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Contents
The edge-transitive polytopes that are not vertex-transitive
Frank Göring, Martin Winter . . . . . . . . . . . . . . . . . . . . . . . . . 191
Double generalized majorization and diagrammatics
Marija Dodig, Marko Stošić . . . . . . . . . . . . . . . . . . . . . . . . . 221
Classification of minimal Frobenius hypermaps
Kai Yuan, Yan Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
A parametrisation for symmetric designs admitting a flag-transitive,
point-primitive automorphism group with a product action
Eugenia O’Reilly-Regueiro, José Emanuel Rodríguez-Fitta . . . . . . . . . 247
Mutually orthogonal cycle systems
Andrea C. Burgess, Nicholas J. Cavenagh, David A. Pike . . . . . . . . . . 261
Some remarks on the square graph of the hypercube
Seyed Morteza Mirafzal . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
A new family of additive designs
Andrea Caggegi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
On metric dimensions of hypercubes
Aleksander Kelenc, Aoden Teo Masa Toshi, Riste Škrekovski,
Ismael G. Yero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Complete forcing numbers of graphs
Xin He, Heping Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
Locally s-arc-transitive graphs arising from product action
Michael Giudici, Eric Swartz . . . . . . . . . . . . . . . . . . . . . . . . . 335
Volume 23, Number 2, Spring/Summer 2023, Pages 191–348
xiii

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.01 / 191–219
https://doi.org/10.26493/1855-3974.2712.6be
(Also available at http://amc-journal.eu)
The edge-transitive polytopes that are not
vertex-transitive*
Frank Göring
Faculty of Mathematics, University of Technology, 09107 Chemnitz, Germany
Martin Winter †
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Received 27 October 2021, accepted 19 March 2022, published online 28 October 2022
Abstract
In 3-dimensional Euclidean space there exist two exceptional polyhedra, the rhombic
dodecahedron and the rhombic triacontahedron, the only known polytopes (besides poly-
gons) that are edge-transitive without being vertex-transitive. We show that these poly-
hedra do not have higher-dimensional analogues, that is, that in dimension d ≥ 4, edge-
transitivity of convex polytopes implies vertex-transitivity.
More generally, we give a classification of all convex polytopes which at the same time
have all edges of the same length, an edge in-sphere and a bipartite edge-graph. We show
that any such polytope in dimension d ≥ 4 is vertex-transitive.
Keywords: Convex polytopes, symmetry of polytopes, vertex-transitive, edge-transitive.
Math. Subj. Class. (2020): 52B15, 52B11
1 Introduction
A d-dimensional (convex) polytope P ⊂ Rd is the convex hull of finitely many points.
P is said to be vertex-transitive resp. edge-transitive if its (orthogonal) symmetry group
Aut(P ) ⊂ O(Rd) acts transitively on its vertices resp. edges. For a general overview over
the state of the art regarding symmetries in convex and abstract polytopes we refer to [9].
It has long been known that there are exactly nine edge-transitive polyhedra in R3 (see
e.g. [6]). These are the five Platonic solids (tetrahedron, cube, octahedron, icosahedron and
dodecahedron) together with the cuboctahedron, the icosidodecahedron, and their duals,
the rhombic dodecahedron and the rhombic triacontahedron (depicted in this order):
*This article appears as Chapter 6 in the second author’s doctoral thesis [11]. The authors thank the anonymous
referees for their careful reading and their many remarks that led to an improvement of the article in several ways.
†Corresponding author.
E-mail addresses: frank.goering@mathematik.tu-chemnitz.de (Frank Göring),
martin.h.winter@warwick.ac.uk (Martin Winter)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
192 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
Little is known about analogous questions in higher dimensions. Branko Grünbaum writes
in “Convex Polytopes” [5, page 413]
No serious consideration seems to have been given to polytopes in dimension d ≥ 4
about which transitivity of the symmetry group is assumed only for faces of suitably
low dimensions, [...].
Even though families of higher-dimensional edge-transitive polytopes have been stud-
ied, to the best of our knowledge, no classification of these has been achieved so far.
Equally striking, all the known examples of such polytopes in dimension at least four are
simultaneously vertex-transitive. In dimension up to three, certain polygons (see Figure 1),
as well as the rhombic dodecahedron and rhombic triacontahedron are edge- but not vertex-
Figure 1: Some examples of edge-transitive 2n-gons with 2n ∈ {4, 6, 8} (the same works
for all n). The polygons depicted with black boundary are not vertex-transitive.
transitive. No higher dimensional example of this kind has been found. In this paper we
prove that this is not for lack of trying:
Theorem 1.1. In dimension d ≥ 4, edge-transitivity of convex polytopes implies vertex-
transitivity.
As immediate consequence, we obtain the classification of all polytopes that are edge-
but not vertex-transitive. The list is quite short:
Corollary 1.2. If P ⊂ Rd, d ≥ 2 is edge- but not vertex-transitive, then P is one of the
following:
(i) a non-regular 2k-gon (see Figure 1),
(ii) the rhombic dodecahedron, or
(iii) the rhombic triacontahedron.
Theorem 1.1 is proven by embedding the class of edge- but not vertex-transitive poly-
topes in a larger class of polytopes, defined by geometric regularities instead of symmetry.
In Theorem 2.4 we show that a polytope P ⊂ Rd which is edge- but not vertex-transitive
must have all of the following properties:
(i) all edges are of the same length,
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 193
(ii) it has a bipartite edge-graph GP = (V1 ·∪ V2, E), and
(iii) there are radii r1 ≤ r2, so that ∥v∥ = ri for all v ∈ Vi.
We compile this into a definition: a polytope that has these three properties shall be called
bipartite (cf. Definition 2.1). The edge- but not vertex-transitive polytopes then form a sub-
class of the bipartite polytopes, but the class of bipartite polytopes is much better behaved.
For example, faces of bipartite polytopes are bipartite (Proposition 2.5), something which
is not true for edge/vertex-transitive polytopes1. Our quest is then to classify all bipartite
polytopes. The surprising result: already being bipartite is very restrictive:
Theorem 1.3. If P ⊂ Rd, d ≥ 2 is bipartite, then P is one of the following:
(i) an edge-transitive 2k-gon (see Figure 1),
(ii) the rhombic dodecahedron,
(iii) the rhombic triacontahedron, or
(iv) a Γ-permutahedron for some finite reflection group Γ ⊂ O(Rd) (see Definition 2.10;
some 3-dimensional examples are shown in Figure 2).
Figure 2: From left to right: the A3-, B3 and H3-permutahedron.
The Γ-permutahedra are vertex-transitive, and all the other entries in the list are of
dimension d ≤ 3. This immediately implies Theorem 1.1.
Remarkably, despite the definition of bipartite polytope being purely geometric, all
bipartite polytopes are highly symmetric, that is, at least vertex- or facet-transitive, and
sometimes even edge-transitive.
Overview
In Section 2 we introduce the central notion of bipartite polytope and prove its most relevant
properties: that being bipartite generalizes being edge- but not vertex-transitive, and that
all faces of bipartite polytopes are again bipartite. We then investigate certain subclasses of
bipartite polytopes: bipartite polygons and inscribed bipartite polytopes. We prove that the
latter coincide with the Γ-permutahedra, a class of vertex-transitive polytopes. It therefore
remains to classify the non-inscribed cases, the so-called strictly bipartite polytopes. We
1For example, consider a vertex-transitive but not uniform antiprism. Its faces are non-regular triangles, which
are thus not vertex-transitive. Alternatively, consider the (n, n)-duoprism, n ̸= 4, that is, the cartesian product
of a regular n-gon with itself. This polytope is edge-transitive, but its facets are n-gonal prisms (the cartesian
product of a regular n-gon with an edge), which are not edge-transitive.
194 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
show that the classification of these reduces to the classification of bipartite polyhedra, i.e.,
the case d = 3.
From Section 3 on the investigation is focused on the class of strictly bipartite poly-
hedra. We successively determine restrictions on the structure of such, e.g. the degrees of
their vertices and the shapes of their faces. This quite elaborate process uses many clas-
sical geometric results and techniques, including spherical polyhedra, the classification of
rhombic isohedra and the realization of graphs as edge-graphs of polyhedra. As a result,
we can exclude all but two cases, namely, the rhombic dodecahedron, and the rhombic
triacontahedron. Additionally, we shall find a remarkable near-miss, that is, a polyhedron
which fails to be bipartite only by a tiny (but quantifiable) amount.
2 Bipartite polytopes
From this section on let P ⊂ Rd, d ≥ 2 denote a d-dimensional polytope of full dimension
(i.e., P is not contained in a proper affine subspace). By F(P ) we denote the face lattice
of P , and by Fδ(P ) ⊂ F(P ) the subset of δ-dimensional faces.
Definition 2.1. P is called bipartite, if
(i) all its edges are of the same length ℓ,
(ii) its edge-graph is bipartite, which we write as GP = (V1 ·∪ V2, E), and
(iii) there are radii r1 ≤ r2 so that ∥v∥ = ri for all v ∈ Vi.
If r1 < r2, then P is called strictly bipartite. A vertex v ∈ Vi is called an i-vertex. The
numbers r1, r2 and ℓ are called the parameters of a bipartite polytope.
Remark 2.2. Since P is full-dimensional by convention, Definition 2.1 only defines full-
dimensional bipartite polytopes.
To extend this notion to not necessarily full-dimensional polytopes, we shall call a
polytope bipartite even if it is just bipartite as a subset of its affine hull where we made
an appropriate choice of origin in the affine hull (note that whether a polytope is bipartite
depends on its placement relative to the origin and that there is at most one such placement
if the polytope is full-dimensional). This comes in handy when we discuss faces of bipartite
polytopes.
Remark 2.3. An alternative definition of bipartite polytope would replace (iii) by the con-
dition that P has an edge in-sphere, that is, a sphere that touches each edge of P in a single
point (this definition was used in the abstract). The configuration depicted below (an edge
of P connecting two vertices v1 ∈ V1 and v2 ∈ V2) shows how any one of the four quanti-
ties r1, r2, ℓ and ρ (the radius of the edge in-sphere) is determined from the other three by
solving the given set of equations:
ρ2 + ℓ21 = r
2
1
ρ2 + ℓ22 = r
2
2
ℓ1 + ℓ2 = ℓ
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 195
There is a subtlety: for the edge in-sphere to actually touch the edge (rather than only
its affine hull outside of the edge) it is necessary that the perpendicular projection of the
origin onto the edge ends up inside the edge (equivalently, that the triangle conv{0, v1, v2}
is acute at v1 and v2). One might regard this as intuitively clear since we are working with
convex polytopes, but this will also follows formally as part of our proof of Proposition 3.7
(as we shall mention there in a footnote).
This alternative characterization of bipartite polytopes via edge in-spheres will become
relevant towards the end of the classification (in Section 3.9). Still, for the larger part of
our investigation, Definition 2.1(iii) is the more convenient version to work with.
2.1 General obsevations
Proposition 2.4. If P is edge- but not vertex-transitive, then P is bipartite.
This is a geometric analogue to the well known fact that every edge- but not vertex-
transitive graph is bipartite. A proof of the graph version can be found in [4]. The following
proof can be seen as a geometric analogue:
Proof of Proposition 2.4. Clearly, all edges of P are of the same length.
Fix some edge e ∈ F1(P ) with end vertices v1, v2 ∈ F0(P ). Let Vi be the orbit of vi
under Aut(P ). We prove that V1 ∪ V2 = F0(P ), V1 ∩ V2 = ∅ and that the edge graph GP
is bipartite with partition V1 ·∪ V2.
Let v ∈ F0(P ) be some vertex and ẽ ∈ F1(P ) an incident edge. By edge-transitivity,
there is a symmetry T ∈ Aut(P ) that maps ẽ onto e, and therefore maps v onto vi for
some i ∈ {1, 2}. Thus, v is in the orbit Vi. This holds for all vertices of P , and therefore
V1 ∪ V2 = F0(P ).
The orbits of v1 and v2 must either be identical or disjoint. Since V1 ∪ V2 = F0(P ),
from V1 = V2 it would follow V1 = F0(P ), stating that P has a single orbit of vertices.
But since P is not vertex-transitive, this cannot be. Thus, V1 ∩ V2 = ∅, and therefore
V1 ·∪ V2 = F0(P ).
Let ẽ ∈ F1(P ) be an edge with end vertices ṽ1 and ṽ2. By edge-transitivity, ẽ can be
mapped onto e by some symmetry T ∈ Aut(P ). Equivalently {T ṽ1, T ṽ2} = {v1, v2}.
Since v1 and v2 belong to different orbits under Aut(P ), so do ṽ1 and ṽ2. Hence ẽ has
one end vertex in V1 and one end vertex in V2. This holds for all edges, and thus, GP is
bipartite with partition V1 ·∪ V2.
It remains to determine the radii r1 ≤ r2. Set ri := ∥vi∥ (assuming w.l.o.g. that
∥v1∥ ≤ ∥v2∥). Then for every v ∈ Vi there is a symmetry T ∈ Aut(P ) ⊂ O(Rd) so that
Tvi = v, and thus
∥v∥ = ∥Tvi∥ = ∥vi∥ = ri.
Bipartite polytopes are more comfortable to work with than edge- but not vertex-
transitive polytopes because their faces are again bipartite polytopes (in the sense as ex-
plained in Remark 2.2). Later, this will enable us to reduce the problem to an investigation
in lower dimensions.
Proposition 2.5. Let σ ∈ F(P ) be a face of P . Then it holds
(i) if P is bipartite, so is σ.
196 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
(ii) if P is strictly bipartite, then so is σ, and v ∈ F0(σ) ⊆ F0(P ) is an i-vertex in P if
and only if it is an i-vertex in σ.
(iii) if r1 ≤ r2 are the radii of P and ρ1 ≤ ρ2 are the radii of σ, then there holds
h2 + ρ2i = r
2
i ,
where h is the height of σ, that is, the distance of aff(σ) from the origin.
Proof. Properties clearly inherited by σ are that all edges are of the same length and that
the edge graph is bipartite. It remains to show the existence of the radii ρ1 ≤ ρ2 compatible
with the bipartition of the edge-graph of σ.
Let c ∈ aff(σ) be the orthogonal projection of 0 onto aff(σ). Then ∥c∥ = h, the height
of σ as defined in (iii). For any vertex v ∈ F0(σ) which is an i-vertex in P , the triangle
∆ := conv{0, c, v} has a right angle at c. Set ρi := ∥v − c∥ and observe
ρ2i := ∥v − c∥2 = ∥v∥2 − ∥c∥2 = r2i − h2. (∗)
In particular, the value ρi does only depend on i. In other words, σ is a bipartite poly-
tope when considered as a subset of its affine hull, where the origin is chosen to be c (cf.
Remark 2.2). This proves (i), and (∗) is equivalent to the equation in (iii). From (∗) also
follows r1 < r2 ⇔ ρ1 < ρ2, which proves (ii).
The following observation will be of use later on.
Observation 2.6. Given two adjacent vertices v1, v2 ∈ F0(P ) with vi ∈ Vi, and if P has
parameters r1, r2 and ℓ, then
ℓ2 = ∥v1 − v2∥2 = ∥v1∥2 + ∥v2∥2 − 2⟨v1, v2⟩ = r21 + r22 − 2r1r2 cos∡(v1, v2),
This can be rearranged for cos∡(v1, v2). While the exact value of this expression is not
of relevance to us, this shows that this angle is determined by the parameters and does not
depend on the choice of the adjacent vertices v1 and v2.
2.2 Bipartite polygons
The easiest to describe (and to explicitly construct) are the bipartite polygons.
Foremost, the edge-graph is bipartite, and thus, a bipartite polygon must be a 2k-gon
for some k ≥ 2. One can show that the bipartite polygons are exactly the edge-transitive
2k-gons (cf. Figure 1), and that such one is strictly bipartite if and only if it is not vertex-
transitive (or equivalently, not regular). We will not make use of these symmetry properties
of bipartite polygons.
The parameters r1, r2 and ℓ uniquely determine a bipartite polygon, as can be seen by
explicit construction:
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 197
One starts with an arbitrary 1-vertex v ∈ R2 placed on the circle Sr1(0). Its neighboring
vertices are then uniquely determined as the intersections Sr2(0) ∩ Sℓ(v). The procedure
is repeated with the new vertices until the edge cycle closes (which only happens if the
parameters are chosen appropriately).
The procedure also makes clear that the interior angle αi ∈ (0, π) at an i-vertex only
depends on i, but not on the chosen vertex v ∈ Vi.
Corollary 2.7. A bipartite polygon P ⊂ R2 is a 2k-gon with alternating interior angles
α1, α2 ∈ (0, π) (αi being the interior angle at an i-vertex), and its shape is uniquely
determined by its parameters (up to congruence).
The exact values for the interior angles are not of relevance. Instead, we only need the
following properties:
Proposition 2.8. The interior angles α1, α2 ∈ (0, π) satisfy
α1 + α2 = 2α
k
reg and α2 ≤ αkreg ≤ α1, (2.1)
where αkreg := (1 − 1/k)π is the interior angle of a regular 2k-gon, and the inequalities
are satisfied with equality if and only r1 = r2.
Proof. The sum of interior angles of a 2k-gon is 2(k−1)π, and thus kα1+kα2 = 2(k−1)π,
which, after division by k, yields the first part of (2.1).
For two adjacent vertices v1, v2 ∈ F0(P ) (where vi ∈ Vi), consider the triangle ∆ :=
conv{0, v1, v2} whose edge lengths are r1, r2 and ℓ, and whose interior angles at v1 resp.
v2 are α1/2 resp. α2/2. From r1 ≤ r2 (resp. r1 < r2) and the law of sine follows α1 ≥ α2
(resp. α1 > α2). With α1 + α2 = 2αkreg this yields the second part of (2.1).
Observation 2.9. For later use (in Corollary 3.18), consider Proposition 2.8 with 2k = 4.
In this case we find,
α2 ≤
π
2
≤ α1,
that is, α1 is never acute, and α2 is never obtuse.
2.3 The case r1 = r2
We classify the inscribed bipartite polytopes, that is, those with coinciding radii r1 = r2.
This case is made especially easy by a classification result from [10]. We need the following
definition:
Definition 2.10. Let Γ ⊂ O(Rd) be a finite reflection group and v ∈ Rd a generic point
w.r.t. Γ (i.e., v is not fixed by a non-identity element of Γ). The orbit polytope
Orb(Γ, v) := conv{Tv | T ∈ Γ} ⊂ Rd
is called a Γ-permutahedron.
The relevant result then reads
Theorem 2.11 (Corollary 4.6. in [10]). If P has only centrally symmetric 2-dimensional
faces (that is, it is a zonotope), has all vertices on a common sphere and all edges of the
same length, then P is a Γ-permutahedron.
198 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
This provides a classification of bipartite polytopes with r1 = r2.
Theorem 2.12. If P ⊂ Rd is bipartite with r1 = r2, then it is a Γ-permutahedron.
Proof. If r1 = r2, then all vertices are on a common sphere (that is, P is inscribed). By
definition, all edges are of the same length. Both statements then also hold for the faces of
P , in particular, the 2-dimensional faces. An inscribed polygon with a unique edge length
is necessarily regular. With Corollary 2.7 the 2-faces are then regular 2k-gons, therefore
centrally symmetric.
Summarizing, P is inscribed, has all edges of the same length, and all 2-dimensional
faces of P are centrally symmetric. By Theorem 2.11, P is a Γ-permutahedron.
Γ-permutahedra are vertex-transitive by definition, hence do not provide examples of
edge- but not vertex-transitive polytopes.
