα, contain-
ing αi. Then αisi = {βi} for some βi ∈ ∆. It follows that for any extension of Xα, each
or none of the two singletons {αi} and {βi} is a fiber of this extension, see [2, Corollary
3.3.6]. Thus,
X ′ = Xα,α2,...,αm = Xα,β2,...,βm ,
which completes the proof of the claim.
Now from formula (3.6) and the fact that βi ∈ ∆ for all i, it easily follows that
X ′∆ =(Xα,α2,...,αm)∆ = (Xα,β2,...,βm)∆
=((Xα,β2,...,βm)∆)β2,...,βm ≥ ((Xα)∆)β2,...,βm .
The coherent configuration on the right-hand side of this relation is partly regular by Corol-
lary 3.6. Therefore, the coherent configuration X ′∆ is also partly regular. By Theorem 2.2,
this implies that X ′∆ is schurian and separable, as required.
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 143
4 A lower bound for indistinguishing number
The main result of this section (Theorem 4.1 below) establishes a lower bound for the
indistinguishing number of a coherent configuration which is not partly regular, cf. [17,
Theorem 3.1]. This bound gives a sufficient condition for a coherent configuration to be
partly regular, and is used to prove Theorems 1.4 and 1.5 in the next section.
Theorem 4.1. Let X be a coherent configuration of degree n, k the maximal cardinality of
a fiber of X , and c = c(X ). If X is not partly regular, then
(2k − 1)c ≥ n. (4.1)
Proof. Let X = (Ω, S). In the sequel, ∆ ∈ F (X ) and |∆| = k. The fiber ∆ contains
at least two points, for otherwise k = 1 and X is the discrete and hence partly regular
configuration in contrast to the hypothesis of the theorem. Set
∆α = {δ ∈ ∆ : nr(α,δ) = 1} and Ω1 = {α ∈ Ω : ∆α = ∆}.
Lemma 4.2. Ω1 is a (possibly empty) union of fibers of cardinality k. Moreover, the co-
herent configuration XΩ1 is semiregular.
Proof. Let Γ be a fiber containing a point of Ω1. The set SΓ,∆ consists of s = r(γ, δ),
where γ ∈ Γ ∩ Ω1 and δ runs over ∆. Consequently, ns = 1 for all s ∈ SΓ,∆. Therefore,
Γ ⊆ Ω1. Thus, Ω1 is a union of fibers of X . Furthermore, given s ∈ SΓ,∆ we have
k ≥ |Γ| = ns |Γ| = ns∗ |∆| ≥ |∆| = k.
This proves the first statement. To prove the second one, let ∆′ and ∆′′ be fibers contained
in Ω1. By the definition of Ω1 and the first statement, any relations s′ ∈ S∆′,∆ and s′′ ∈
S∆,∆′′ are matchings. Therefore, s′ · s′′ is a matching contained in S∆′,∆′′ . Thus, SΩ1
consists of matchings and we are done.
By Lemma 4.2 and the hypothesis of the theorem, the complement Ω′ of the set Ω1 in Ω
contains at least two distinct points.
Lemma 4.3. For each γ ∈ Ω′, ∑
s∈Sγ
ns ≥
k
2
,
where Sγ = {r(γ, δ) : δ ∈ ∆ and nr(γ,δ) > 1}.
Proof. We have ∑
s∈Sγ
ns =
∑
s∈Sγ
|γs| = |∆| − |∆γ |. (4.2)
Since |∆| = k, this proves the required inequality if ∆γ = ∅. Let δ ∈ ∆γ . It is easily seen
that r(δ, λ) = r(δ, γ) · r(γ, λ) is a matching of X∆ for each λ ∈ ∆γ . Therefore,
∆γ ⊆ {λ ∈ ∆ : nr(δ,λ) = 1}. (4.3)
On the other hand, denote by e the union of all matchings of the scheme X∆. Then e
is a relation of this scheme. Moreover, e is an equivalence relation on ∆ (see [2, Theo-
rem 2.1.25(4)]) and the set on the right-hand side of (4.3) is a class of e. In view of [2,
144 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
Corollary 2.1.23], the cardinality a of this class divides |∆| = k. Furthermore, a ̸= k, for
otherwise, ∆γ = ∆ and then γ ∈ Ω1, a contradiction. Thus,
|∆γ | ≤ a ≤
k
2
and the required statement follows from (4.2).
Lemma 4.4. Let ε = 2(k−1)2k−1 . Assume that |Ω
′| ≥ εn. Then inequality (4.1) holds.