2.4 Strictly bipartite polytopes
It remains to classify the strictly bipartite polytopes. This problem is divided into two
independent cases: dimension d = 3, and dimension d ≥ 4. The detailed study of the case
d = 3 (which turns out to be the actual hard work) is postponed until Section 3, the result
of which is the following theorem:
Theorem 2.13. If P ⊂ R3 is strictly bipartite, then P is the rhombic dodecahedron or the
rhombic triacontahedron.
Presupposing Theorem 2.13, the case d ≥ 4 is done quickly.
Theorem 2.14. There are no strictly bipartite polytopes in dimension d ≥ 4.
Proof. It suffices to show that there are no strictly bipartite polytopes in dimension d = 4,
as any higher-dimensional example has a strictly bipartite 4-face (by Proposition 2.5).
Let P ⊂ R4 be a strictly bipartite 4-polytope. Let e ∈ F1(P ) be an edge of P .
Then there are s ≥ 3 cells (aka. 3-faces) σ1, ..., σs ∈ F3(P ) incident to e, each of which
is again strictly bipartite (by Proposition 2.5). By Theorem 2.13 each σi is a rhombic
dodecahedron or rhombic triacontahedron.
The dihedral angle of the rhombic dodecahedron resp. triacontahedron is 120◦ resp.
144◦ at every edge [3]. However, the dihedral angles meeting at e must sum up to less than
2π. With the given dihedral angles this is impossible.
3 Strictly bipartite polyhedra
In this section we derive the classification of strictly bipartite polyhedra. The main goal is to
show that there are only two: the rhombic dodecahedron and the rhombic triacontahedron.
From this section on, let P ⊂ R3 denote a fixed strictly bipartite polyhedron with radii
r1 < r2 and edge length ℓ. The 2-faces of P will be shortly referred to as just faces of P .
Since they are bipartite, they are necessarily 2k-gons.
Definition 3.1. We use the following terminology:
(i) a face of P is of type 2k (or called a 2k-face) if it is a 2k-gonal polygon.
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 199
(ii) an edge of P is of type (2k1, 2k2) (or called a (2k1, 2k2)-edge) if the two incident
faces are of type 2k1 and 2k2 respectively.
(iii) a vertex of P is of type (2k1, ..., 2ks) (or called a (2k1, ..., 2ks)-vertex) if its incident
faces can be enumerated as σ1, ..., σs so that σi is a 2ki-face (note, the order of the
numbers does not matter).
We write τ(v) for the type of a vertex v ∈ F0(P ).
3.1 General observations
In a given bipartite polyhedron, the type of a vertex, edge or face already determines much
of its metric properties. We prove this for faces:
Proposition 3.2. For some face σ ∈ F2(P ), any of the following properties of σ determines
the other two:
(i) its type 2k,
(ii) its interior angles α1 > α2.
(iii) its height h (that is, the distance of aff(σ) from the origin).
Corollary 3.3. Any two faces of P of the same height, or the same type, or the same interior
angles, are congruent.
Proof of Proposition 3.2. Fix a face σ ∈ F2(P ).
Suppose that the height h of σ is known. By Proposition 2.5, a face of P of height h is
bipartite with radii ρ2i := r
2
i − h2 and edge length ℓ. By Corollary 2.7, these parameters
then uniquely determine the shape of σ, which includes its type and its interior angles. This
shows (iii) =⇒ (i), (ii).
Suppose now that we know the interior angles α1 > α2 of σ (it actually suffices to
know one of these, say α1). Fix a 1-vertex v ∈ V1 of σ and let w1, w2 ∈ V2 be its two
adjacent 2-vertices in σ. Consider the simplex S := conv{0, v, w1, w2}. The length of
each edge of S is already determined, either by the parameters alone, or by additionally
using the known interior angles via
∥w1 − w2∥2 = ∥w1 − v∥2 + ∥w2 − v∥2 − 2⟨w1 − v, w2 − v⟩
= 2ℓ2(1− cos∡(w1 − v, w2 − v)︸ ︷︷ ︸
α1
).
Thus, the shape of S is determined. In particular, this determines the height of the face
conv{v, w1, w2} ⊂ S over the vertex 0 ∈ S. Since aff{v, w1, w2} = aff(σ), this deter-
mines the height of σ in P . This proves (ii) =⇒ (iii).
Finally, suppose that the type 2k is known. We then want to show that the height h
is uniquely determined.2 For the sake of contradiction, suppose that the type 2k does not
2The reader motivated to prove this himself should know the following: it is indeed possible to write down
a polynomial in h of degree four whose coefficients involve only r1, r2, ℓ and cos(π/k), and whose zeroes
include all possible heights of any 2k-face of P . However, it turns out to be quite tricky to work out which zeroes
correspond to feasible solutions. For certain values of the coefficients there are multiple positive solutions for h,
some of which correspond to non-convex 2k-faces. There seems to be no easy way to tell them apart.
200 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
uniquely determine the height of the face. Then there is another 2k-face σ′ ∈ F2(P ) of
some height h′ ̸= h. W.l.o.g. assume h > h′.
Visualize both faces embedded in R2, on top of each other and centered at the origin as
shown in the figure below:
The vertices in both polygons are equally spaced by an angle of π/k (cf. Observation 2.6)
and we can therefore assume that the vertex vi of σ (resp. v′i of σ
′) is a positive multiple
of (sin(iπ/k), cos(iπ/k)) ∈ R2 for i ∈ {1, ..., 2k}. There are then factors δi ∈ R+ with
v′i = δvi.
The norms of vectors v1, v2, δ1v1 and δ2v2 are the radii of the bipartite polygons σ and
σ′. With Proposition 2.5(iii) from h > h′ follows ∥v1∥ < ∥δ1v1∥ and ∥v2∥ < ∥δ2v2∥, and
thus, (∗) δ1, δ2 > 1. W.l.o.g. assume δ1 ≤ δ2.
Since both faces have edge length ℓ, we have ∥v1 − v2∥ = ∥δ1v1 − δ2v2∥ = ℓ. Our
goal is to derive the following contradiction:
ℓ = ∥v1 − v2∥
(∗)
< δ1∥v1 − v2∥ = ∥δ1v1 − δ1v2∥
(∗∗)
< ∥δ1v1 − δ2v2∥ = ℓ,
To prove (∗∗), consider the triangle ∆ with vertices δ1v1, δ2v2 and δ1v2:
Since σ is convex, the angle α is smaller than 90◦. It follows that the interior angle of ∆ at
δ1v2 is obtuse (here we are using δ1 ≤ δ2). Hence, by the sine law, the edge of ∆ opposite
to δ1v2 is the longest, which translates to (∗∗).
As a consequence of Proposition 3.2, the interior angles of a face of P do only depend
on the type of the face (and the parameters), and so we can introduce the notion of the
interior angle αki ∈ (0, π) of a 2k-face at an i-vertex. Furthermore, set ϵk := (αk1−αk2)/2π.
By Proposition 2.8 we have ϵk > 0 and
αk1 =
(
1− 1
k
+ ϵk
)
π, αk2 =
(
1− 1
k
− ϵk
)
π.
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 201
Definition 3.4. If τ = (2k1, ..., 2ks) is the type of a vertex, then define
K(τ) :=
s∑
i=1
1
ki
, E(τ) :=
s∑
i=1
ϵki .
Both quantities are strictly positive.
Proposition 3.5. Let v ∈ F0(P ) be a vertex of type τ = (2k1, ..., 2ks).
(i) If v ∈ V1, then E(τ) < K(τ)− 1 and s = 3.
(ii) If v ∈ V2, then E(τ) > s− 2−K(τ).
Proof. Let σ1, ..., σs ∈ F2(P ) be the faces incident to v, so that σj is a 2kj-face. The
interior angle of σj at v is αkji , and the sum of these must be smaller than 2π. In formulas
2π >
s∑
j=1
αkji =
s∑
j=1
(
1− 1
kj
± ϵkj
)
π = (s−K(τ)± E(τ))π,
where ± is the plus sign for i = 1, and the minus sign for i = 2. Rearranging for E(v)
yields (∗) ∓E(τ) > s − 2 −K(τ). If i = 2, this proves (ii). If i = 1, note that from the
implication kj ≥ 2 =⇒ K(τ) ≤ s/2 follows
s
(∗)
< −E(τ) +K(τ) + 2 ≤ 0 + s
2
+ 2 =⇒ s < 4.
The minimum degree of a vertex in a polyhedron is at least three, hence s = 3, and (∗)
becomes (i).
This allows us to exclude all but a manageable list of types for 1-vertices. Note that a
vertex v ∈ V1 has a type of some form (2k1, 2k2, 2k3).
Corollary 3.6. For a 1-vertex v ∈ V1 of type τ holds K(τ) > 1 + E(τ) > 1. One checks
that this leaves exactly the options in Table 1.
τ K(τ) Γ
(4, 4, 4) 3/2 I1 ⊕ I1 ⊕ I1
(4, 4, 6) 4/3 I1 ⊕ I2(3)
(4, 4, 8) 5/4 I1 ⊕ I2(4)
...
...
...
(4, 4, 2k) 1 + 1/k I1 ⊕ I2(k)
(4, 6, 6) 7/6 A3 = D3
(4, 6, 8) 13/12 B3
(4, 6, 10) 31/30 H3
Table 1: Possible types of 1-vertices, their K-values and the Γ of the Γ-permutahedron in
which all vertices have this type.
202 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
The types in Table 1 are called the possible types of 1-vertices. Each of the possi-
ble types is realizable in the sense that there exists a bipartite polyhedron in which all
1-vertices have this type. Examples are provided by the Γ-permutahedra (the Γ of that
Γ-permutahedron is listed in the right column of Table 1). These are not strictly bipartite
though.
The convenient thing about Γ-permutahedra is that all their vertices are of the same
type. We cannot assume this for general strictly bipartite polyhedra, not even for all 1-
vertices.
3.2 Spherical polyhedra
The purpose of this section is to define a second notion of interior angle for each face. These
angles can be defined in several equivalent ways, one of which is via spherical polyhedra.
A spherical polyhedron is an embedding of a planar graph into the unit sphere, so that
all edges are embedded as great circle arcs, and all regions are convex3. If 0 ∈ int(P ), we
can associate to P a spherical polyhedron PS by applying central projection
R3 \ {0} → S1(0), x 7→
x
∥x∥
to all its vertices and edges (this process is visualized below).
The vertices, edges and faces of P have spherical counterparts in PS obtained as pro-
jections onto the unit sphere. Those will be denoted with a superscript “S ”. For example,
if e ∈ F1(P ) is an edge of P , then eS denotes the corresponding “spherical edge”, which
is a great circle arc obtained as the projection of e onto the sphere.
We still need to justify that the spherical polyhedron of P is well-defined, by proving
that P contains the origin:
Proposition 3.7. 0 ∈ int(P ).
Proof. The proof proceeds in several steps.
Step 1: Fix a 1-vertex v ∈ V1 with neighbors w1, w2, w3 ∈ V2, and let ui := wi − v
be the direction of the edge conv{v, wi} emanating from v. Let σij ∈ F2(P ) denote the
2k-face containing v, wi and wj . The interior angle of σij at v is then ∡(ui, uj), which by
Proposition 2.8 and k ≥ 2 satisfies
∡(ui, uj) >
(
1− 1
k
)
π ≥ π
2
=⇒ ⟨ui, uj⟩ < 0.
Step 2: Besides v, the polyhedron P contains another 1-vertex v′ ∈ V1. It then
holds v′ ∈ v + cone{u1, u2, u3}, which means that there are non-negative coefficients
3Convexity on the sphere means that the shortest great circle arc connecting any two points in the region is
also contained in the region.
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 203
a1, a2, a3 ≥ 0, at least one positive, so that v + a1u1 + a2u2 + a3u3 = v′. Rearranging
and applying ⟨v, ·⟩ yields
a1⟨v, u1⟩+ a2⟨v, u2⟩+ a3⟨v, u3⟩ = ⟨v, v′⟩ − ⟨v, v⟩ (∗)
= r21 cos∡(v, v
′)− r21 < 0.
The value ⟨v, ui⟩ is independent of i (see Observation 2.6). Since there is at least one
positive coefficient ai, from (∗) follows ⟨v, ui⟩ < 0.4
Step 3: By the previous steps, {v, u1, u2, u3} is a set of four vectors with pair-wise neg-
ative inner product. The convex hull of such an arrangement in 3-dimensional Euclidean
space does necessarily contain the origin in its interior, or equivalently, there are positive
coefficients a0, ..., a3 > 0 with a0v + a1u1 + a2u2 + a3u3 = 0 (for a proof, see Proposi-
tion A.1). In other words: 0 ∈ v + int(cone{u1, u2, u3}).
Step 4: If H(σ) denotes the half-space associated with the face σ ∈ F2(P ), then
0 ∈ v + int(cone{u1, u2, u3}) =
⋂
σ∼v
int(H(σ)).
Thus, 0 ∈ int(H(σ)) for all faces σ incident to v. But since every face is incident to a
1-vertex, we obtain 0 ∈ int(H(σ)) for all σ ∈ F2(P ), and thus 0 ∈ int(P ) as well.
The main reason for introducing spherical polyhedra is that we can talk about the spher-
ical interior angles of their faces.
Let σ ∈ F2(P ) be a face, and v ∈ F0(σ) one of its vertices. Let α(σ, v) denote the
interior angle of σ at v, and β(σ, v) the spherical interior angle of σS at vS . It only needs
a straight-forward computation (involving some spherical geometry) to establish a direct
relation between these angles: e.g. if v is a 1-vertex, then
sin2(ℓS) · (1− cosβ(σ, v)) =
( ℓ
r2
)2
· (1− cosα(σ, v)),
where ℓS denotes the arc-length of an edge of PS (indeed, all edges are of the same length).
An equivalent formula exists for 2-vertices. The details of the computation are not of
relevance, but can be found in Appendix A.2.
The core message is that the value of α(σ, v) uniquely determines the value of β(σ, v)
and vice versa. In particular, since the value of α(σ, v) = αki does only depend on the
type of the face and the partition class of the vertex, so does β(σ, v), and it makes sense to
introduce the notion βki for the spherical interior angle of a 2k-gonal spherical face of P
S
at (the projection of) an i-vertex. Thus, we have
βk1i = β
k2
i ⇐⇒ α
k1
i = α
k2
i
3.2⇐⇒ k1 = k2, (3.1)
where we use Proposition 3.2 for the last equivalence.
Observation 3.8. The spherical interior angles βki have the following properties:
(i) The spherical interior angles surrounding a vertex add up to exactly 2π. That is, for
an i-vertex v ∈ F0(P ) of type (2k1, ..., 2ks) holds
βk1i + · · ·+ β
ks
i = 2π.
4Note that this provides the formal proof mentioned in Remark 2.3, namely, that the triangle conv{0, v1, v2}
is acute at v1 and v2.
204 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
(ii) The sum of interior angles of a spherical polygon always exceed the interior angle
sum of a respective flat polygon. That is, it holds
kβk1 + kβ
k
2 > 2(k − 1)π =⇒ βk1 + βk2 > 2
(
1− 1
k
)
π.
This has some consequences for the strictly bipartite polyhedron P :
Corollary 3.9. P contains at most two different types of 1-vertices, and if there are two,
then one is of the form (4, 4, 2k), and the other one is of the form (4, 6, 2k′) for distinct
k ̸= k′ and 2k′ ∈ {6, 8, 10}.
Proof. Each possible type listed in Table 1 is either of the form (4, 4, 2k) or of the form
(4, 6, 2k′) for some 2k ≥ 4 or 2k′ ∈ {6, 8, 10}.
If P contains simultaneously 1-vertices of type (4, 4, 2k1) and (4, 4, 2k2), apply Ob-
servation 3.8(i) to see
β21 + β
2
1 + β
k1
1
(i)
= β21 + β
2
1 + β
k2
1 =⇒ β
k1
1 = β
k2
1
(3.1)
=⇒ k1 = k2.
If P contains simultaneously 1-vertices of type (4, 6, 2k′1) and (4, 6, 2k
′
2), then
β21 + β
3
1 + β
k′1
1
(i)
= β21 + β
3
1 + β
k′2
1 =⇒ β
k′1
1 = β
k′2
1
(3.1)
=⇒ k′1 = k′2.
Finally, if P contains simultaneously 1-vertices of type (4, 4, 2k) and (4, 6, 2k′), then
β21 + β
2
1 + β
k
1
(i)
= β21 + β
3
1 + β
k′
1 =⇒ βk1 − βk
′
1 = β
3
1 − β21︸ ︷︷ ︸
̸= 0 by (3.1)
(3.1)
=⇒ k ̸= k′.
Since each edge of P is incident to a 1-vertex, we obtain
Observation 3.10. If P has only 1-vertices of types (4, 4, 2k) and (4, 6, 2k′), then each
edge of P is of one of the types
(4, 4), (4, 2k)︸ ︷︷ ︸
from a (4, 4, 2k)-vertex
, (4, 6), (4, 2k′) or (6, 2k′)︸ ︷︷ ︸
from a (4, 6, 2k′)-vertex
.
Corollary 3.11. The dihedral angle of an edge e ∈ F1(P ) of P only depends on its type.
Proof. Suppose that e is a (2k1, 2k2)-edge. Then e is incident to a 1-vertex v ∈ V1 of
type (2k1, 2k2, 2k3). By Observation 3.8(i) holds βk31 = 2π − βk11 − βk21 , which further
determines k3. By Proposition 3.2 we have uniquely determined interior angles αk11 , α
k2
1
and αk31 .
It is known that for a simple vertex (that is, a vertex of degree three) the interior angles
of the incident faces already determine the dihedral angles at the incident edges (for a
proof, see the Appendix, Proposition A.2). Consequently, the dihedral angle at e is already
determined.
The next result shows that Γ-permutahedra are the only bipartite polytopes in which a
1-vertex and a 2-vertex can have the same type.
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 205
Corollary 3.12. P cannot contain a 1-vertex and a 2-vertex of the same type.
Proof. Let v ∈ F0(P ) be a vertex of type (2k1, 2k2, 2k3). The incident edges are of type
(2k1, 2k2), (2k2, 2k3) and (2k3, 2k1) respectively. By Corollary 3.11 the dihedral angles
of these edges are uniquely determined, and since v is simple (that is, has degree three), the
interior angles of the incident faces are also uniquely determined (cf. Appendix, Proposition
A.2). In particular, we obtain the same angles independent of whether v is a 1-vertex or a
2-vertex.
A 1-vertex is always simple, and thus, a 1-vertex and a 2-vertex of the same type would
have the same interior angles at all incident faces, that is, αk1 = α
k
2 for each incident
2k-face. But this is not possible if P is strictly bipartite (by Proposition 2.5(ii) and Propo-
sition 2.8).
3.3 Adjacent pairs
Given a 1-vertex v ∈ V1 of type τ1 = (2k1, 2k2, 2k3), for any two distinct i, j ∈ {1, 2, 3},
v has a neighbor w ∈ V2 of type τ2 = (2ki, 2kj , ∗, ..., ∗), where ∗ are placeholders for
unknown entries. The pair of types
(τ1, τ2) = ((2k1, 2k2, 2k3), (2ki, 2kj , ∗, ..., ∗))
is called an adjacent pair of P . It is the purpose of this section to show that certain adjacent
pairs cannot occur in P . Excluding enough adjacent pairs for fixed τ1 then proves that the
type τ1 cannot occur as the type of a 1-vertex.
Our main tools for achieving this will be the inequalities established in Proposition 3.5 (i)
and (ii), that is
E(τ1)
(i)
< K(τ1)− 1 and E(τ2)
(ii)
> s− 2−K(τ2),
where s is the number of elements in τ2. For a warmup, and as a template for further
calculations, we prove that the adjacent pair (τ1, τ2) = ((4, 6, 8), (6, 8, 8)) will not occur
in P .
Example 3.13. By Proposition 3.5(i) we have
(∗) ϵ2 + ϵ3 + ϵ4 = E(τ1)
(i)
< K(τ1)− 1 =
1
2
+
1
3
+
1
4
− 1 = 1
12
.
On the other hand, by Proposition 3.5(ii) we have
(∗∗) 2
12
= 3− 2−
(1
3
+
1
4
+
1
4
)
= s− 2−K(τ2)
(ii)
< E(τ2) = ϵ3 + ϵ4︸ ︷︷ ︸
<1/12
+ ϵ4︸︷︷︸
<1/12
<
2
12
,
which is a contradiction. Hence this adjacent pair cannot occur. Note that we used (∗) to
upperbound certain sums of ϵi in (∗∗).
An adjacent pair excluded by using the inequalities from Proposition 3.5(i) and (ii) as
demonstrated in Example 3.13 will be called infeasible.
206 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
The argument applied in Example 3.13 will be repeated many times for many different
adjacent pairs in the upcoming Sections 3.5, 3.4, 3.6, 3.8, and we shall therefore use a
tabular form to abbreviate it. After fixing, τ1 = (4, 6, 8), the argument to refute the adjacent
pair (τ1, τ2) = ((4, 6, 8), (6, 8, 8)) is abbreviated in the first row of the following table:
τ2 s− 2−K(τ2)
?
< E(τ2)
(6, 8, 8) 2/12 ̸< (ϵ3 + ϵ4) + ϵ4 < 2/12
(6, 8, 6, 6) 9/12 ̸< (ϵ3 + ϵ4) + ϵ3 + ϵ3 < 3/12
The second row displays the analogue argument for another example, namely, the pair
((4, 6, 8), (6, 8, 6, 6)), showing that it is infeasible as well. Both rows will reappear in the
table of Section 3.5 where we exclude (4, 6, 8) as a type for 1-vertices entirely. Note that
the terms in the column below E(τ2) are grouped by parenthesis to indicate which subsums
are upper bounded via Proposition 3.5(i). In this example, if there are n groups, then the
sum is upper bounded by n/12.
The placeholders in an adjacent pair ((2k1, 2k2, 2k3), (2ki, 2kj , ∗, ..., ∗)) can, in theory,
be replaced by an arbitrary sequence of even numbers, and each such pair has to be refuted
separately. The following fact will make this task tractable: write τ ⊂ τ ′ if τ is a subtype
of τ ′, that is, a vertex type that can be obtained from τ ′ by removing some of its entries.
We then can prove
Proposition 3.14. If (τ1, τ2) is an infeasible adjacent pair, then the pair (τ1, τ ′2) is infea-
sible as well, for every τ ′2 ⊃ τ2.
Proof. Suppose τ2 = (2k1, ..., 2ks), τ ′2 = (2k1, ..., 2ks, 2ks+1, ..., 2ks′) ⊃ τ2, and that the
pair (τ1, τ ′2) is not infeasible. Then τ
′
2 satisfies Proposition 3.5(ii)
E(τ ′2) > s
′ − 2−K(τ ′2)
=⇒ E(τ2) > s− 2−K(τ2) +
s′∑
i=s+1
α
ki
2 /π>0︷ ︸︸ ︷(
1− 1
ki
− ϵki
)
> s− 2−K(τ2).
But this is exactly the statement that τ2 satisfies Proposition 3.5(ii) as well, i.e., that the pair
(τ1, τ2) is also not infeasible.
By Proposition 3.14 it is sufficient to exclude so-called minimal infeasible adjacent
pairs, that is, infeasible adjacent pairs (τ1, τ2) for which (τ1, τ ′2) is not infeasible for any
τ ′2 ⊂ τ2.
A second potential problem is, that we know little about the values that might replace
the placeholders in τ2 = (2ki, 2kj , ∗, ..., ∗). For our immediate goal, dealing with the
following special case is sufficient:
Proposition 3.15. The placeholders in an adjacent pair ((4, 6, 2k′), (6, 2k′, ∗, ..., ∗)) can
only contain 4, 6 and 2k′.
Proof. Suppose that P contains an adjacent pair
(τ1, τ2) = ((4, 6, 2k
′), (6, 2k′, 2k, ∗, ..., ∗))
induced by a 1-vertex v ∈ V1 of type τ1 with neighbor w ∈ V2 of type τ2. Suppose further
that 2k ̸∈ {4, 6, 2k′}. The vertex w is then incident to a 2k-face, and therefore also to a
1-vertex u ∈ V1 of type (4, 4, 2k) (u cannot be of type (4, 6, 2k) because of k ̸= k′ and
Corollary 3.9). This configuration is depicted below:
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 207
Note that w is also incident to a 4-face, and thus (6, 2k′, 2k, 4) ⊆ τ2.
By Proposition 3.5(i) the existence of 1-vertices of type (4, 4, 2k) and (4, 6, 2k′) yields
inequalities
ϵ2 + ϵ2 + ϵk <
1
k
and ϵ2 + ϵ3 + ϵk′ <
1
k′
− 1
6
. (3.2)
Since τ2 has τ := (6, 2k′, 2k, 4) as a subtype, by Proposition 3.14 it suffices to show that
the pair ((4, 6, 2k′), (6, 2k′, 2k, 4)) is infeasible. This follows via Proposition 3.5(ii):
7
6
− 1
k
− 1
k′
= 4− 2−K(τ)
(ii)
< E(τ) = ϵ2 + ϵ3 + ϵk′︸ ︷︷ ︸
<1/k′−1/6
+ ϵk︸︷︷︸
<1/k
(3.2)
<
1
k
+
1
k′
− 1
6
,
which rearranges to 1/k + 1/k′ > 2/3. Recalling 2k′ ∈ {6, 8, 10} =⇒ k′ ≥ 3 (from
Corollary 3.9) and 2k ̸∈ {4, 6, 2k′} =⇒ k ≥ 4 shows that this is not possible.
3.4 The case τ1 = (4, 6, 10)
If P contains a 1-vertex of type (4, 6, 10), then it contains an adjacent pair of the form
(τ1, τ2) = ((4, 6, 10), (6, 10, ∗, ..., ∗)).
We proceed as demonstrated in Example 3.13. Proposition 3.5(i) yields ϵ2+ϵ3+ϵ5 < 1/30.
By Proposition 3.15 the placeholders can only take on values 4, 6 or 10. The following table
lists the minimally infeasible adjacent pairs and proves their infeasibility.
τ2 s− 2−K(τ2)
?
< E(τ2)
(6, 10, 6) 4/30 ̸< (ϵ3 + ϵ5) + ϵ3 < 2/30
(6, 10, 10) 8/30 ̸< (ϵ3 + ϵ5) + ϵ5 < 2/30
(6, 10, 4, 4) 14/30 ̸< (ϵ2 + ϵ3 + ϵ5) + ϵ2 < 2/30
By Proposition 3.14 we conclude: the placeholder in τ2 = (6, 10, ∗, ..., ∗) can contain
no 6 or 10, and at most one 4. This leaves us with the option τ2 = (4, 6, 10), which is the
same as τ1 and therefore not possible by Corollary 3.12. Therefore, P cannot contain a
1-vertex of type (4, 6, 10).
3.5 The case τ1 = (4, 6, 8)
If P contains a 1-vertex of type (4, 6, 8), then it also contains an adjacent pair of the form
(τ1, τ2) = ((4, 6, 8), (6, 8, ∗, ..., ∗)).
We proceed as demonstrated in Example 3.13. Proposition 3.5(i) yields ϵ2+ϵ3+ϵ4 < 1/12.
By Proposition 3.15 the placeholders can only take on values 4, 6 or 8. The following table
lists the minimally infeasible adjacent pairs and proves their infeasibility.
208 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
τ2 s− 2−K(τ2)
?
< E(τ2)
(6, 8, 8) 2/12 ̸< (ϵ3 + ϵ4) + ϵ3 < 2/12
(6, 8, 4, 4) 5/12 ̸< (ϵ2 + ϵ3 + ϵ4) + ϵ2 < 2/12
(6, 8, 4, 6) 7/12 ̸< (ϵ2 + ϵ3 + ϵ4) + ϵ3 < 2/12
(6, 8, 6, 6) 9/12 ̸< (ϵ2 + ϵ3 + ϵ4) + ϵ3 + ϵ3 < 3/12
By Proposition 3.14 we conclude: the placeholder in τ2 = (6, 8, ∗, ..., ∗) can contain no 8,
and at most one 4 or 6, but not both at the same time.
This leaves us with the options τ2 = (4, 6, 8) and τ2 = (6, 6, 8). In the first case,
τ1 = τ2 which not possible by Corollary 3.12. In the second case, there would be two
adjacent 6-faces, but P does not contain (6, 6)-edges by Observation 3.10 with 2k′ = 8.
Therefore, P cannot contain a 1-vertex of type (4, 6, 8).
3.6 The case τ1 = (4, 6, 6)
If P contains a 1-vertex of type (4, 6, 6), then it also contains an adjacent pair of the form
(τ1, τ2) = ((4, 6, 6), (6, 6, ∗, ..., ∗)).
We proceed as demonstrated in Example 3.13. Proposition 3.5(i) yields ϵ2+ϵ3+ϵ3 < 1/6.
By Proposition 3.15 the placeholders can only take on values 4 or 6. The following table
lists the minimally infeasible adjacent pairs and proves their infeasibility.
τ2 s− 2−K(τ2)
?
< E(τ2)
(6, 6, 4, 4) 2/6 ̸< (ϵ2 + ϵ3 + ϵ3) + ϵ2 < 2/6
(6, 6, 6, 4) 3/6 ̸< (ϵ2 + ϵ3 + ϵ3) + ϵ3 < 2/6
(6, 6, 6, 6) 4/6 ̸< (ϵ3 + ϵ3) + (ϵ3 + ϵ3) < 2/6
By Proposition 3.14 we conclude: the placeholder in τ2 = (6, 6, ∗, ..., ∗) can contain at
most one 4 or 6, but not both at the same time.
This leaves us with the options τ2 = (4, 6, 6) and τ2 = (6, 6, 6). In the first case we
have τ1 = τ2, which is not possible by Corollary 3.12. Excluding (6, 6, 6) needs more
work: fix a 6-gon σ ∈ F2(P ). Each edge of σ is either of type (4, 6) or of type (6, 6) (by
Observation 3.10). Each 1-vertex of σ (which must be of type (4, 6, 6)) is then incident to
exactly one of these (6, 6)-edges of σ. Thus, there are exactly three (6, 6)-edges incident
to σ (see Figure 3). On the other hand, each 2-vertex of σ is incident to an even number
of (6, 6)-edges of σ (since if a 2-vertex is incident to at least one (6, 6)-edge, then we
have previously shown that its type must be (6, 6, 6), implying another incident (6, 6)-
edge). Therefore the number of (6, 6)-edges incident to σ must be even (see Figure 3), in
contradiction to the previously obtained number three of such edges.
Consequently, P cannot contain a 1-vertex of type (4, 6, 6).
Observation 3.16. It is a consequence of Sections 3.6, 3.5, 3.4 that P cannot have a 1-
vertex of a type (4, 6, 2k′) for a 2k′ ∈ {6, 8, 10}. By Corollary 3.9 this means that all
1-vertices of P are of the same type τ1 = (4, 4, 2k) for some fixed 2k ≥ 4.
It is worth to distinguish the case (4, 4, 4) from the cases (4, 4, 2k) with 2k ≥ 6.
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 209
Figure 3: Possible distributions of (4, 6)-edges (gray) and (6, 6)-edges (thick) around a
6-gon as discussed in Section 3.6. The top row shows configurations compatible with the
conditions set by 1-vertices (black), and the bottom row shows the configurations compat-
ible with the conditions set by the 2-vertices (white).
3.7 The case τ1 = (4, 4, 4)
In this case, all 2-faces are 4-gons, and all 4-gons are congruent by Proposition 3.2. A
4-gon with all edges of the same length is known as a rhombus, and the polyhedra with
congruent rhombic faces are known as rhombic isohedra (from german Rhombenisoeder).
These have a known classification:
Theorem 3.17 (S. Bilinksi, 1960 [2]). If P is a polyhedron with congruent rhombic faces,
then P is one of the following:
(i) a member of the infinite family of rhombic hexahedra, i.e., P can be obtained from a
cube by stretching or squeezing it along a long diagonal,
(ii) the rhombic dodecahedron,
(iii) the Bilinski dodecahedron,
(iv) the rhombic icosahedron, or
(v) the rhombic triacontahedron.
The figure below depicts these polyhedra in the given order (from left to right; including
only one instance from the family (i)):
The rhombic dodecahedron and triacontahedron are known edge- but not vertex-transitive
polytopes. We show that the others are not even strictly bipartite.
Corollary 3.18. If P is strictly bipartite with all 1-vertices of type (4, 4, 4), then P is one
of the following:
(i) the rhombic dodecahedron,
210 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
(ii) the rhombic triacontahedron.
Proof. The listed ones are edge-transitive but not vertex-transitive. Also they are not in-
scribed. By Proposition 2.4 they are therefore strictly bipartite.
We then have to exclude the other polyhedra listed in Theorem 3.17. The rhombic
hexahedra include the cube, which is inscribed, hence not strictly bipartite. In all the other
cases, there exist vertices where acute and obtuse angles meet (see the figure). So this vertex
cannot be assigned to either V1 or V2 (cf. Observation 2.9), and the polyhedron cannot be
bipartite.
These are the only strictly bipartite polyhedra we will find, and both are edge-transitive
without being vertex-transitive.
3.8 The case τ1 = (4, 4, 2k), 2k ≥ 6
If P contains a 1-vertex of type (4, 4, 2k) with 2k ≥ 6, then it also has an adjacent pair of
the form
(τ1, τ2) = ((4, 4, 2k), (4, 2k, ∗, ..., ∗)).
We proceed as demonstrated in Example 3.13. Proposition 3.5(i) yields ϵ2+ϵ2+ϵk < 1/k.
Since (4, 4, 2k) is the only type of 1-vertex of P , there are only 4-faces and 2k-faces and the
placeholders can only take on the values 4 and 2k (note that we do not use Proposition 3.15
for this). The following table lists some inequalities derived for infeasible pairs:
τ2 s− 2−K(τ2)
?
< E(τ2)
(4, 2k, 4, 4, 4) 1− 1/k < (ϵ2 + ϵ2 + ϵk) + (ϵ2 + ϵ2) < 2/k
(4, 2k, 4, 4, 2k) 3/2− 2/k < (ϵ2 + ϵ2 + ϵk) + (ϵ2 + ϵk) < 2/k
One checks that these inequalities are not satisfied for 2k ≥ 6. Proposition 3.14 then states
that the placeholders can contain at most two 4-s, and if exactly two, then nothing else.
Moreover, τ2 must contain at least as many 4-s as it contains 2k-s, as otherwise we would
find two adjacent 2k-faces while P cannot contain a (2k, 2k)-edge by Observation 3.10.
We are therefore left with the following options for τ2:
(4, 4, 2k), (4, 4, 4, 2k) and (4, 2k, 4, 2k).
The case τ2 = (4, 4, 2k) is impossible by Corollary 3.12. We show that τ2 = (4, 4, 4, 2k)
is also not possible: consider the local neighborhood of a (4, 4, 4, 2k)-vertex (the high-
lighted vertex) in the following figure:
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 211
Since the 1-vertices (black dots) are of type (4, 4, 6), this configuration forces on us the ex-
istence of the two gray 6-faces. These two faces intersect in a 2-vertex, which is then
incident to two 2k-faces and must be of type (4, 2k, 4, 2k). But we can show that the types
(4, 4, 4, 2k) and (4, 2k, 4, 2k) are incompatible by Observation 3.8(i):
β22 + β
2
2 + β
2
2 + β
k
2
(i)
= β22 + β
k
2 + β
2
2 + β
k
2 =⇒ β22 = βk2
(3.1)
=⇒ 4 = 2k ≥ 6.
Thus, (4, 4, 4, 2k) cannot occur.
We conclude that every 2-vertex incident to a 2k-face must be of type (4, 2k, 4, 2k).
Consider then the following table:
τ2 s− 2−K(τ2)
?
< E(τ2)
(4, 2k, 4, 2k) 1− 2/k < (ϵ2 + ϵ2 + ϵk) + ϵ2 < 2/k
The established inequality yields 2k ≤ 6, and hence 2k = 6. We found that then all 1-
vertices must be of type (4, 4, 6), and all 2-vertices incident to a 6-face must be of type
(4, 6, 4, 6).
3.9 The case τ1 = (4, 4, 6)
At this point we can now assume that all 1-vertices of P are of type (4, 4, 6) and that each
2-vertex of P that is incident to a 6-face is of type (4, 6, 4, 6). In particular, P contains a
2-vertex w ∈ V2 of this type. Since there is no (6, 6)-edge in P , the two 6-faces incident to
w cannot be adjacent. In other words, the faces around w must occur alternatingly of type
4 and type 6, which is the reason for writing the type (4, 6, 4, 6) with alternating entries.
On the other hand, P contains (4, 4)-edges, and none of these is incident to a (4, 6, 4, 6)-
vertex surrounded by alternating faces. Thus, there must be further 2-vertices of a type
other than (4, 6, 4, 6), necessarily not incident to any 6-face. These must then be of type
(4r) := (4, ..., 4︸ ︷︷ ︸
r
), for some r ≥ 3.
Proposition 3.19. r = 5.
Proof. If there is a (4r)-vertex, Observation 3.8(i) yields β22 = 2π/r. Analogously, from the
existence of a (4, 6, 4, 6)-vertex follows
2β22 + 2β
3
2
(i)
= 2π =⇒ β32 =
2π − 2β22
2
=
(
1− 2
r
)
π.
Recall kβk1 + kβ
k
2 > 2π(k − 1) from Observation 3.8(ii). Together with the previously
established values for β22 and β
3
2 , this yields
β21 >
2π(2− 1)− 2β22
2
=
(
1− 2
r
)
π, and
β31 >
2π(3− 1)− 3β32
3
=
(1
3
+
2
r
)
π.
(3.3)
Since the 1-vertices are of type (4, 4, 6), Observation 3.8(i) yields
2π
(i)
= 2β21 + β
3
1
(3.3)
> 2
(
1− 2
r
)
π +
(1
3
+
2
r
)
π =
(7
3
− 2
r
)
π.
212 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
Figure 4: The edge-graph of the final candidate polyhedron.
And one checks that this rearranges to r < 6.
This leaves us with the options r ∈ {3, 4, 5}. If r = 4, then β32 = π/2 = β22 ,
which is impossible by Equation (3.1). And if r = 3, then (3.3) yields β31 > π, which is
also impossible for a convex face of a spherical polyhedron. We are left with r = 5.
To summarize: P is a strictly bipartite polyhedron in which all 1-vertices are of type
(4, 4, 6), and all 2-vertices are of types (4, 6, 4, 6) or (45), and both types actually occur in
P . This information turns out to be sufficient to uniquely determine the edge-graph of P ,
which is shown in Figure 4.
This graph can be constructed by starting with a hexagon in the center with vertices
of alternating colors (indicating the partition classes). One then successively adds further
faces (according to the structural properties determined above), layer by layer. This process
involves no choice and thus the result is unique.