Proof. Denote by N the cardinality of the set
{(α, β, γ) ∈ ∆×∆× Ω′ : α ̸= β, γ ∈ c(α, β)}, (4.4)
where c(α, β) is as in formula (2.3). The number of (α, β) ∈ ∆×∆ with α ̸= β is equal
to k(k − 1). Therefore there exists at least one such pair for which
|c(α, β)| ≥ N
k(k − 1)
. (4.5)
On the other hand, let γ ∈ Ω′, and let Sγ be as in Lemma 4.3. Then ns ≥ 2 for
all s ∈ Sγ . For every such s there are exactly ns(ns − 1) triples (α, β, γ) with distinct
α, β ∈ γs, and all these triples belong to the set (4.4). By Lemma 4.3 this implies that
N =
∑
γ∈Ω′
∑
s∈SΓ,∆
ns(ns − 1) ≥
∑
γ∈Ω′
∑
s∈Sγ
ns ≥
∑
γ∈Ω′
k
2
=
|Ω′| k
2
,
where Γ is the fiber containing γ. By formula (4.5) and the lemma assumption, we obtain
c ≥ |c(α, β)| ≥ N
k(k − 1)
≥ |Ω
′|
2(k − 1)
≥ 2(k − 1)n
2k − 1
· 1
2(k − 1)
=
n
2k − 1
,
as required.
By Lemma 4.4, we may assume that |Ω′| < εn. The coherent configuration X is not
partly regular. Therefore no point δ ∈ Ω1 is regular and there exist distinct α, β ∈ Ω′ such
that δ ∈ c(α, β). Since the coherent configuration XΩ1 is semiregular (Lemma 4.2), the
relation s = r(δ, λ) is a matching for all λ ∈ Ω1. It follows that
r(α, λ) = r(α, δ) · s = r(β, δ) · s = r(β, λ).
Consequently, Ω1 ⊆ c(α, β). This implies that
c ≥ |c(α, β)| ≥ |Ω1| = n− εn > n
(
1− 2(k − 1)
2k − 1
)
=
n
2k − 1
which completes the proof of Theorem 4.1.
Corollary 4.5. Let X be a coherent configuration of degree n, c = c(X ), and t an irreflex-
ive basis relation of X . Assume that (2mt − 1) c < n, where
mt = max
r,s∈S
ctrs. (4.6)
Then the extension of X with respect to any two points forming a pair from t is partly
regular.
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 145
Proof. Let X ′ be the extension of X with respect to the points α, β such that (α, β) ∈ t.
Then each fiber ∆ of X ′ different from both {α} and {β} is contained in the set αr ∩ βs∗
for appropriate r, s ∈ S. It follows that
|∆| ≤ |αr ∩ βs∗| = ctrs ≤ mt.
Thus the maximal cardinality k′ of a fiber of X ′ is less than or equal to mt. Since obviously
c′ = c(X ′) is less than or equal to c, the condition of the corollary implies that
(2k′ − 1) c′ ≤ (2mt − 1) c < n.
Thus X ′ is partly regular by Theorem 4.1.
5 Small Hollmann and Passman schemes
5.1 Algebraic fusion
Let X = (Ω, S) be a coherent configuration, and let Φ be a group of algebraic automor-
phisms of X . For each s ∈ S, set
sΦ =
⋃
φ∈Φ
φ(s).
Clearly, (1Ω)Φ = 1Ω. Moreover the set SΦ = {sΦ : s ∈ S} forms a partition of the
Cartesian square Ω2. According to [2, Lemma 2.3.26], the pair
XΦ = (Ω, SΦ)
is a coherent configuration called the algebraic fusion of X with respect to Φ. In the
following lemma, we establish a simple upper bound for the intersection numbers of an
algebraic fusion.
Lemma 5.1. In the above notation, let r, s, t ∈ S and mt be as in (4.6). Then
ct
Φ
rΦsΦ ≤ mt |Φ|
2.
Proof. We have ct
Φ
rΦsΦ =
∑
φ,ψ∈Φ
ctφ(r)ψ(s) ≤ mt |Φ|
2.
5.2 Proof of Theorem 1.4
Let X be a small Hollmann scheme. Then the degree of X is equal to n = q(q − 1)/2,
where q = 2d for a prime d ≥ 3. Moreover, X is associated with the permutation group
G = PΓL(2, q) of degree n from Theorem 1.1(1). As in Subsection 3.1, one can see that
X coincides with symmetric pseudocyclic scheme X ′ of degree n and valency d(q + 1),
associated with the group PΓL(2, q) and studied in [8]. In particular, X is obtained from
the large Hollmann scheme of degree n by merging the basis relations via the Frobenius
map x 7→ x2, x ∈ Fq . In other words, X is the algebraic fusion of the large Hollmann
scheme of degree n with respect the induced action of Aut(Fq) on its basis relations.