As mentioned in Remark 2.3, a bipartite polyhedron has an edge in-sphere. Thus, P is
a polyhedral realization of the graph in Figure 4 with an edge in-sphere. It is known that
any two such realizations are related by a projective transformation [8]. One representative
Q ⊂ R3 from this class (which we do not yet claim to coincide with P ) can be constructed
by applying the following operation ⋆ to each vertex of the regular icosahedron:
The operation is performed in such a way, so that
• the five new “outer” vertices of the new 4-gons are positioned in the centers of edges
of the icosahedron.
• the edges of each new 4-gon are tangent to a common sphere centered at the center
of the icosahedron
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 213
The resulting polyhedron Q looks as follows:
One can verify that Q has indeed the desired edge-graph.
It is clear from the construction that Q has an edge in-sphere, and any two of its 4-
gonal or 6-gonal faces are congruent (as we would expect from a bipartite polyhedron).
Like-wise, P has an edge in-sphere and the same edge-graph. Hence, P must be a pro-
jective transformation of Q. However, any projective transformation that is not just a re-
orientation or a uniform rescaling will inevitably destroy the property of congruent faces.
In conclusion, we can assume that P is identical to Q (up to scale and orientation).
It remains to check whether Q is indeed a bipartite polyhedron. For this, recall that any
two of the following properties imply the third (cf. Remark 2.3):
(i) Q has an edge in-sphere.
(ii) Q has all edges of the same length.
(iii) for each vertex v ∈ F0(Q), the distance ∥v∥ only depends on the partition class of
the vertex.
Now, Q satisfies (i) by construction, and it would need to satisfy both (ii) and (iii) in order
to be bipartite. The figure certainly suggests that all edges of Q are of the same length.
However, as we shall show now, Q cannot satisfy both (ii) and (iii) at the same time, and
thus, can satisfy neither. In particular, the edges must have a tiny difference in length that
cannot be spotted visually, making Q into a remarkable near-miss (we will quantify this
below).
For what follows, let us assume that (ii) holds, that is, that all edges of Q are of the same
length, in particular, that all 4-gons are rhombuses. Our goal is to show that ∥v∥ depends
on the type of the vertex v ∈ V2 (not only its partition class), establishing that (iii) does not
hold.
For this, start from the following well-known construction of the regular icosahedron
from the cube of edge-length 2 centered at the origin.
214 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
The construction is as follows: insert a line segment in the center of each face of the cube
as shown in the left image. Each line segment is of length 2φ, where φ ≈ 0.61803 is the
positive solution of φ2 = 1−φ (one of the numbers commonly knows as the golden ratio).
The convex hull of these line segments gives the icosahedron with edge length 2φ.
It is now sufficient to consider a single vertex of the icosahedron together with its inci-
dent faces. The image below shows this vertex after we applied ⋆.
The image on the right is the orthogonal projection of the configuration on the left onto
the yz-plane. This projection makes it especially easy to give 2D-coordinates for several
important points:
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 215
The points A and C are 2-vertices of Q of type (45) and (4, 6, 4, 6) respectively. Both
points and the origin O are contained in the yz-plane onto which we projected. Conse-
quently, distances between these points are preserved during the projection, and assuming
that Q is bipartite, we would expect to find |OA| = |OC| = r2. We shall see that this
is not the case, by explicitly computing the coordinates of A and C in the new coordinate
system (y, z).
By construction, C = (0, 1) and |OC| = 1. Other points with easily determined
coordinates are P, Q, R, S, T (the midpoint of R and S) and U (the midpoint of Q and
S).
By construction, the point B lies on the line segment QT. The parallel projection
of a rhombus is a (potentially degenerated) parallelogram, and thus, opposite edges in the
projection are still parallel. Hence, the gray edges in the figure are parallel. For that reason,
the segment UB is parallel to PQ. This information suffices to determine the coordinates
of B, which is now the intersection of QT with the parallel of PQ through U. The
coordinates are given in the figure.
The rhombus containing the vertices A, B and C degenerated to a line. Its fourth
vertex is also located at B. Therefore, the segments CB and BA are translates of each
other. Since the point B and the segment CB are known, this allows the computation of
the coordinates of A as given in the figure.
We can finally compute |OA|. For this, recall (∗)φ2n = F2n−2 − φF2n−1, where Fn
denotes the n-th Fibonacci number with initial conditions F0 = F1 = 1. Then
|OA|2 = (4φ− 3)2 + (3φ− 1)2
= 25φ2 − 30φ+ 10
(∗)
= 25(1− φ)− 30φ+ 10
= 35− 55φ
= 1 + (34− 55φ) (∗)= 1 + φ10 > 1,
and thus, Q cannot be bipartite. Remarkably, we find that
|OA| =
√
1 + φ10 ≈ 1.00405707
is only about 0.4% larger than |OC| = 1, and so while Q is not bipartite, it is a remarkable
near-miss.
216 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
Since P was assumed to be bipartite, but was also shown to be identical to Q, we
reached a contradiction, which finally proves Theorem 2.13, and the goal of the paper is
achieved.
4 Conclusions and open questions
In this paper we have shown that any edge-transitive (convex) polytope in four or more
dimensions is necessarily vertex-transitive. We have done this by classifying all polytopes
which simultaneously have all edges of the same length, an edge in-sphere and a bipartite
edge graph (which we named bipartite polytopes).
The obstructions we derived for being edge-transitive without being vertex-transitive
have been primarily geometric and less a matter of symmetry (a detailed investigation of
the Euclidean symmetry groups was not necessary, but it might be interesting to view the
problem from this perspective). We suspect that dropping convexity or considering com-
binatorial symmetries instead of geometric ones will quickly lead to examples of “just
edge-transitive structures”. For example, it is easy to find embeddings of graphs into Rd
with these properties.
Slightly stronger than being simultaneously vertex- and edge-transitive is being tran-
sitive on arcs, that is, on incident vertex-edge pairs. This additional degree of symmetry
allows an edge to be not only mapped onto any other edge, but also onto itself with reversed
orientation. While there are graphs that are vertex- and edge-transitive without being arc-
transitive (the so-called half-transitive graphs, see [7]), we believe it is unlikely that this
distinction is necessary for convex polytopes.
Question 4.1. Is there a polytope P ⊂ Rd that is edge-transitive and vertex-transitive,
but not arc-transitive?
In a different direction, the questions of this paper naturally generalize to faces of higher
dimensions. In general, the interactions between transitivities of faces of different dimen-
sions have been little investigated. For example, already the following question seems to
be open:
Question 4.2. For fixed k ∈ {2, ..., d − 3}, are there convex d-polytopes for arbitrarily
large d ∈ N that are transitive on k-dimensional faces without being transitive on either
vertices or facets?
Of course, any such question could be attacked by attempting to classify the k-face-
transitive (convex) polytopes for some k ∈ {1, ..., d− 2}. It seems to be unclear for which
k this problem is tractable (for comparison, k = 0 is intractable, see [1]), and it appears
that there are no techniques applicable to all (or many) k at the same time.
ORCID iDs
Frank Göring https://orcid.org/0000-0001-8331-2138
Martin Winter https://orcid.org/0000-0002-3817-9494
References
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218 Ars Math. Contemp. 23 (2023) #P2.01 / 191–219
A
A.1 Geometry
Proposition A.1. Given a set x0, ..., xd ∈ Rd\{0} of d+1 vectors with pair-wise negative
inner product, then there are positive coefficients α0, ..., αd > 0 with
α0x0 + · · ·+ αdxd = 0.
Proof. We proceed by induction. The induction base d = 1 which is trivially true.
Now suppose d ≥ 2, and, W.l.o.g. assume ∥x0∥ = 1. Let π0 be the orthogonal projec-
tion onto x⊥0 , that is, π0(u) := u− x0⟨x0, u⟩. In particular, for i ̸= j and i, j > 0
⟨π0(xi), π0(xj)⟩ = ⟨xi, xj⟩︸ ︷︷ ︸
<0
−⟨x0, xi⟩︸ ︷︷ ︸
<0
⟨x0, xj⟩︸ ︷︷ ︸
<0
< 0.
Then {π(x1), ..., π0(xd)} is a set of d vectors in x⊥0 ∼= Rd−1 with pair-wise negative inner
product. By induction assumption there are positive coefficients α1, ..., αd > 0 so that
α1π0(x1) + · · ·+ αdπ0(xd) = 0.
Set α0 := −⟨x0, α1x1+ · · ·+αdxd⟩ > 0. We claim that x := x0α0+ · · ·+αdxd = 0.
Since Rd = span{x0} ⊕ x⊥0 , it suffices to check that ⟨x0, x⟩ = 0 as well as π0(x) = 0.
This follows:
⟨x0, x⟩ = α0 ⟨x0, x0⟩︸ ︷︷ ︸
=1
+ ⟨x0, α1x1 + · · ·+ αdxd⟩︸ ︷︷ ︸
=−α0
= 0,
π0(x) = α0 π0(x0)︸ ︷︷ ︸
=0
+α1π0(x1) + · · ·+ αdπ0(xd)︸ ︷︷ ︸
=0
= 0.
Proposition A.2. Let P ⊂ R3 be a polyhedron with v ∈ F0(P ) a vertex of degree three.
The interior angles of the faces incident to v determine the dihedral angles at the edges
incident to v and vice versa.
Proof. For w1, w2, w3 ∈ F0(P ) the neighbors of v, let ui := wi − v denote the direction
of the edge ei from v to wi. Let σij be the face that contains v, wi and wj . Then ∡(ui, uj)
is the interior angle of σij at v.
The set {u1, u2, u3} is uniquely determined (up to some orthogonal transformation) by
the angles ∡(ui, uj). Furthermore, since P is convex, {u1, u2, u3} forms a basis of R3, and
this uniquely determines the dual basis {n12, n23, n31} for which ⟨nij , ui⟩ = ⟨nij , uj⟩ =
0. In other words, nij is a normal vector to σij . The dihedral angle at the edge ej is
then π − ∡(nij , njk), hence uniquely determined. The other direction is analogous, via
constructing {u1, u2, u3} as the dual basis to the set of normal vectors.
A.2 Computations
The edge lengths in a spherical polyhedron are measured as angles between its end ver-
tices. Consider adjacent vertices vS1 , v
S
2 ∈ F0(PS), then the incident edge has (arc-)length
ℓS := ∡(vS1 , v
S
2 ) = ∡(v1, v2).
It follows from Observation 2.6 that these angles are completely determined by the
parameters, hence the same for all edges of PS .
F. Göring and M. Winter: The edge-transitive polytopes that are not vertex-transitive 219
Proposition A.3. For a face σ ∈ F2(P ) and a vertex v ∈ F0(σ), there is a direct relation-
ship between the value of α(σ, v) and the value of β(σ, v).
Proof. Let w1, w2 ∈ V2 be the neighbors of v in the 2k-face σ, and set ui := wi − v. Then
∡(u1, u2) = α(σ, v). W.l.o.g. assume that v is a 1-vertex (the argument is equivalent for a
2-vertex).
For convenience, we introduce the notation χ(θ) := 1− cos(θ). We find that
(∗) 2ℓ2 · χ(α(σ, v)) = ℓ2 + ℓ2 − 2ℓ2 cos(∡(u1, u2))
= ∥u1∥2 + ∥u2∥2 − 2⟨u1, u2⟩
= ∥u1 − u2∥2 = ∥w1 − w2∥2
= ∥w1∥2 + ∥w2∥2 − 2⟨w1, w2⟩
= r22 + r
2
2 − r22 cos∡(w1, w2) = 2r22 · χ(∡(w1, w2)).
The side lengths of the spherical triangle wS1 v
SwS2 are ∡(w1, w2), ℓ
S and ℓS . By the
spherical law of cosine5 we obtain
cos∡(w1, w2) = cos(ℓ
S) cos(ℓS) + sin(ℓS) sin(ℓS) cos(β(σ, v))
= cos2(ℓS) + sin2(ℓS)(cos(β(σ, v))− 1 + 1)
= [cos2(ℓS) + sin2(ℓS)] + sin2(ℓS)(cos(β(σ, v))− 1)
= 1− sin2(ℓs) · χ(β(σ, v))
=⇒ sin2(ℓS) · χ(β(σ, v)) = χ(∡(w1, w2))
(∗)
=
( ℓ
r2
)2
· χ(α(σ, v)).
5cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(γ), where a, b and c are the side lengths (arc-lengths) of a
spherical triangle, and γ is the interior angle opposite to the side of length c.

ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.02 / 221–237
https://doi.org/10.26493/1855-3974.2691.0b7
(Also available at http://amc-journal.eu)
Double generalized majorization and
diagrammatics*
Marija Dodig †
CEAFEL, Departamento de Matématica, Universidade de Lisboa, Edificio C6,
Campo Grande, 1749-016 Lisbon, Portugal, and
Mathematical Institute SANU, Knez Mihajlova 36, 11000 Belgrade, Serbia
Marko Stošić
CAMGSD, Departamento de Matemática, Instituto Superior Técnico,
Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal, and
Mathematical Institute SANU, Knez Mihajlova 36, 11000 Belgrade, Serbia
Received 7 September 2021, accepted 16 May 2022, published online 11 November 2022
Abstract
In this paper we show that the generalized majorization of partitions of integers has a
surprising completing-squares property. Together with the previously obtained transitivity-
like property, this enables a compelling diagrammatical interpretation. Apart from purely
combinatorial interest, the main result has applications in matrix completion problems, and
representation theory of quivers.
Keywords: Partitions, majorization, diagrammatics, inequalities.
Math. Subj. Class. (2020): 05A17
1 Introduction
By a partition we mean a finite non-increasing sequence of integers. Let a1 ≥ . . . ≥ as be
integers, then we can define the corresponding partition a = (a1, . . . , as). For a partition
*We would like to thank the Referee for the valuable comments and suggestions. This work was done within
the activities of CEAFEL and was partially supported by FCT, project UIDB/04721/2020, and Exploratory Grant
EXPL/MAT-PUR/0584/2021. In the final stages, this work was also partially supported by the Science Fund of
the Republic of Serbia, Projects no. 7744592, MEGIC – “Integrability and Extremal Problems in Mechanics,
Geometry and Combinatorics” (M.D.) and no. 7749891, GWORDS – “Graphical Languages” (M.S.).
†Corresponding author.
E-mail addresses: msdodig@fc.ul.pt (Marija Dodig), mstosic@isr.ist.utl.pt (Marko Stošić)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
222 Ars Math. Contemp. 23 (2023) #P2.02 / 221–237
a = (a1, . . . , as) we shall assume that ai := +∞, for i ≤ 0, and ai := −∞, for i > s.
The following notation will be used throughout the paper
a = (a1, . . . , as), (1.1)
b = (b1, . . . , bk), (1.2)
c = (c1, . . . , cm), (1.3)
d = (d1, . . . , dm+s), (1.4)
g = (g1, . . . , gm+k), (1.5)
f = (f1, . . . , fm+k+s). (1.6)
Arguably, the most famous comparison between two partitions of integers is a classical
majorization in Hardy-Littlewood-Polya sense [16]. In this paper we deal with its gener-
alisation given in [2, 9, 10]. More precisely, we compare three partitions of integers in the
following way:
Definition 1.1. Let b, c and g be partitions (1.2), (1.3) and (1.5), respectively. If
ci ≥ gi+k, i = 1, . . . ,m, (1.7)
hj∑
i=1
gi −
hj−j∑
i=1
ci ≤
j∑
i=1
bi, j = 1, . . . , k (1.8)
m+k∑
i=1
gi =
m∑
i=1
ci +
k∑
i=1
bi, (1.9)
where
hj := min{i|ci−j+1 < gi}, j = 1, . . . , k,
then we say that g is majorized by c and b. This type of majorization we call the general-
ized majorization, and we write
g ≺′ (c,b).
The generalized majorization generalizes the classical majorization. Indeed, if m = 0,
i.e. if the partition c is empty, the generalized majorization becomes the classical majoriza-
tion between the partitions g and b. Many intrinsic, purely combinatorial properties of
generalized majorization, including generalizations of some of the well-known properties
of the classical majorization, have been obtained in [10, 11, 14]. These results demonstrate
rich structure of generalized majorization as an independent combinatorial object.
Apart from purely combinatorial interest, this relationship between three partitions of
integers naturally appears in Matrix and Matrix Pencils completion problems [2, 7, 9, 12],
as well as in Representation Theory of Quivers [22], and Perturbation Theory [1, 12].
In this paper we go further, and show that generalized majorization, apart from transitivity-
like property that has been shown in [10, Theorem 8], also has certain completing-squares
property. This novel property of generalized majorization is motivated by the study of two
problems given below, that naturally appear both from matrix pencils completions, and
representation theory of quivers point of view.
The first problem has appeared in [9, 11] and turned out to be very challenging and the
key point in solving many perturbation and completion problems of Matrix Pencils, see
e.g. [6, 7, 9, 12, 13].
M. Dodig and M. Stošić: Double generalized majorization and diagrammatics 223
Problem 1.2 (Double general majorization problem). Let a, b, d, and g be partitions
(1.1), (1.2), (1.4) and (1.5). Find necessary and sufficient conditions for the existence of a
partition f = (f1, . . . , fm+k+s), such that
f ≺′ (d,b) and f ≺′ (g,a). (1.10)
We note that in the case of classical majorization there always exists a minimal partition
of a given sum, i.e. for any two partitions of the same length and total sum, there exists
a partition that is majorized by both of them. However, here the problem is much more
complicated, and involved. A complete solution to Problem 1.2 was obtained in [11, 14]:
Theorem 1.3 ([14, Theorem 3]). Let a, b, d, and g be partitions (1.1), (1.2), (1.4) and
(1.5). There exists a partition f = (f1, . . . , fm+k+s), such that
f ≺′ (d,b) and f ≺′ (g,a)
if and only if
m+s∑
i=1
di +
k∑
i=1
bi =
m+k∑
i=1
gi +
s∑
i=1
ai
and the condition Ω̄(g,d,b,a) holds.
The explicit form of the condition Ω̄(g,d,b,a) is given in [11, 14], and consists of
inequalities between the elements of the partitions g,d,b,a. These involve very technical
explicit definition of certain sets S and ∆, and we dismiss it here. We refer the interested
reader to [11, 14] for all details and properties on these sets, and for the explicit form of Ω̄.
The second problem has showed its importance when studying bounded rank one per-
turbations of matrix pencil [12]. Also, it naturally appears in the study of the possible
Kronecker invariants of a partially prescribed Matrix Pencil, see e.g. [13, 17]. Apart of the
case k = s = 1 which has been solved in [12], the following problem is still open:
Problem 1.4 (Pseudo double majorization problem). Let a, b, d, and g be partitions (1.1),
(1.2), (1.4) and (1.5). Find necessary and sufficient conditions for the existence of a parti-
tion c = (c1, . . . , cm), such that
g ≺′ (c,b) and d ≺′ (c,a). (1.11)
The goal of the paper is to prove the relationship between the double majorization Prob-
lem 1.2 and pseudo double majorization Problem 1.4. In Theorem 3.2, as the main result
of the paper, we prove that Problem 1.4 implies Problem 1.2. That is, we prove that for
four partitions a, b, d, and g as in (1.1), (1.2), (1.4) and (1.5), the existence of a partition
c satisfying (1.11) implies the existence of a partition f satisfying (1.10). In addition, we
explicitly construct such partition f . This is a surprising, and nontrivial property of the
generalized majorization. Also, in Section 4 we give a counterexample that the converse
does not hold.
224 Ars Math. Contemp. 23 (2023) #P2.02 / 221–237
This purely combinatorial result has several interpretations. First, let us introduce some
diagrammatics into the story, and denote general majorization by an arrow, i.e. let us denote
g ≺′ (c,b)
by
c
g
b
Now, as a direct corollary to our main result we obtain the following commutative diamond-
like diagram:
c
f
g d
b
a
a
b
In other words, the lower half of the square (represented by full lines) can always be com-
pleted to a full square. More details on diagrammatics are given in Section 4.2.
In addition, the above completion up to a commutative diagram is related to various
classical Linear Algebra problems. First of all, both Problems 1.2 and 1.4 naturally appear
as cornerstones in solving the classical General Matrix Pencils Completion Problem [17].
In particular, a solution to Problem 1.2 is a key result in obtaining a full description of the
possible Kronecker invariants of a quasi-regular matrix pencil with a prescribed subpencil
in [13]. For similar contributions and importance of Problems 1.2 and 1.4 in matrix pencils
completion problems see [6, 7, 9]. The close relationship between Problems 1.2 and 1.4
obtained in this paper, should have a significant impact in obtaining a complete solution of
the General Matrix Pencils Completion Problem. Similar applications are expected in the
study of representation of Kronecker quivers, [22].
Another area of applications of results on generalized majorizations is in Bounded Rank
Perturbation problems [3–5,18–21]. In the case when partitions a and b are both of length
one (i.e. when s = k = 1), Problems 1.2 and 1.4 have been addressed and solved separately
in [12], and were crucial in solving the rank one perturbation problem for matrix pencils. It
is expected that the main result of this paper should lead to a solution of the arbitrary rank
perturbation problem in the future. Some steps in this direction have already been done
in [8]. Indeed, in [8] we have studied and resolved the classical bounded rank perturbation
M. Dodig and M. Stošić: Double generalized majorization and diagrammatics 225
problem for quasi-regular matrix pencils (pencils with full normal rank). For all details
on matrix pencils see [15]. This is a very general result in low rank perturbation theory,
and has been open for a long time. The milestone in its solution is the main result of the
paper – Theorem 3.2. It allows to choose a special, preferred form of the low rank matrix
pencil that performs the perturbation. We expect more impact of Theorem 3.2 in the study
of bounded rank perturbations of different classes of matrix pencils in the future.
2 Partitions and generalized majorization
For any two partitions a = (a1, . . . , as) and b = (b1, . . . , bk) by a∪b we mean a partition
obtained as a non-increasing ordering of {a1, . . . , as, b1, . . . , bk}. If a > b are nonnegative
integers, then we assume
∑b
i=a ai := 0.
Now we shall list some of the basic properties of the auxiliary numbers, hj , that ap-
pear in the definition of the generalized majorization. Below we use the notation from
Definition 1.1.
Since hj = min{i|ci−j+1 < gi}, for j = 1, . . . , k, we have
m+ k + 1 > hk > · · · > h2 > h1 > 0, (2.1)
and so in particular
hj ≥ j, j = 1, . . . ,m+ k. (2.2)
Also from the definition of hj we have
ci−j+1 ≥ gi, for i < hj , for any j = 1, . . . , k. (2.3)
We notice that in Definition 1.1, if (1.9) is satisfied, then (1.8) is equivalent to the following:
m+k∑
i=hj+1
gi ≥
m∑
i=hj−j+1
ci +
k∑
i=j+1
bi, j = 1, . . . , k. (2.4)
The generalized majorization implies weak majorization given by the following definition:
Definition 2.1. If partitions b, c, and g from (1.2), (1.3), and (1.5), respectively, satisfy
conditions (1.7), (2.4) and
m+k∑
i=1
gi ≥
m∑
i=1
ci +
k∑
i=1
bi,
then we say that g is weakly majorized by c and b, and we write
g ≺′′ (c,b).
Lemma 2.2 ([7, Theorem 2.5]). Let a, b, d, and g from (1.1), (1.2), (1.4), and (1.5),
respectively, satisfy
m+s∑
i=1
di +
k∑
i=1
bi =
m+k∑
i=1
gi +
s∑
i=1
ai.
226 Ars Math. Contemp. 23 (2023) #P2.02 / 221–237
If there exists a partition f̄ = (f̄1, . . . , f̄m+k+s) such that
f̄ ≺′′ (d,b) and f̄ ≺′′ (g,a), (2.5)
then there exists a partition f = (f1, . . . , fm+k+s) such that
f ≺′ (d,b) and f ≺′ (g,a). (2.6)
Moreover, if the partition f̄ satisfying (2.5) consists of nonnegative integers, and
m+s∑
i=1
di +
k∑
i=1
bi ≥ 0,
then there exists a partition f consisting of nonnegative integers satisfying (2.6).
We also cite the result from [10] which shows the transitivity property of generalized
majorization. More on this topic is given in Section 4.
Theorem 2.3 ([10]). Let a, b, d and f be partitions (1.1), (1.2), (1.4) and (1.6), respec-
tively. If
f ≺′ (d,b) and d ≺′ (c,a),
then
f ≺′ (c,a ∪ b).
3 Main result
We start this section by giving one auxiliary result:
Lemma 3.1. Let a, b, d and g be the partitions (1.1), (1.2), (1.4) and (1.5), respectively.
Let c = (c1, . . . , cm) be a partition such that
d ≺′ (c,a) and g ≺′ (c,b). (3.1)
Let hj = min{i|ci−j+1 < gi}, j = 1, . . . , k, and h̄j = min{i|ci−j+1 < di}, j = 1, . . . , s.
Let g′ = (g′1, . . . , g
′
m) be a partition obtained from g after removing gh1 , . . . , ghk , i.e.
{g′1, . . . , g′m} = {g1, . . . , gm+k} \ {gh1 , . . . , ghk},
and let d′ = (d′1, . . . , d
′
m) be a partition obtained from d after removing dh̄1 , . . . , dh̄s , i.e.
{d′1, . . . , d′m} = {d1, . . . , dm+s} \ {dh̄1 , . . . , dh̄s}.
Then
ci ≥ max(g′i, d′i), i = 1, . . . ,m. (3.2)
Proof. Fix i ∈ {1, . . . ,m}. Let h0 := 0, hk+1 := m + k + 1. Then there exists j ∈
{0, . . . , k} such that
hj+1 − (j + 1) ≥ i > hj − j.
This is true since hu+1 > hu, and so hu+1 − (u + 1) ≥ hu − u, for all u = 0, . . . , k, as
well as h0 − 0 = 0 and hk+1 − (k + 1) = m.
M. Dodig and M. Stošić: Double generalized majorization and diagrammatics 227
Then
hj+1 > i+ j > hj ,
and so by the definition of g′ we have
gi+j = g
′
i.
If j < k, by (2.3) we have that cl−j ≥ gl for all l < hj+1, and so
ci ≥ gi+j = g′i.
If j = k, by (3.1) and definition of the generalized majorization, we again obtain
ci ≥ gi+k = gi+j = g′i.
By replacing the partitions g′ by d′ we shall also obtain
ci ≥ d′i.
Altogether we have obtained (3.2), as desired.
Now we can give the main result of the paper:
Theorem 3.2. Let a, b, d and g be partitions (1.1), (1.2), (1.4) and (1.5), respectively. If
there exists a partition c = (c1, . . . , cm) such that
d ≺′ (c,a) and g ≺′ (c,b), (3.3)
then there exists a partition f = (f1, . . . , fm+k+s) such that
f ≺′ (d,b) and f ≺′ (g,a). (3.4)
Proof. By the definition of the generalized majorization (Definition 1.1) and by (2.4), we
have that (3.3) is equivalent to:
ci ≥ gi+k, i = 1, . . . ,m, (3.5)
m+k∑
i=hj+1
gi −
m∑
i=hj−j+1
ci ≥
k∑
i=j+1
bi, j = 1, . . . , k, (3.6)
m+k∑
i=1
gi =
m∑
i=1
ci +
k∑
i=1
bi, (3.7)
and
ci ≥ di+s, i = 1, . . . ,m, (3.8)
m+s∑
i=h̄j+1
di −
m∑
i=h̄j−j+1
ci ≥
s∑
i=j+1
ai, j = 1, . . . , s, (3.9)
m+s∑
i=1
di =
m∑
i=1
ci +
s∑
i=1
ai, (3.10)
228 Ars Math. Contemp. 23 (2023) #P2.02 / 221–237
where
hj := min{i|ci−j+1 < gi}, j = 1, . . . , k,
and
h̄j := min{i|ci−j+1 < di}, j = 1, . . . , s.
Equalities (3.7) and (3.10) together give
m+s∑
i=1
di +
k∑
i=1
bi =
m+k∑
i=1
gi +
s∑
i=1
ai. (3.11)
Let us denote by g′ = (g′1, . . . , g
′
m) a partition obtained from g after removing
{gh1 , . . . , ghk}. Also, let us denote by d′ = (d′1, . . . , d′m), a partition obtained from d
after removing {dh̄1 , . . . , dh̄s}. By Lemma 3.1 we have that
ci ≥ max(g′i, d′i), i = 1, . . . ,m. (3.12)
In order to prove the existence of a partition f = (f1, . . . , fm+k+s) satisfying (3.4), by
(3.11) and by Lemma 2.2 it is enough to prove the existence of a partition f̄ = (f̄1, . . . ,
f̄m+k+s) satisfying
f̄ ≺′′ (d,b) and f̄ ≺′′ (g,a). (3.13)
We shall define the partition f̄ = (f̄1, . . . , f̄m+k+s) as a non-increasing ordering of
integers min(g′1, d
′
1), . . . ,min(g
′
m, d
′
m), gh1 , . . . , ghk , dh̄1 , . . . , dh̄s , i.e.
f̄ := {min(g′1, d′1), . . . ,min(g′m, d′m)} ∪ {gh1 , . . . , ghk} ∪ {dh̄1 , . . . , dh̄s}.
By Definition 2.1, we are left with proving the following:
gi ≥ f̄i+s, i = 1, . . . ,m+ k, (3.14)
m+k+s∑
i=lj+1
f̄i ≥
m+k∑
i=lj−j+1
gi +
s∑
i=j+1
ai, j = 1, . . . , s, (3.15)
m+k+s∑
i=1
f̄i ≥
m+k∑
i=1
gi +
s∑
i=1
ai, (3.16)
di ≥ f̄i+k, i = 1, . . . ,m+ s, (3.17)
m+k+s∑
i=l̄j+1
f̄i ≥
m+s∑
i=l̄j−j+1
di +
k∑
i=j+1
bi, j = 1, . . . , k, (3.18)
m+k+s∑
i=1
f̄i ≥
m+s∑
i=1
di +
k∑
i=1
bi, (3.19)
where
lj := min{i|gi−j+1 < f̄i}, j = 1, . . . , s,
M. Dodig and M. Stošić: Double generalized majorization and diagrammatics 229
and
l̄j := min{i|di−j+1 < f̄i}, j = 1, . . . , k.
In fact, we shall prove only (3.14) – (3.16). By replacing the partition g by d, and the
partition a by b, the formulas (3.17) – (3.19) will follow.
To that end, let us denote by f̄ ′ = (f̄ ′1, . . . , f̄
′
m+k) the following partition:
f̄ ′ := {min(g′1, d′1), . . . ,min(g′m, d′m)} ∪ {gh1 , . . . , ghk}.
Then
f̄ = f̄ ′ ∪ {dh̄1 , . . . , dh̄s},
and so
f̄ ′i ≥ f̄i+s, i = 1, . . . ,m+ k. (3.20)
Since
g = g′ ∪ {gh1 , . . . , ghk},
we also have
gi ≥ f̄ ′i , i = 1, . . . ,m+ k. (3.21)
Altogether, (3.20) and (3.21) give (3.14). By the definition of f̄ we have
m+k+s∑
i=1
f̄i =
m∑
i=1
min(g′i, d
′
i) +
k∑
i=1
ghi +
s∑
i=1
dh̄i
=
m∑
i=1
g′i +
m∑
i=1
d′i −
m∑
i=1
max(g′i, d
′
i) +
k∑
i=1
ghi +
s∑
i=1
dh̄i
=
m+k∑
i=1
gi +
m+s∑
i=1
di −
m∑
i=1
max(g′i, d
′
i).
By applying (3.12), we get
m+k+s∑
i=1
f̄i ≥
m+k∑
i=1
gi +
m+s∑
i=1
di −
m∑
i=1
ci,
which by (3.10) gives (3.16), as desired.
Hence, we are left with proving (3.15). First, we introduce by convention h0 := 0, and
hk+1 := m+k+1. Now, fix j ∈ {1, . . . , s}. Let uj ∈ {0, . . . , k} and αj ∈ {0, . . . ,m+k}
be such that
ghuj ≥ dh̄j > ghuj+1 , (3.22)
gαj ≥ dh̄j > gαj+1. (3.23)
Then
huj+1 > αj ≥ huj . (3.24)
From the definition of hi we have that hi ≥ i, for all i = 1, . . . , k, (see (2.2)). This,
together with (3.24) gives
αj ≥ uj .
230 Ars Math. Contemp. 23 (2023) #P2.02 / 221–237
Also, by the definition of g′, from (3.22) and (3.23) we obtain that
g′αj−uj ≥ dh̄j > g
′
αj−uj+1. (3.25)
Moreover, from the definition of h̄j , and from (3.12), we have that
dh̄j > ch̄j−j+1 ≥ g
′
h̄j−j+1.
Thus,
g′αj−uj > g
′
h̄j−j+1,
and so
αj − uj ≤ h̄j − j.
Hence,
min(αj − uj , h̄j − j) = αj − uj . (3.26)
Next, we shall prove that
lj = αj + j, (3.27)
and
f̄lj = dh̄j . (3.28)
(Recall that lj = min{i|gi−j+1 < f̄i}). Indeed, we have:
gh1 ≥ · · · ≥ ghuj ≥ dh̄j > gαj+1, (3.29)
dh̄1 ≥ · · · ≥ dh̄j−1 ≥ dh̄j > gαj+1, (3.30)
g′1 ≥ · · · ≥ g′αj−uj ≥ dh̄j > gαj+1, (3.31)
d′1 ≥ · · · ≥ d′h̄j−j ≥ dh̄j > gαj+1. (3.32)
From the definition of f̄ , and by (3.26), we have that there are at least uj + j +
min(αj − uj , h̄j − j) = αj + j elements of f̄ that are bigger or equal than dh̄j . Therefore
f̄αj+j ≥ dh̄j > gαj+1, and so lj ≤ αj + j.
For the other inequality, first suppose that f̄lj > dh̄j . Then among {f̄1, . . . , f̄lj}, there
would be at most j − 1 dh̄i ’s, while all other elements would be less than or equal to some
of the elements of the partition g. Therefore, we would have that for all i = 1, . . . , lj ,
f̄i ≤ gi−(j−1), and so f̄lj ≤ glj−j+1, which contradicts the definition of lj .
Hence f̄lj ≤ dh̄j , and so by (3.23) and the definition of lj
gαj ≥ dh̄j ≥ f̄lj > glj−j+1,
and so lj ≥ αj + j. Altogether, this proves (3.27) and (3.28).
In addition, by (3.29) – (3.32), we have also shown that
lj∑
i=1
f̄i =
αj+j∑
i=1
f̄i =
j∑
i=1
dh̄i +
uj∑
i=1
ghi +
αj−uj∑
i=1
min(g′i, d
′
i). (3.33)
M. Dodig and M. Stošić: Double generalized majorization and diagrammatics 231
Now, we have
m+k+s∑
i=lj+1
f̄i =
m+k+s∑
i=αj+j+1
f̄i =
s∑
i=j+1
dh̄i +
k∑
i=uj+1
ghi +
m∑
i=αj−uj+1
min(g′i, d
′
i)
=
s∑
i=j+1
dh̄i +
k∑
i=uj+1
ghi +
m∑
i=αj−uj+1
g′i +
m∑
i=αj−uj+1
d′i
−
m∑
i=αj−uj+1
max(g′i, d
′
i).
We note that by (3.22), (3.23) and (3.25) we have
k∑
i=uj+1
ghi +
m∑
i=αj−uj+1
g′i =
m+k∑
i=αj+1
gi =
m+k∑
i=lj−j+1
gi.
Also,
s∑
i=j+1
dh̄i +
m∑
i=αj−uj+1
d′i −
m∑
i=αj−uj+1
max(g′i, d
′
i) =
s∑
i=j+1
dh̄i +
h̄j−j∑
i=αj−uj+1
d′i +
m∑
i=h̄j−j+1
d′i
−
h̄j−j∑
i=αj−uj+1
max(g′i, d
′
i)−
m∑
i=h̄j−j+1
max(g′i, d
′
i).
For all i ∈ {αj − uj + 1, . . . , h̄j − j}, by (3.32) and (3.25) we have
d′i ≥ dh̄j > g
′
i,
and so
max(g′i, d
′
i) = d
′
i.
We also have
s∑
i=j+1
dh̄i +
m∑
i=h̄j−j+1
d′i =
m+s∑
i=h̄j+1
di.
Altogether we have
m+k+s∑
i=lj+1
f̄i =
m+k∑
i=lj−j+1
gi +
m+s∑
i=h̄j+1
di −
m∑
i=h̄j−j+1
max(g′i, d
′
i)
≥
m+k∑
i=lj−j+1
gi +
m+s∑
i=h̄j+1
di −
m∑
i=h̄j−j+1
ci,
where the last inequality follows from (3.12). Finally by (3.9) we obtain (3.15), as desired.
This finishes our proof.
232 Ars Math. Contemp. 23 (2023) #P2.02 / 221–237
Remark 3.3. We note that if both d and g are partitions consisting of nonnegative integers,
such that
m+s∑
i=1
di +
k∑
i=1
bi ≥ 0,
then by Lemma 2.2 the partition f also consists of nonnegative integers.
In the course of the proof of Theorem 3.2, we have also proved the following result
Corollary 3.4. Let a, b, d and g be partitions (1.1), (1.2), (1.4) and (1.5), respectively.
Let c = (c1, . . . , cm) be a partition such that
d ≺′′ (c,a) and g ≺′′ (c,b), (3.34)
then there exists a partition f = (f1, . . . , fm+k+s) such that
f ≺′′ (d,b) and f ≺′′ (g,a). (3.35)
Also, by Theorem 2.3 we have
Corollary 3.5. Let a, b, d and g be partitions (1.1), (1.2), (1.4) and (1.5), respectively. If
there exists a partition c = (c1, . . . , cm), such that
d ≺′ (c,a) and g ≺′ (c,b)
then there exists a partition f = (f1, . . . , fm+k+s) such that
f ≺′ (c,a ∪ b).
Finally, by combining Theorem 1.3 with the result of Corollary 3.4, we obtain necessary
conditions for the pseudo double majorization problem.
Corollary 3.6. Let a, b, d and g be partitions (1.1), (1.2), (1.4) and (1.5), respectively. If
there exists a partition c = (c1, . . . , cm), such that
d ≺′′ (c,a) and g ≺′′ (c,b)
then the condition Ω̄(g,d,b,a) holds.