Proposition 5.2. Let X and Xq be the small and large Hollmann schemes of degree q(q −
1)/2, respectively. Then
146 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
(1) X = XqΦ, where Φ ≤ Autalg(Xq) is a group of order d,
(2) X is a pseudocyclic scheme of valency d(q + 1),
(3) for each irreflexive t ∈ S(X ), we have mt ≤ 4d2, where mt is as in (4.6).
Proof. Statements (1) and (2) follow from the above discussion. Next, from the formulas
for the intersection numbers of the scheme Xq , given in [8, Theorem 2.2], it follows that
czxy ≤ 4 for all irreflexive x, y, z ∈ S(Xq). In particular,
mz ≤ 4.
On the other hand, by statement (1), each irreflexive t ∈ S(X ) is of the form zΦ for some
irreflexive z. Thus by Lemma 5.1,
mt = max
x,y∈S(Xq)
cz
Φ
xΦyΦ ≤ mz|Φ|
2 ≤ 4d2,
which proves statement (3).
Let us prove Theorem 1.4. If d = 3, then a straightforward calculation shows that X is
trivial scheme and hence c(X ) = 1. Let d > 3. By Lemma 2.1 and Theorem 2.2, it suffices
to verify that the extension of X with respect to at least two points is partly regular.
Theorem 5.3. Let q = 2d, where d > 3 is a prime and d ̸= 7, 11, 13. Then every extension
of the small Hollmann scheme of degree q(q − 1)/2 with respect to at least two points is
partly regular.
Proof. Let X be the small Hollmann scheme of degree n = q(q − 1)/2. By Proposi-
tion 5.2(2), the number c = c(X ) is equal to d(2d + 1) − 1. By statement (3) of the same
proposition, mt ≤ 4 d2 for any irreflexive t ∈ S(X ). Now if d > 16, then
(2mt − 1) c < (8d2 − 1) (d (2d + 1)− 1) < 2d−1(2d − 1) = n.
By Corollary 4.5, this proves the required statement for all (prime) d > 13. In the remain-
ing case, d = 5, the required statement has been checked with the help of the computer
package COCO2P [9].
5.3 Proof of Theorem 1.5
Let q be an odd prime power. The permutation group G ≤ AGL(2, q) defined in Theo-
rem 1.1(2) has a Frobenius subgroup H consisting of the permutations(
x
y
)
→
(
a 0
0 a−1
)(
x
y
)
+
(
b
c
)
, x, y ∈ Fq, (5.1)
where a, b, c ∈ Fq and a ̸= 0. The group D ≤ GL(2, q) consisting of the eight matrices(
±1 0
0 ±1
)
and
(
0 ±1
±1 0
)
is contained in G, normalizes H , and, moreover, G = H ⋊ D. In particular, D acts
on the basis relations of the Frobenius scheme Y = Inv(H) as a group Φ of algebraic
automorphisms.
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 147
Proposition 5.4. Let X = Inv(G) be the Passman scheme of degree q2. Then
(1) X = YΦ, where Φ ≤ Autalg(Y) is a group of order 2,
(2) X is a pseudocyclic scheme of valency 2(q − 1),
(3) there is an irreflexive t ∈ S(X ) such that mt ≤ 8, where mt is as in (4.6).
Proof. Statements (1) and (2) follow from [13, Section 4.3]. To prove statement (3), let u
be the basis relation of Y , containing the pair (α, β), where
α = (0, 0) and β = (1, 1).
It suffices to verify that mu ≤ 2; indeed, then t = uΦ is the required relation by state-
ment (1) and Lemma 5.1.
We need to verify that curs ≤ 2 for all r, s ∈ S(Y). Without loss of generality, we may
assume that r and s are such that curs ̸= 0. Then there is γ ∈ αr ∩ βs∗. It follows that
αr = γHα , βs∗ = γHβ , and
curs = |γHα ∩ γHβ |. (5.2)
Let us calculate the number on the right-hand side. Using the explicit form (5.1) of the
elements of H , one can easily find that the groups Hα and Hβ consist of permutations the
parameters a, b, c of which satisfy the relations
b = c = 0 and a+ b = a−1 + c = 1,
respectively. Consequently, assuming γ = (x, y), we have
γHα =
{(
ax,
y
a
)
: a ∈ F∗q
}
and γHβ =
{(
a′x+ 1− a′, y
a′
+ 1− 1
a′
)
: a′ ∈ F∗q
}
.
In view of (5.2), the intersection number curs is equal to the number of elements a ∈ F∗q
such that
ax = a′x+ 1− a′ and y
a
=
y
a′
+ 1− 1
a′
.