4 Some comments and more on diagrammatics of generalized ma-
jorization
4.1 A counter example for the converse of Theorem 3.2
In the following example we show that the converse of Theorem 3.2 does not hold:
Example 4.1. Let us consider the following partitions of integers:
d = (7, 2, 1) (4.1)
g = (7, 2, 1) (4.2)
a = (3, 1) (4.3)
b = (2, 2) (4.4)
M. Dodig and M. Stošić: Double generalized majorization and diagrammatics 233
The partition
f = (4, 4, 3, 2, 1) (4.5)
satisfies
f ≺′ (g,a) and f ≺′ (d,b). (4.6)
Indeed, (4.6) is equivalent to
min(gi, di) ≥ fi+2, i = 1, . . . , 3, (4.7)
5∑
i=lj+1
fi ≥
3∑
i=lj−j+1
gi +
2∑
i=j+1
ai, j = 1, 2, (4.8)
5∑
i=1
fi =
3∑
i=1
gi +
2∑
i=1
ai =
3∑
i=1
di +
2∑
i=1
bi, (4.9)
5∑
i=l̄j+1
fi ≥
3∑
i=l̄j−j+1
di +
2∑
i=j+1
bi, j = 1, 2, (4.10)
where
l1 = l̄1 = 2, l2 = l̄2 = 3.
By (4.1) – (4.5) we directly get that all of (4.7) – (4.10) hold. Hence we have (4.6), as
announced.
However, there is no partition c satisfying
g ≺′ (c,b) and d ≺′ (c,a). (4.11)
Indeed, by the definition of generalized majorization, we would have that such a partition
c would be of length one, i.e. c = (c1) for certain integer c1, and that
c1 =
3∑
i=1
gi −
2∑
i=1
bi =
3∑
i=1
di −
2∑
i=1
ai = 6
Then
h1 = min{i|ci < gi} = 2,
and hence we would need that
3∑
i=h1+1
gi ≥
1∑
i=h1−1+1
ci +
2∑
i=1+1
bi
which gives
g3 = 1 ≥ b2 = 2,
which is a contradiction. Hence there is no partition c satisfying (4.11), as announced.
234 Ars Math. Contemp. 23 (2023) #P2.02 / 221–237
4.2 Diagrammatics
By using diagrammatics introduced in Section 1, Theorem 2.3 implies the following transitivity-
like property of the generalized majorization:
a1
a2
⇒g
d
f f
d
a1 ∪ a2
The main result of the paper, Theorem 3.2, can be described diagrammatically, by stating
that every diagram of the form
c
g d
b a
can be completed to a square
c
f
g d
b a
a b
The two properties allow various combinations. For example, by combining the result from
Theorem 3.2 with the result from Theorem 2.3 we can get the following. Let c, u, w, g, d,
a1, a2, b1 and b2 be partitions such that
u ≺′ (g,a1), u ≺′ (c,b1), w ≺′ (c,a2), w ≺′ (d,b2),
i.e.
cg d
u w
b1 a2
a1 b2
Then by Theorem 3.2 there exists a partition f such that
f ≺′ (u,a2) and f ≺′ (w,b1).
Diagrammatically this gives
M. Dodig and M. Stošić: Double generalized majorization and diagrammatics 235
cg d
f
u w
b1
a2
a2
b1
a1 b2
Finally, by Theorem 2.3, such f satisfies
g d
f
a b
where
a = a1 ∪ a2, b = b1 ∪ b2.
5 Conclusions
In this paper we study new properties of generalized majorization. The main result of
the paper is the proof that the generalized majorization has a completing-squares prop-
erty. More precisely, we have introduced pseudo double majorization problem for two
pairs of partitions (Problem 1.4), and we relate it with double majorization problem (Prob-
lem 1.2). In particular, we prove that the existence of a partition c satisfying (1.11) implies
the existence of a partition f satisfying (1.10). By introducing diagrammatical interpreta-
tion of generalized majorization, our main result has an elegant geometric interpretation,
which also complements the previous results on transitivity-like property of generalized
majorization [10].
Finally, the obtained results are expected to have strong impact in solving the General
Matrix Pencil Completion Problem, as well as in solving Bounded Rank Perturbation Prob-
lems for matrix pencils.
ORCID iDs
Marija Dodig https://orcid.org/0000-0001-8209-6920
Marko Stošić https://orcid.org/0000-0002-4464-396X
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.03 / 239–246
https://doi.org/10.26493/1855-3974.2415.fd1
(Also available at http://amc-journal.eu)
Classification of minimal Frobenius hypermaps*
Kai Yuan , Yan Wang †
School of Mathematics and Information Science, Yan Tai University, Yan Tai, P.R.C.
Received 23 August 2020, accepted 24 May 2022, published online 11 November 2022
Abstract
In this paper, we give a classification of orientably regular hypermaps with an automor-
phism group that is a minimal Frobenius group. A Frobenius group G is called minimal if
it has no nontrivial normal subgroup N such that G{N is a Frobenius group. An orientably
regular hypermap H is called a Frobenius hypermap if AutpHq acting on the hyperfaces
is a Frobenius group. A minimal Frobenius hypermap is a Frobenius hypermap whose
automorphism group is a minimal Frobenius group with cyclic point stabilizers. Every
Frobenius hypermap covers a minimal Frobenius hypermap. The main theorem of this
paper generalizes the main result of Breda D’Azevedo and Fernandes in 2011.
Keywords: Frobenius hypermap, Frobenius group.
Math. Subj. Class. (2020): 57M15, 05C25, 20F05
1 Introduction
Let S be a compact and connected orientable surface. A topological hypermap H on S is a
triple pS;V ;Eq, where V and E denote closed subsets of S with the following properties:
(1) B “ V X E is a finite set. Its elements are called the brins of H;
(2) V Y E is connected;
(3) the components of V (called the hypervertices) and of E (called the hyperedges), are
homeomorphic to closed discs;
(4) the components of the complement SzpV YEq are homeomorphic to open discs, and
they are called the hyperfaces of H.
*The Authors thank the referees for their helpful comments.
†Corresponding author. Supported by NSFC (No. 12101535) and NSFS (No. ZR2020MA044).
E-mail addresses: pktide@163.com (Kai Yuan), wang´yan@pku.org.cn (Yan Wang)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
240 Ars Math. Contemp. 23 (2023) #P2.03 / 239–246
The following Figure 1 shows a topological hypermap on torus with 9 brins, 3 hy-
pervertices (black components), 3 hyperedges (grey components) and 3 hyperfaces (white
components).
Figure 1: A hypermap on torus.
An important and convenient way to visualize hypermaps was introduced by Walsh in
[13]. The Walsh representation of a hypermap as a bipartite graph embedding on S can
be described as follows. At the centre of each hypervertex place a white vertex and at the
centre of each hyperedge place a black vertex. If a hypervertex intersects a hyperedge then
we join the corresponding white vertex and black vertex by an edge. In this way we obtain
a bipartite graph. This bipartite graph is said to be the underlying graph of H. Figure 2 is
the Walsh representation of the hypermap in Figure 1.
Figure 2: The Walsh representation.
An algebraic hypermap is a quadruple H “ pG,B, ρ0, ρ1q, where G is a finite group
which is generated by two elements ρ0, ρ1 and acts transitively on a finite set B. By [3],
there is a one-to-one correspondence between topological and algebraic hypermaps. The
finite group G is the monodromy group of H, denoted by MonpHq. In the Walsh repre-
sentation, G is a permutation group acting on the set of edges, ρ0, ρ1 generate the cyclic
permutations of the edges going around the white resp. black vertices in a positive sense,
and each cycle of ρ0ρ1 bounds a hyperface in a negative direction. A permutation α of B
is called an automorphism of the hypermap H “ pG,B, ρ0, ρ1q if it is G-equivariant, i.e.
if
αpgpbqq “ gpαpbqq
K. Yuan and Y. Wang: Classification of minimal Frobenius hypermaps 241
for every b P B and g P G.
Since αρ0α´1 “ ρ0 and αρ1α´1 “ ρ1, α induces a permutation on the cycles of ρ0 and
ρ1. So, in the Walsh representation, AutpHq induces a subgroup of the automorphism group
of the underlying graph of H, and AutpHq preserves the hypervertex set and hyperedge set,
respectively. A hypermap is called regular if G acts regularly on B. In this case, AutpHq
is isomorphic to G which acts regularly on B as well.
For a regular hypermap H “ pG,B, ρ0, ρ1q, the set B can be replaced by G, so that
MonpHq and AutpHq can be viewed as the right and left regular multiplications of G,
respectively. So, H can be denoted by a triple H “ pG; ρ0, ρ1q, where G “ xρ0, ρ1y. In
this way, the hypervertices (resp. hyperedges and hyperfaces ) correspond to right cosets
of G relative to xρ0y, (resp. xρ1y and xρ0ρ1y). In [4], the hypermap H “ pG; ρ0, ρ1q is
denoted by pG; a, bq where a “ ρ1´1ρ0´1 and b “ ρ0. From now on, we denote a regular
hypermap H by the triple H “ pG; a, bq, and then the hyperfaces (resp. hypervertices and
hyperedges) correspond to left cosets of G relative to subgroups xay (resp. xby and xaby).
Let H “ pG; a, bq and H1 “ pG1; a1, b1q be two orientably regular hypermaps. If there is an
epimorphism ρ from G to G1 such that aρ “ a1 and bρ “ b1, then H is called a covering of
H1 or H covers H1. Given a group G, pG; a1, b1q – pG; a2, b2q if and only if there exists
an automorphism σ of G such that aσ1 “ a2 and bσ1 “ b2.
A (face-)primer hypermap is an orientably regular hypermap whose automorphism
group induces faithful actions on its hyperfaces, see [4]. The classification of regular hy-
permaps with given automorphism groups isomorphic to PSLp2, qq or PGLp2, qq can be
extracted from [12] by Sah. Moreover, Conder, Potočnik and Širáň extended Sah’s investi-
gation to reflexible hypermaps, on both orientable and nonorientable surfaces, and provided
explicit generating sets for projective linear groups, see [1]. In [2], Conder described all
regular hypermaps of genus 2 to 101, and all non-orientable regular hypermaps of genus 3
to 202.
The study of primer hypermaps was initiated by Breda d’Azevedo and Fernandes in
2011. In [4], the authors classified the primer hypermaps with p-hyperfaces for a prime
number p, where their automorphism groups are Frobenius groups. Thereafter, they de-
termined all regular hypermaps with p-hyperfaces, see [5]. In [7], Du and Hu classified
primer hypermaps with a product of two primes number of hyperfaces. Recently, Du and
Yuan characterized primer hypermaps with nilpotent automorphism groups and prime hy-
pervertex valency, see [8].
A Frobenius group is a transitive permutation group G on a set Ω which is not regular
on Ω , but has the property that the only element of G which fixes more than one point is
the identity element. A Frobenius group G is called minimal if it does not have a nontrivial
normal subgroup N such that G{N is a Frobenius group. A regular hypermap H is called
a Frobenius hypermap if AutpHq acting on the hyperfaces is a Frobenius group. Clearly,
H is a primer hypermap. A minimal Frobenius hypermap is a Frobenius hypermap whose
automorphism group is a minimal Frobenius group with a cyclic point stabilizer. Clearly,
every Frobenius hypermap covers a minimal Frobenius hypermap.
This paper has three sections. In the first section, a quick overview of orientably regu-
lar hypermaps is given. In Section 2, we introduce minimal Frobenius groups. In the last
section, we give a classification of orientably regular minimal Frobenius hypermaps. Fur-
thermore, the main theorem of this paper generalizes the main result of Breda D’Azevedo
and Fernandes, see [4].
242 Ars Math. Contemp. 23 (2023) #P2.03 / 239–246
2 Minimal Frobenius groups
We refer the readers to [10] for standard notation and results in group theory. Set pr, sq
to denote the greatest common divisor of two positive integers r and s. We denote the
orders of an element x and of a subgroup H of G as |x| and |H|, respectively. A semidirect
product of a group N by a group H is denoted by N : H . Let Zm “ t0, 1, ¨ ¨ ¨ ,m ´ 1u
and Z˚m “ tk
ˇ
ˇ k P Zm and pk,mq “ 1u.
Let G be a Frobenius group on Ω. A subgroup K of G is called the Frobenius kernel
if K acts regularly on Ω. Each point stabilizer is called a Frobenius complement of K in
G. In the following, we give some interesting results about Frobenius groups and primitive
groups.
Proposition 2.1 ([6, P86]). Let G be a Frobenius group on Ω and α P Ω, K be the
Frobenius kernel, and H be a Frobenius complement. Then:
(i) K is a normal and regular subgroup of G.
(ii) For each odd prime number p, the Sylow p-subgroups of H are cyclic, and the Sylow
2-subgroups are either cyclic or quaternion groups. If G is not solvable, then it has
exactly one nonabelian composition factor, namely A5.
(iii) K is a nilpotent group.
Proposition 2.2 ([6, Corollary 1.5A.]). Let G be a group acting transitively on a set Ω
with at least two points. Then G is primitive if and only if each point stabilizer Gα is a
maximal subgroup of G.
Lemma 2.3. Assume G ď SympΩq has a regular normal subgroup R, where Ω has at least
two points. Then G is primitive if and only if no nontrivial subgroup of R is normalized by
Gα, for each α.
Proof. By Proposition 2.2, G is primitive if and only if Gα is a maximal subgroup of G.
Because R is a regular normal subgroup of G, G “ GαR and Gα X R “ t1u.
We claim that Gα is maximal if and only if no nontrivial subgroup of R is normalized
by Gα. Suppose Gα is not maximal, then there exists a proper subgroup K of G such that
Gα ă K. It follows that K “ K XG “ K XGαR “ GαpK XRq. In this case, K XR is
a proper subgroup of R which is normalized by Gα. Conversely, suppose that there exists
a proper subgroup H , normalized by Gα, of R. Thus GαH is a proper subgroup of G and
so Gα is not maximal. l
Corollary 2.4 follows directly from Lemma 2.3.
Corollary 2.4. Assume G ď SympΩq has a regular normal subgroup R, where Ω has at
least two points. If R is abelian, then G is primitive if and only if no nontrivial normal
subgroup of G is contained in R.
Lemma 2.5. Let K be the Frobenius kernel of a Frobenius group G which acts on a set Ω.
If N is a normal subgroup of G, then either N ď K or K ă N .
Proof. Assume that N is not a subgroup of K. Set α P Ω. Since N is a normal subgroup
of G, we have N “ p
Ť
gPK
Ngαq Y pN X Kq and so N is a subgroup of NαK. Let |Nα| “
K. Yuan and Y. Wang: Classification of minimal Frobenius hypermaps 243
m, |K| “ n and |N XK| “ t. Then, |N | “ npm´1q ` t. Since N ď NαK and Nα ď N ,
we get N “ N X NαK “ NαpN X Kq. So, |N | “ mt which implies npm ´ 1q ` t “
mt. Note that m ą 1, then n “ t. Therefore, N X K “ K and K is a proper subgroup of
N . l
Proposition 2.6 ([11, Lemma 2.3]). Let K be the Frobenius kernel of a Frobenius group
G. If N is a normal subgroup of G and N ă K, then G{N is a Frobenius group.
Proposition 2.7 ([11, Corollary 2.6]). Let G “ KH be a Frobenius group, where K is the
Frobenius kernel and H is a Frobenius complement. For each h P H,h ‰ 1, and for each
k P K, the orders of h, kh and hk are equal, that is |h| “ |kh| “ |hk|.
Based on Lemma 2.5 and Proposition 2.6, we give the following definition of minimal
Frobenius groups.
Definition 2.8. A Frobenius group G is called minimal if it does not have a nontrivial
normal subgroup N such that G{N is a Frobenius group.
Lemma 2.9. If G is a minimal Frobenius group acting on a set Ω with the Frobenius kernel
K, then K is an elementary abelian p-group and G is primitive.
Proof. If G is minimal, then by Proposition 2.6 no nontrivial normal subgroup of G exists
in K. Note that K is a nilpotent group. Let P be a Sylow p-group of K, ΦpP q be the
Frattini subgroup of P and L be the p1-Hall group of K. Both ΦpP q and L are characteristic
subgroups of K. So, L “ ΦpP q “ 1 which implies that K is an elementary abelian p-
group.
Because no nontrivial normal subgroup of G is contained in K and K is abelian, it
follows that G is primitive by Corollary 2.4. l
Lemma 2.10. If G is a primitive group acting on a set Ω with non-trivial abelian point sta-
bilizers, then G is a Frobenius group and its Frobenius kernel K is an elementary abelian
p-group.
Proof. It suffices to show that for any two distinct points α, β P Ω, Gα XGβ “ 1. Let J “
Gα X Gβ . Since G is primitive, G “ xGα, Gβy. Note that Gα and Gβ are abelian, so J is
a normal subgroup of G. Because αJ “ tαu, for any g P G, we have αgJ “ αJg “ tαgu.
That is to say J fixes every point of Ω, so J “ 1 and G is a Frobenius group. Furthermore,
as point stabilizers are maximal, the Frobenius kernel K must be an elementary abelian
p-group . l
Corollary 2.11 follows from Lemma 2.9 and 2.10 directly.
Corollary 2.11. Let G be a permutation group with cyclic point stabilizers. Then, G is a
minimal Frobenius group if and only if G is a primitive group.
For a prime number p and an integer n, an integer m pm ą 1q is called a primitive
divisor of pn ´ 1 if m divides pn ´ 1, but it does not divide ps ´ 1 for any s ă n.
The following Proposition 2.12 can be obtained from some results in
[10, Kapitel II: 3.10, 3.11, 7.3].
Proposition 2.12. For a prime number p and a positive integer n, set G “ GLpn, pq.
244 Ars Math. Contemp. 23 (2023) #P2.03 / 239–246
(i) The group G contains a cyclic Singer-Zyklus group S “ xxy of order pn ´ 1, and
CGpSq “ S. Moreover, NGpSq “ S : xyy “ xx, y
ˇ
ˇ xp
n´1 “ yn “ 1, xy “ xpy,
and |NGpSq| “ nppn ´ 1q. Take an element g P S, if |g| is a primitive divisor of
pn ´ 1, then NGpxgyq “ NGpSq, CGpxgyq “ S and xgy is an irreducible subgroup.
(ii) Let L be a cyclic irreducible subgroup of G. Then L is conjugate to a subgroup of
S, and |L| is a primitive divisor of pn ´ 1.
The following lemma generalizes Lemma 3.3 in [9]. The proof is similar to that of
Lemma 3.3, so we omit it.
Lemma 2.13. Let X “ T : xxy and Y “ T : xyy be two subgroups of A “ AGLpn, pq “
T : G, where G “ GLpn, pq, T is the translation subgroup, and x, y are nontrivial ele-
ments in G. If σ is an isomorphism from X to Y mapping xxy to xyy, then, there exists an
element u P G such that σ “ Ipuq|X , where Ipuq is the inner automorphism of A induced
by u. In particular, u P NGpxxyq if xxy “ xyy.
3 Classification of minimal Frobenius hypermaps
For a prime number p, an integer n ě 1 (n ě 2 if p “ 2) and a primitive divisor m of
pn ´ 1, let S be the cyclic Singer-Zyklus group of GLpn, pq, xay be a subgroup of S with
order m and T be the translation subgroup of AGLpn, pq. Define a group M of order mpn
as
M “ T : xay ď T : S ď AGLpn, pq “ T : GLpn, pq.
By Proposition 2.12, xay is an irreducible subgroup. Hence M is a primitive group, and
consequently M is a Frobenius group by Lemma 2.10.
Let F be a minimal Frobenius group acting on a set Ω (|Ω| ą 2) with cyclic point
stabilizers, and K be its Frobenius kernel. By Lemma 2.9, K is an elementary abelian
p-group and F is a primitive group. Set |K| “ pn, and then |Ω| “ pn. Take an element
α P Ω and assume |Fα| “ k. By Proposition 2.12, k is a primitive divisor of pn ´ 1, and
GLpn, pq has only one conjugacy class of irreducible cyclic subgroups of order k. Hence
AGLpn, pq has only one conjugacy class of subgroups isomorphic to F which implies
F – M “ T : xay when k “ m. These discussions give the following Theorem 3.1.