If exactly one of x, y equals 0, then these equations are satisfied for a = 1 = a′ only, and
curs = 1. Assume that x ̸= 0 ̸= y. Then
a = a′
(
1− 1
x
)
+
1
x
and
1
a
=
1
a′
(
1− 1
y
)
+
1
y
,
and hence
1 =
(
1− 1
x
)(
1− 1
y
)
+
a′
y
(
1− 1
x
)
+
1
a′x
(
1− 1
y
)
+
1
xy
.
This gives a quadratic equation with unknown a′, and at most two possible values for a′.
Thus, curs ≤ 2.
Let us prove Theorem 1.5. By Lemma 2.1 and Theorem 2.2, it suffices to verify that
the extension of X with respect to at least two points is partly regular.
148 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
Theorem 5.5. Let q be an odd prime power. Then every extension of the Passman scheme
of degree q2 with respect to at least two points is partly regular.
Proof. Let X be the Passman scheme of degree n = q2. By Proposition 5.4(2), the number
c = c(X ) is equal to 2(q − 1) − 1. Let t be the basis relation of X , defined in Proposi-
tion 5.4(3). Then mt ≤ 8. Now if q ≥ 29, then
(2mt − 1) c ≤ 15 (2q − 3) < q2 = n.
By Corollary 4.5, this proves the required statement for all q ≥ 29. In the remain-
ing cases, the required statement has been checked with the help of the computer pack-
age COCO2P [9].
6 Concluding remarks and open problems
6.1 Pseudocyclic schemes
A scheme is said to be k-equivalenced (and just equivalenced if k is irrelevant) if all irreflex-
ive basis relations of it have valency k. It is known that every k-equivalenced scheme is
pseudocyclic for 1 ≤ k ≤ 4; this follows from results obtained in [12,14,15] and [13, The-
orem 3.1].
By Theorem 1.1, if X is a schurian equivalenced scheme of sufficiently large degree,
which is not trivial or Frobenius, then either X is exceptional, i.e., the Hollmann or Passman
scheme, or the inclusion Aut(X ) ≤ AΓL(1, q) holds for some q. In all cases except for the
last one, X is pseudocyclic, see Propositions 3.2, 5.2(2), and 5.4(2), and [13, Theorem 3.1].
In the latter case, the group AΓL(1, q) can contain 32 -transitive subgroups which are not 2-
equivalent to Frobenius groups (such a subgroup is always primitive). We do not know
whether the scheme of at least one of these subgroups is not pseudocyclic.
6.2 Separability number
Finding the exact values of s(X ) for an exceptional scheme X is still an open problem. A
direct calculation shows that these schemes are separable for small q.
6.3 Superschurian schemes
The following concept was first formulated many years ago in discussions of the third
author with Sergei Evdokimov. A scheme X is said to be superschurian if the extension of
X with respect to every set of points is schurian. In particular, all superschurian schemes are
schurian. In fact, only a few families of superschurian schemes are known; these include
partly regular schemes, cyclotomic schemes over finite fields, normal circulant schemes,
and some TI-schemes [3, 5, 13]. Theorem 3.7 implies that any large Hollmann scheme is
superschurian. We do not know whether other exceptional schemes are superschurian.
6.4 Base number
The base number b(X ) of a coherent configuration X is defined to be the smallest number
of points such that the extension of X with respect to them is the discrete configuration,
see [2, Section 3.3.2]. In general, the base number of the group Aut(X ) is less than or
equal to b(X ). The equality is attained, for example, if X is a partly regular coherent
configuration. By virtue of this observation, Theorems 3.7, 5.3, and 5.5 imply that except,
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 149
possibly, for several small Hollmann schemes the equality holds also for all exceptional
pseudocyclic schemes.
In fact, the base number of an exceptional scheme of enough large degree is bounded by
3. This fact can be used to construct a polynomial-time algorithm recognizing whether or
not a given scheme is exceptional. Taking the above discussion into account, this reduces
the recognition problem for the class of schurian equivalenced schemes (see [18, p.281]) to
the non-Frobenius schemes X for which Aut(X ) ≤ AΓL(1, q).
6.5 Bound in Theorem 4.1
Denote by f(n) the maximum of the ratio nc(X )k(X ) taken over all non-partly-regular co-
herent configurations X of degree n, where k(X ) is the maximal cardinality of a fiber
of X . Clearly, f(n) > 0. Theorem 4.1 states that f(n) < 2. It would be interesting to
find the function f(n) explicitly. We have found a (schurian) non-partly-regular coherent
configuration with parameters
n = 24, k = 8, c = 4,
which shows that f(24) ≥ 3/4.
ORCID iDs
Gang Chen https://orcid.org/0000-0001-5173-6710
Jiawei He https://orcid.org/0000-0001-7526-1246
Ilia Ponomarenko https://orcid.org/0000-0003-2444-731X
Andrey Vasil’ev https://orcid.org/0000-0002-6498-0138
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