Theorem 3.1. Let F be a minimal Frobenius group with cyclic point stabilizers of order
m. Then, F – T : xay, where T is elementary abelian of order pn for some prime number
p and an integer n ě 1, m is a primitive divisor of pn ´ 1 and |xay| “ m. Clearly,
|F | “ mpn.
Lemma 3.2. Let M “ T : xay be the group defined as in the first paragraph of this section.
If H “ pM ;R,Lq is a Frobenius hypermap, then H is isomorphic to
Hpp, n,m, i, jq “ pM ; ai, ajbq,
where 1 ‰ b P T , m is a primitive divisor of pn ´ 1, j P Zm, i P Z˚m and pi, pq “ 1. More-
over, different parameter pairs pi, jq give non-isomorphic hypermaps with pn hyperfaces,
each of valency m. Furthermore, there are mϕpmqn non-isomorphic hypermaps, where ϕ is
the Euler’s totient function.
K. Yuan and Y. Wang: Classification of minimal Frobenius hypermaps 245
Proof. Let G “ GLpn, pq and then M ď AGLpn, pq “ T : G. Since M is a Frobenius
group, M has only one conjugacy class of subgroups of order m. So we can assume R “ ai
for some i P Z˚m. Remember that S is the cyclic Singer-Zyklus group of GLpn, pq and xay
is a subgroup of S. So, M is a normal subgroup of T : S. Since S fixes a and acts
transitively on T zt1u by conjugation, we may fix L “ ajb, where j is calculated modular
m.
If there exists an automorphism σ of M such that paiqσ “ ai1 and pajbqσ “ aj1b, then
bσ “ aϵb for some ϵ P Zm. Clearly, the orders of b and aϵb are equal. While according to
Proposition 2.7, the two elements aϵb and aϵ have the same order which is coprime with
that of b if ϵ ‰ 0 modulo m. So, bσ “ b. By Lemma 2.13, there exists an element u P G
such that σ “ Ipuq|F , where u P NGpxayq. According to Proposition 2.12,
NGpxayq “ S : xyy “ xx, y
ˇ
ˇ xp
n´1 “ yn “ 1, xy “ xpy,
where S “ xxy. Because bσ “ b, it follows that u “ yt, where t is calculated modular n.
So, aσ “ ayt “ apt . As a result, we may assume pi, pq “ 1 in R “ ai. As a result, we get
mϕpmq
n non-isomorphic hypermaps pM ; a
i, ajbq, where ϕ is the Euler’s totient function.
Clearly, pM ; ai, ajbq has pn hyperfaces, each of valency m. l
By Theorem 3.1, the automorphism group of a minimal Frobenius hypermap is isomor-
phic to M “ T : xay, where |T | “ pn and |xay| “ m. Consequently, we give the following
classification theorem of minimal Frobenius hypermaps.
Theorem 3.3. H is a minimal Frobenius hypermap if and only if H is isomorphic to
Hpp, n,m, i, jq “ pM ; ai, ajbq,
where M is a group defined as in the first paragraph of this section, m is a primitive divisor
of pn ´ 1, j P Zm, i P Z˚m and pi, pq “ 1. Moreover, different parameter pairs pi, jq give
non-isomorphic hypermaps with pn hyperfaces, each of valency m. And, there are mϕpmqn
non-isomorphic minimal Frobenius hypermaps, where ϕ is the Euler’s totient function.
According to Corollary 2.11, we have the following Proposition 3.4.
Proposition 3.4. If H is a regular hypermap, then H is a minimal Frobenius hypermap if
and only if AutpHq acts primitively on the hyperfaces.
The next Proposition 3.5 follows from Lemma 2.5.
Proposition 3.5. Every Frobenius hypermap covers a minimal Frobenius hypermap.
The H-sequence of a hypermap H is a sequence r|v|, |e|, |f |;V,E, F ; |AutpHq|s, where
|v|, |e|, |f |, V, E and F stand for the hypervertex valency, hyperedge valency, hyperface
valency, number of hypervertices, number of hyperedges and number of hyperfaces of H,
respectively.
Corollary 3.6. The H-sequence of the minimal Frobenius hypermap Hpp, n,m, i, jq “
pM ; ai, ajbq is
(i) rp,m,m;mpn´1, pn, pn;mpns for j “ 0;
(ii) rm, p,m; pn,mpn´1, pn;mpns for j “ m ´ i;
246 Ars Math. Contemp. 23 (2023) #P2.03 / 239–246
(iii) r mpm,jq ,
m
pm,i`jq ,m; pm, jqp
n, pm, i ` jqpn, pn;mpns for j ‰ 0 and j ‰ m ´ i.
Proof. The sequence is determined by the first three entries, namely |ajb|, |ai`jb| and |ai|.
These entries can be easily calculated according to Proposition 2.7. l
ORCID iDs
Kai Yuan https://orcid.org/0000-0003-1858-3083
Yan Wang https://orcid.org/0000-0002-0148-2932
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.04 / 247–260
https://doi.org/10.26493/1855-3974.2507.a1d
(Also available at http://amc-journal.eu)
A parametrisation for symmetric designs
admitting a flag-transitive, point-primitive
automorphism group with a product action*
Eugenia O’Reilly-Regueiro †
Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de
la Investigación Cientı́fica, Circuito Exterior, Ciudad Universitaria, Coyoacán, 04510,
Mexico City, Mexico
José Emanuel Rodrı́guez-Fitta
Facultad de Estudios Superiores Acatlán, Universidad Nacional Autónoma de México
(UNAM), Av. Alcanfores y San Juan Totoltepec s/n, Santa Cruz Acatlán, Naucalpan,
53150, Edo. de México, Mexico
Received 17 December 2020, accepted 2 May 2022, published online 11 November 2022
Abstract
We study (v, k, λ)-symmetric designs having a flag-transitive, point-primitive automor-
phism group, with v = m2 and (k, λ) = t > 1, and prove that if D is such a design with
m even admitting a flag-transitive, point-primitive automorphism group G, then either:
(1) D is a design with parameters
(
(2t+ s− 1)2, 2t
2−(2−s)t
s ,
t2−t
s2
)
with s ≥ 1 odd, or
(2) G does not have a non-trivial product action.
We observe that the parameters in (1), when s = 1, correspond to Menon designs.
We also prove that if D is a (v, k, λ)-symmetric design with a flag-transitive, point-
primitive automorphism group of product action type with v = ml and l ≥ 2 then the
complement of D does not admit a flag-transitive automorphism group.
Keywords: Symmetric-designs, flag-transitivity, primitive groups, automorphism groups of designs.
Math. Subj. Class. (2020): 05B05, 51E05, 20B15, 20B25
*The authors would like to express their gratitude to the referee who made very helpful comments and sugges-
tions that improved our paper.
†Corresponding author.
E-mail addresses: eugenia@im.unam.mx (Eugenia O’Reilly-Regueiro), 887191@pcpuma.acatlan.unam.mx
(José Emanuel Rodrı́guez-Fitta)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
248 Ars Math. Contemp. 23 (2023) #P2.04 / 247–260
1 Introduction
If D = (P,B) is a (v, k, λ)-symmetric design, a flag of D is an ordered pair (p,B) such
that p ∈ P is a point of D, B ∈ B is a block of D, and p ∈ B. The order of D is n = k−λ.
There are some symmetric designs in which the parameters are related in some special
way, such as Hadamard designs in which v = 4n + 3, k = 2n + 1, and λ = n (n ∈ Z+),
and Menon designs, in which v = 4t2, k = 2t2 − t, and λ = t2 − t for some positive
integer t. These last ones will be relevant in the present work.
If G = Aut(D), then G is point-transitive if it is transitive on P (the set of points of
D), and it is flag-transitive if it is transitive on the set of flags of D. If G is point-transitive,
it can either be point-primitive, that is, there is no G-invariant non-trivial partition of P , or
point-imprimitive, which is when there is a non-trivial partition of the points of D invariant
under the action of G.
Primitive groups are classified by the O’Nan-Scott Theorem, we will use the classifi-
cation in [4] by Liebeck, Praeger, and Saxl, with five types, namely affine, almost simple,
product, simple diagonal, and twisted wreath.
Buekenhout, Delandtsheer, and Doyen proved in [1] that if a 2-design with λ = 1
(linear space) admits a point-primitive, flag-transitive automorphism group G, then it must
be of affine or almost simple type. O’Reilly-Regueiro proved the same result for symmetric
2-designs with 2 ≤ λ ≤ 4 in [5, 6]. All designs in this paper will be 2-designs.
In [7], Tian and Zhou extended this result to λ ≤ 100, and conjectured that it holds
for all values of λ. Having an upper bound on λ, in [7] they ruled out the simple diagonal,
product, and twisted wreath action by finding possible groups and/or sets of parameters
of designs and then ruling them out by arithmetic constraints and the use of GAP [2].
Additionally, in [3, 8, 9, 10], Zhou et al. have tackled this issue from different perspectives,
and have ruled out the product action for flag-transitive (v, k, λ) symmetric designs in
which λ ≥ (k, λ)2, as well as, for those cases in which λ is prime.
We have tried to prove that if D is a (v, k, λ)-symmetric design with v = m2 even and
any λ admitting a point-primitve, flag-transitive automorphism group G, then G does not
have a product action. In this paper we present our results, namely, a parametrisation for
such designs which in some cases correspond to Menon designs.
In 1998, Zieschang proved in [11] that if a (not necessarily symmetric) 2-design in
which (r, λ) = 1 (where r is the number of blocks incident with any given point) admits a
flag-transitive group G, then G is of affine or almost simple type. Given this result, in our
work we will assume (k, λ) = t > 1.
2 Product action
We start with a result from [5], which will be useful later.
Corollary 2.1. If G is a flag-transitive automorphism group of a (v, k, λ)-symmetric design
D = (P,B), then k divides λ(v − 1, |Gx|) for every point-stabiliser Gx.
The next lemma gives us an arithmetic condition that will be used throughout this work.
Suppose that the group G has a product action on the set of points P . Then there is a
finite set Γ with |Γ| ≥ 5 and a group H acting primitively on Γ, with an almost simple or
simple diagonal action, such that
P = Γl and G ≤ H l ⋊ Sl = HwrSl, with l ≥ 2.
E. O’Reilly-Regueiro et al.: A parametrisation for symmetric designs . . . 249
Lemma 2.2. If G is a point-primitive group acting flag-transitively on a (v, k, λ)-symmetric
design D = (P,B), with a product action on P , then k divides λl(|Γ| − 1) and v = |Γ|l ≤
λl2(|Γ| − 1)2.
Proof. Take x ∈ P = Γl. If x = (γ1, ..., γl), define for 1 ≥ j ≥ l the Cartesian line of the
jth parallel class through x to be the set:
Gx,j = {(γ1, ..., γj−1, γ, γj+1, ..., γl)|γ ∈ Γ},
(So there are l Cartesian lines through x).
Denote |Γ| = m.
Since G is primitive, Gx is transitive on the l Cartesian lines through x. Denote by ∆
the union of those lines (excluding x). Then ∆ is a union of orbits of Gx, and so every
block through x intersects it in the same number of points. Hence k divides λl(|Γ| − 1).
Also, k2 > λ(ml − 1), so (ml − 1) < λl2(m− 1)2.
Hence
v = ml ≤ λl2(m− 1)2. (2.1)
3 Results
In this section we will only consider l = 2, further work may be done for greater values
of l.
When l = 2, m = r
2+2r+4λ
4λ−r2 so
(m+ 1)r2 + 2r − 4λ(m− 1) = 0 (3.1)
solving for r we have
r =
−2±
√
4 + 16λ(m− 1)(m+ 1)
2(m+ 1)
=
−1±
√
1 + 4λ(m2 − 1)
m+ 1
therefore
r =
2(k − 1)
m+ 1
. (3.2)
Suppose that (k, λ) = t > 1 (the case where (k, λ) = 1 was done by Paul-Hermann
Zieschang [11]), so there exist positive integers a and b such that
k = at, λ = bt. (3.3)
Then, by Lemma 2.2 we have
k =
2λ(m− 1)
r
, (3.4)
and substituting (3.3) in the last one and also in k(k − 1) = λ(v − 1) we obtain
a =
2b(m− 1)
r
, (3.5)
a(at− 1) = b(m2 − 1). (3.6)
250 Ars Math. Contemp. 23 (2023) #P2.04 / 247–260
From (3.5) we can see that a divides b(m − 1). But (k, λ) = t so t = (at, bt) implies
(a, b) = 1. Therefore a divides m − 1, that is, there exists a positive integer s such that
m− 1 = as and substituting in (3.5), we obtain r = 2bs. Then since (a, b) = 1, this forces
s = (m− 1, r2 ).
We have te following results with respect to the new parameters a and s.
Lemma 3.1. Let D be a (v, k, λ)-symmetric design with v = m2 admitting a flag-transitive,
point-primitive automorphism group with product action. If k = at and λ = bt with
t = (k, λ), then a ̸= 1.
Proof. If a = 1 then k = t and λ = kb with b ≥ 1. This is a contradiction because k > λ,
therefore a ̸= 1.
Lemma 3.2. Let D be a (v, k, λ)-symmetric design with v = m2 admitting a flag-transitive,
point-primitive automorphism group with product action. If k = at and λ = bt with
t = (k, λ), then (a, s) = 1 where s is a positive integer such that m− 1 = as and r = 2bs.
Proof. Note (3.2) can be rewritten as:
r + 1 = k − (m− 1)r
2
.
Using the expressions k = at, λ = bt, m − 1 = as and r = 2bs we obtain 1 =
a(t− bs2)− 2bs and here we can see that (a, s) = 1 .
The fact that the parameter s = 1 is a necessary and sufficient condition for Menon
designs is seen in the following result:
Lemma 3.3. Let D be a (v, k, λ)-symmetric design with v = m2 admitting a flag-transitive,
point-primitive automorphism group with product action. If t = (k, λ) and s ∈ Z+ is such
that m − 1 = as and r = 2bs, then s = 1 if and only if v = 4t2, k = 2t2 − t, and
λ = t2 − t.
Proof. Suppose first that s = 1, so m− 1 = a which implies k = (m− 1)t. We also have
r
2 = b, so λ =
r
2 t. Now from m =
r2+2r+4λ
4λ−r2 we obtain
m =
b+ t+ 1
t− b
(3.7)
then
a = m− 1 = 2b+ 1
t− b
. (3.8)
Now if t = b then λ = tb = b2 =
r2
4
, and substituting in (3.1) we obtain
r2(m+ 1) + 2r − r2(m− 1) = 0, so r(r + 1) = 0.
This forces r = 0 or r = −1, which is a contradiction and so t ̸= b. From (3.8) we
have t− b ≥ 1.
E. O’Reilly-Regueiro et al.: A parametrisation for symmetric designs . . . 251
Suppose that t− b > 1, and let x > 1 be an integer such that t = b+ x. We will prove
that x is an odd number. If x = 2y for some y ∈ Z, then t = b + 2y, and substituting in
(3.8) we obtain
a =
2b+ 1
2y
,
which is a contradiction since a ∈ Z, so x is odd. Therefore there exists a positive integer
y such that x = 2y + 1 > 1 and with this we obtain t = b + 2y + 1, substituting in (3.8)
results in
a =
2b+ 1
2y + 1
.
Using the last expression for a together with a > b we obtain 2b+1 > b(2y+1) which
results in
1 > b(2y − 1). (3.9)
But we assumed t− b > 1 so x = 2y+1 > 1, that is, 2y− 1 > −1. This together with
the expression (3.9) implies that the equation 2y − 1 = 0 should hold. But that implies
y = 12 , which is a contradiction since we assumed y ∈ Z.
From the above we can conclude that b = t − 1 and this implies λ = t(t − 1). Then
substituting this expression in (3.8) we have
a = 2(t− 1) + 1 = 2t− 1,
so k = t(2t− 1) and m = a+ 1 = 2t, therefore v = m2 = 4t2.
Now, suppose that we have a symmetric design with parameters v = 4t2, k = 2t2 − t
and λ = t2 − t. Then a = 2t − 1 and b = t − 1. In addition, we have m = 2t and all of
these combined imply m − 1 = 2t − 1 = a. But m − 1 = as, and so s = 1. Hence the
result.
Remark 3.4. When we fix m − 1 and we vary r2 we get many possible values for λ that
satisfy the equation m = r
2+2r+4λ
4λ−r2 , at this point we observe that if m− 1 is a power of an
odd prime, then the parameters satisfy the conditions of Menon designs.
Lemma 3.5. Let D be a (v, k, λ)-symmetric design with v = m2 admitting a flag-transitive,
point-primitive automorphism group with product action such that t = (k, λ) and s ∈ Z+
such that m − 1 = as and r = 2bs. If m − 1 = pd with p an odd prime and d ∈ N, then
v = 4t2, k = 2t2 − t, and λ = t2 − t.
Proof. From m− 1 = as = pd then we have the following possible cases:
1. s = pi and a = pd−i for some natural number i < d.
This case is not possible because this would imply that (a, s) = pj for some natural
number j, and this contradicts Lemma 3.2.
2. a = pi and s = pd−i for some natural number i < d.
This case is not possible because this would imply that (a, s) = pj for some natural
number j, contradicting Lemma 3.2.
3. s = pd and a = 1.
This is not possible because it contradicts Lemma 3.1.
252 Ars Math. Contemp. 23 (2023) #P2.04 / 247–260
4. a = pd and s = 1.
Recall that s = 1 (Lemma 3.3), so in this case v = 4t2, k = 2t2 − t and λ =
t2 − t (these are the conditions for Menon designs). With this we have proved the
lemma.
Remark 3.6. We cannot claim the previous result for any odd m− 1 because the parame-
ters (4900, 3267, 2178), (16900, 2752, 448) and (44100, 8019, 1458) are counterexamples
to that possible generalisation. However we have neither confirmed nor discarded the exis-
tence of designs with these parameters. These (and Menon designs) are the only admissible
parameters for v ≤ (210)2.
Recall the definition of the Cartesian lines from Lemma 2.2. In the case we are study-
ing, when l = 2, there are two Cartesian lines through any point in the design, so we have
two possibilities. Either:
(i) there exists a point x and a block that contains it such that it intersects only one
Cartesian line through x, or
(ii) for any point x in the design, every block that contains it intersects each one of the
Cartesian lines through x.
We now study these cases separately. Although there are similarities between both
proofs, due to their length and enough differences we present two theorems for clarity.
Theorem 3.7 (Case (i)). Let D be a (v, k, λ)-symmetric design with v = m2 admitting a
flag-transitive, point-primitive automorphism group with product action. If there exists a
flag (x,A) in the design such that A intersects only one Cartesian line through x then r+1
divides k.
Proof. Let (x,A) be the flag such that A intersects a Cartesian line through x := (a0, b0).
Suppose that A intersects the second Cartesian line through x.
First, let us prove that for any element of the block A, A intersects only the second
Cartesian line through that point. We have, two subcases: either a point y ∈ A is in the sec-
ond Cartesian line through x, or a point y ∈ A is not in the second Cartesian line through x.
First subcase: we can see that if we take a point y ∈ A so that it is also in the second
Cartesian line through x, then y = (a0, ν) for some ν ∈ Γ. In this way, the set of elements
in the second Cartesian line through y which are also in A is the same as the intersection
of A with the second Cartesian line through x. Also, since by Lemma 2.2 the size of the
intersection of A with the second Cartesian line through x is r + 1, the size of the set of
elements in the second Cartesian line through y which are also in A is r + 1, and since the
size of the intersection of A with the Cartesian lines through any point is r + 1, there are
no more elements of any of the two Cartesian lines through y in A. In particular, there are
no elements of the first Cartesian line through y in A and so the statement is proved for this
subcase.
Second subcase: Now we are going to take y ∈ A such that it is not in the second Cartesian
line through x, in particular y ̸= x, so if y := (a1, b1) then a1 ̸= a0. Let us consider the flag
(y,A). Since the group G is flag-transitive, there is a g ∈ G such that g(x,A) = (y,A),
that is, g(x) = y, so
g(a0, b0) = (a1, b1), (3.10)
E. O’Reilly-Regueiro et al.: A parametrisation for symmetric designs . . . 253
this implies g|Γ(a0) = a1. So, for any µ ∈ Γ such that (a0, µ) ∈ A we have g(a0, µ) =
(a1, ν) for some ν ∈ Γ. Thus the element g ∈ G sends every element of the second
Cartesian line through x which is also in A to an element of the second Cartesian line
through y which is also in A. In this way A intersects only the second Cartesian line
through y. This is true for any y which is not in the second Cartesian line through x and
by Lemma 2.2 the size of this intersection is r + 1 and with this the statement the second
subcase is proved.
Let A0 be the set of points in the second Cartesian line through x which are also in A,
including x, the size of this set is r + 1. Now let us take an element x1 ∈ A \ A0. By
previous arguments A intersects only the second Cartesian line through x1, therefore, if A1
is the set of points in the second Cartesian line through x1 that are in A including x1 then
the size of this set is also r + 1. In the same way as before, we take x2 ∈ A \ (A0 ∪ A1)
and define the set A2 as the set of points in the second Cartesian line through x2 that are in
A including x2 and again its size is r + 1.
The process is continued in this way until no more points can be taken in A, thus we
get a set of points x0, x1, ..., xi ∈ A along with a collection of sets A0, A1, ..., Ai for some
natural number i, such that Aj is the intersection of the second Cartesian line through xj
with A. So, the size of Aj is r + 1 for all j. Also, A =
⋃j=i
j=0 Aj and by construction if
xg ̸= xh then Ag ̸= Ah with 1 ≤ g, h ≤ i.
It remains to prove that each pair of sets in this collection is disjoint, that is, if Ae ̸= Af
are two sets in the collection that was previously constructed, we must prove that Ae∩Af =
∅ with 1 ≤ e, f ≤ i, e ̸= f . Suppose that there exists an element p ∈ Ae ∩ Af , with
xe := (ae, be) and xf := (af , bf ). Then p = (ae, µ) = (af , ν) for some µ, ν ∈ Γ. We can
see that ae = af , which implies that xe is in the second Cartesian line through xf . This is
a contradiction since Ae ̸= Af .
From all of the above we can conclude that we obtain a partition of the block A. We
know that the size of A is k, and on the other hand A =
⋃j=i
j=0 Aj .
They are all disjoint and the size of each Aj is r+1, so k = i(r+1), and r+1 divides
k.
Let (x,A) be a flag such as in Theorem 3.7, that is, A intersects only the second Carte-
sian line through x. We count the number of flags (y, C) such that x ∈ C and y ̸= x is in
the second Cartesian line through x.
The number of these flags is the same as the number of blocks that contain x as well as
elements of the second Cartesian line through x, (we denote this number by z), multiplied
by the number of elements of the second Cartesian line through x (excluding x) which are
in these blocks, that is, r, therefore the number of such flags (y, C) is zr.
On the other hand, x and y are together in λ blocks and there are m − 1 points of the
second Cartesian line through x, so when we count these flags (y, C) we obtain λ(m− 1).
The above implies the equation zr = λ(m − 1), but the equation kr = 2λ(m − 1)
also holds, hence z = k2 and since z ∈ N, k is even. This means that half of the blocks
that contain x intersect with the second Cartesian line through x and the other half intersect
with the first Cartesian line through x. This is possible since the previous argument is also
valid for the first Cartesian line through x.
In the following theorem we examine Case (ii).
Theorem 3.8 (Case (ii)). Let D be a (v, k, λ)-symmetric design with v = m2 admitting a
flag-transitive, point-primitive automorphism group with product action. If for every point
254 Ars Math. Contemp. 23 (2023) #P2.04 / 247–260
x in the design, every block that contains it intersects with the two Cartesian lines through
x, then r2 + 1 divides k.
Proof. Let x = (a0, b0) be an arbitrary point in the design, and let A be a block containing
x, then there are r1 elements of the first Cartesian line through x (excluding x) in A and
there are r2 elements of the second Cartesian line through x (excluding x) in A. The
numbers r1 and r2 satisfy the equation r = r1 + r2, by Lemma 2.2.
If C is another block containing x, then it must intersect the two Cartesian lines through
x. Since G acts transitively on the flags, there is an element g ∈ G such that g(x,A) =
(x,C) and from this we can see that g(x) = x, that is, g|Γ fixes a0 and b0.
First, we will prove that g sends the elements of the first Cartesian line through x
which are also in A to elements of the first Cartesian line through x which are also in
C. Let (µ, b0) be an element of the first Cartesian line through x which is also in A.
Then g(µ, b0) = (ν, b0) ∈ C for some ν ∈ Γ since g|Γ fixes b0. Similarly g sends the
elements of the second Cartesian line through x which are also in A to elements of the
second Cartesian line through x which are also in C. Let(a0, µ) be an element of the first
Cartesian line through x which is also in A, then g(a0, µ) = (a0, ν) ∈ C for some ν ∈ Γ
since g|Γ fixes a0. Therefore the block C has as many elements of the first Cartesian line
through x as A, and as many elements of the second Cartesian line through x as A. The
above is true for every block that contains x.
Now let us count the number of flags (y, C) of the design such that y ̸= x is an element
of the first Cartesian line through x and C is a block containing x. Every block contains r1
elements of the first Cartesian line through x, when we exclude x, and there are k blocks
containing x. All of them intersect the first Cartesian line through x, therefore there are kr1
flags of this type. On the other hand y and x are together in λ blocks and there are m − 1
elements of the first Cartesian line through x (excluding x), hence there are λ(m − 1)
flags of this type. This yields the equation kr1 = λ(m − 1), but from Lemma 2.2 the
equation kr = 2λ(m− 1) also holds and we conclude that r1 = r2 . However r1 + r2 = r,
so the intersection of every block containing x with the second Cartesian line through x
(excluding x) has r2 = r2 elements.
The above is true for every x, that is, for every point in the design, every block that
contains it intersects the first Cartesian line through that point in r2 other points and the
same holds for the second Cartesian line through that point (excluding the point itself).
In what follows we will consider A0 to be the set of points of the second Cartesian
line through x which are also in A including x itself. The number of elements in that set
is r2 + 1. Let us consider x1 ∈ A \ A0 so from the previous paragraphs A intersects the
second Cartesian line through x1 in r2 elements, thus if A1 is the set of points of the second
Cartesian line through x1 which are also in A including x1 itself, the number of elements
in A1 is r2 + 1. Now we take an element x2 ∈ A \ (A0 ∪ A1) in the same way as before,
and let A2 be the set of points of the second Cartesian line through x2 which are also in A
including x2 itself. The number of elements in A2 is r2 + 1.
We can continue this process in this way until there are no more elements in A, (ev-
erything is finite), so we obtain a collection of points x0, x1, ..., xi ∈ A and a collection
of sets A0, A1, ..., Ai for some natural number i such that for all j = 0, . . . , i Aj is the
intersection of the second Cartesian line through xj with A, and Aj has r2 +1 elements. By
construction, A =
⋃j=i
j=0 Aj and the construction implies that if xg ̸= xh then Ag ̸= Ah
with 1 ≤ g, h ≤ i.
E. O’Reilly-Regueiro et al.: A parametrisation for symmetric designs . . . 255
It remains to prove that every two sets in this collection are disjoint, that is, we must
prove that if Ae ̸= Af then Ae∩Af = ∅ (with 1 ≤ e, f ≤ i and e ̸= f ). Suppose there is an
element p ∈ Ae ∩Af , with xe := (ae, be) and xf := (af , bf ). Then p = (ae, µ) = (af , ν)
for some µ, ν ∈ Γ. We can see that ae = af , which implies that xe is in the second
Cartesian line through xf , a contradiction since Ae ̸= Af .
Therefore we have a partition of the block A =
⋃j=i
j=0 Aj . The size of
⋃j=i
j=0 Aj is
i( r2 +1) since they are all disjoint, and the size of A is k, therefore k = i(
r
2 +1) and
r
2 +1
divides k.
Now we will present some consequences of Theorem 3.8.
Corollary 3.9. With the same hypotheses of Theorem 3.8, r2 + 1 divides m.
Proof. From (3.2) we have
k =
r
2
m+
r
2
+ 1,
and there is an integer p such that k = p( r2 + 1). Substituting in the previous equation we
obtain (p− 1)
(
r
2 + 1
)
= r2m. Since
(
r
2 + 1,
r
2
)
= 1, r2 + 1 necessarily divides m.
Corollary 3.10. With the same hypotheses of Theorem 3.8, r2 + 1 divides λ.
Proof. There is an integer p such that k = p( r2 + 1), and substituting this and (3.2) in
k(k − 1) = λ(m− 1)(m+ 1), we obtain
p
r
2
(r
2
+ 1
)
= λ(m− 1).
By Corollary 3.9, r2 + 1 divides m, so
(
r
2 + 1,m− 1
)
= 1 and r2 + 1 divides λ.
Since t is the greatest common divisor of k and λ, the following holds:
Corollary 3.11. With the same hypotheses of Theorem 3.8, r2 + 1 divides t.
Proof. Since r2 + 1 divides k and λ, and also (k, λ) = t we conclude
r
2 + 1 divides t.
In the next results, we will introduce a particular case in which we have obtained the
parameters of a Menon design, as a consequence of Corollary 3.11. Since r2 + 1 divides
t > 1, we will first consider the case in which t is a prime number. The following result is
a first approach to our main result.
Lemma 3.12. Let D be a (v, k, λ)-symmetric design with (k, λ) = t > 1 a prime number
and v = m2, admitting a flag-transitive, point-primitive automorphism group G. If for
every point x in the design, every block that contains it intersects the two Cartesian lines
through x, then either G does not have a product action or D is a Menon design.
Proof. From Corollary 3.11, we have r2 + 1 divides t. Since t is a prime we obtain
r
2 = 0
or r2 + 1 = t.
If r2 = 0 then from (3.2) we have k − 1 = 0 and this is impossible.
If on the other hand r2 + 1 = t, then m =
r2+2r+4λ
4λ−r2 so
m =
bs2 + s+ t
t− bs2
(3.11)
256 Ars Math. Contemp. 23 (2023) #P2.04 / 247–260
which implies t ≥ bs2. If t = bs2 then λ = b2s2 = r
2
4 . Substituting this in (3.1) we
obtain
r2(m+ 1) + 2r − r2(m− 1) = 0, therefore r(r + 1) = 0,
so r = 0 or r = −1 which is a contradiction, so t > bs2.
This forces t = r2 + 1 = bs+ 1 > bs
2, so 1 > bs(s− 1) and s = 1. From Lemma 3.3,
v = 4t2, k = 2t2 − t and λ = t2 − t, which are the parameters of a Menon design.
Remark 3.13. The triples of parameters (4900, 3267, 2178), (16900, 2752, 448), and
(44100, 8019, 1458), do not correspond to Menon designs but they satisfy all known nec-
essary arithmetic conditions on the existence of a symmetric design with v even, so we do
not prove the conjeture for v ≤ (210)2 (we have not tried computational methods).
Lemma 3.14. Let D be a (v, k, λ)-symmetric design with (k, λ) = t > 1 and v = m2 ≤
(210)2 with m even admitting a flag-transitive point-primitive automorphism group G,
then either G does not have a non-trivial product action or one of the following conditions
holds:
1. D is a Menon design with parameters (4t2, 2t2 − t, t2 − t), where t > 1, or
2. D has parameters (16900, 2752, 448).
Proof. For m ≤ 210 the admissible parameters that do not satisfy the conditions of Menon
designs and in which k − λ is a square are (4900, 3267, 2178), (16900, 2752, 448), and
(44100, 8019, 1458). For these, r2 + 1 is 47, 22 and 39 respectively, so they do not satisfy
Theorem 3.8. Now for those parameters r + 1 is 93, 43 and 78 respectively. The first and
third of them do not satisfy Theorem 3.7, but the parameters (16900, 2752, 448) do. Thus,
these are the only possible parameters for m2 ≤ (210)2.
In this case, k is even, which is consistent with Theorem 3.7. We also have s = 3 so
from Lemma 3.3 we know that these parameters cannot correspond to a Menon design.
Remark 3.15. The triple (16900, 2752, 448) does not correspond to a Menon design since
s = 3, although it satisfies all the arithmetic conditions for a symmetric design. We make
no claim as to whether such a design exists, but perhaps it is not the case that when l = 2
only Menon designs are possible (if at all).
The following is our main result, the proof follows Cases (i) and (ii) from
Theorems 3.7 and 3.8, that is, either: there is a flag (x,A), such that the block A inter-
sects only one Cartesian line through x (Case (i)), or for every point x, every block that
contains it intersects both of the Cartesian lines through x (Case (ii)). The proof based on
Case (i) is similar to the proof of the case in which m − 1 is the power of an odd prime.
In this sense it is a generalisation of this proof, but because of the existence of the param-
eter s this generalisation was not obtained in an obvious way. For this reason, we need an
additional arithmetic condition, which is found in Corollary 3.11.
In the proof based on Case (ii) we also obtain an arithmetic condition for a and so
also for k. We believe we do not necessarily obtain parameters for Menon designs for an
arbitrary λ when we study symmetric designs admitting a flag-transitive point-primitive
E. O’Reilly-Regueiro et al.: A parametrisation for symmetric designs . . . 257
automorphism group with product action when l = 2. This case also gives us a parametri-
sation of (v, k, λ) in terms of t and s, and if s = 1 then the parameters correspond to Menon
designs, that is, our parameterisation is a generalisation of the parameterisation of Menon
designs.
Theorem 3.16. Let D a (v, k, λ)-symmetric design admitting a flag-transitive, point-
primitive, automorphism group G with (k, λ) = t > 1 and v = m2 with m even. Then
either:
(i) G does not have a non-trivial product action, or
(ii) D is a design with parameters
(
(2t+ s− 1)2, 2t
2−(2−s)t
s ,
t2 − t
s2
)
with s ≥ 1 odd.
When s = 1 D is a Menon design and if s > 1 then t is even.
Proof. From the hypotheses and from Lemma 2.2 there are two possible cases. For any
given point, either each block that contains it intersects the two Cartesian lines through it,
or there is a point such that a block containing it only intersects one Cartesian line through
it. As we have seen, the latter implies that every block intersects only one Cartesian line
through each point it contains.
First we will study this last case. Here, Theorem 3.7 is satisfied, so r+1 divides k with
r > 1 an integer such that kr = 2λ(m − 1). This implies there is an integer p such that
k = p(r + 1). Also k − 1 = r2 (m+ 1) holds, so k =
r
2m+
r
2 + 1, and therefore
m− 1 = (r + 1)(m+ 1− 2p).
Then r + 1 divides m − 1, but m − 1 = as = x(r + 1) where x := m + 1 − 2p and
since r = 2bs we have (r+1, s) = 1 so r+1 divides a. Also a divides r+1 which forces
r+1 = a, and this implies k = (r+1)t. This all implies t− bs2 = 1, so b = t−1s2 , and we
obtain the parameters λ = t−1s2 t, k =
2t+s−2
s t, v = (2t+ s− 1)
2.
The proof of Theorem 3.7 states that k should be an even number and since r + 1 is an
odd number then t should be an even number. Since m = 2t + s − 1 and m is an even
number then s is an odd number.
The triple (16900, 2752, 448) satisfies the conditions we obtained, with t = 64 and
s = 3, so this is not a Menon design.
When s = 1 we obtain the parametrisation for Menon designs with t an even number.
Now suppose that for every point, every block that contains it intersects both Cartesian
lines through it. Here the hypotheses of Theorem 3.8 hold and from Corollary 3.11, there
exists x ≥ 1 such that
t =
(r
2
+ 1
)
x = (bs+ 1)x. (3.12)
From (3.11) we obtain m = s(bs+1)+tt−bs2 =
s(bs+1)+(bs+1)x
t−bs2 , that is
m =
(
s+ x
t− bs2
)(r
2
+ 1
)
. (3.13)
Using (3.12) we obtain
t− bs2 = x+ bs(x− s), (3.14)
which we divide into the following cases:
258 Ars Math. Contemp. 23 (2023) #P2.04 / 247–260
1. x < s
From Lemma 3.12, we have t−bs2 > 0, and from (3.14) x > bs(s−x) > bx(s−x).
Since x < s then 1 > b(s−x) > 0, but this cannot be the case since b(s−x) should
be an integer.
2. s < x
From (3.13), s+x ≥ t−bs2. If s+x = t−bs2 we have from (3.13) that m = r
2
+1
so as = m − 1 = r
2
= bs and therefore a = b, but (a, b) = 1 so a = 1 this is
impossible by Lemma 3.1. Therefore s + x > t − bs2, and from (3.14) we have
s+x > x+ bs(x− s) so 1 > b(x− s) > 0 and this is also imposisble since b(x− s)
is an integer.
3. s = x
From (3.12), s divides t, and from (3.2) at − bs(as + 2) = 1, so (t, s) = 1 and
therefore s = 1. Also from Lemma 3.3, v = 4t2, k = 2t2 − t, and λ = t2 − t
This concludes the proof.
The parameters of Menon designs are not the only ones we can obtain when we assume
that the automorphism group of the design has a product action on the points of the design,
and the parameters (16900, 2752, 448) are an example of this. However we note that a
design with the possible parameters which arise and do not correspond to Menon designs
must satisfy that each block only intersects one Cartesian line through each point in that
block.
It is not the case that the way in which we consider product action to obtain possi-
ble Menon designs does not work because here is a potential counterexample, but rather
that with this theorem we give explicit expressions for the parameters v, k, λ, in terms of
parameters s, t, and when s = 1 they do correspond to Menon designs.
4 One further result
Here we present an additional result for any l ≥ 2.
Theorem 4.1. Let D a (v, k, λ)-symmetric design with v = ml admitting a flag-transitive,
point-primitive, automorphism group G with a non-trivial product action. Then the com-
plement of the design is not flag-transitive.
Proof. Suppose that D′ is the complement of the design D, so its parameters are
(v′, k′, λ′) = (v, v − k, v − 2k + λ). If we also assume D′ is flag-transitive, then the
following equation holds:
(v − k)(v − k − 1) = (v − 2k + λ)(m− 1)(ml−1 +ml−2 + ...+ 1). (4.1)
If D has a point-primitive automorphisms group G, then D′ has the same point-primitive
automorphisms group G and we can consider the Cartesian lines through a point, since G
is transitive on the points of D′. Thus k′ divides λ′l(m − 1), so there is an integer p such
that
(v − k)p = l(v − 2k + λ)(m− 1). (4.2)
E. O’Reilly-Regueiro et al.: A parametrisation for symmetric designs . . . 259
Substituting this in (4.1) we obtain
l(v − k)(v − 1− k) = (v − k)p(ml−1 +ml−2 + ...+ 1)
so l((m− 1)(ml−1 +ml−2 + ...+ 1)− k) = p(ml−1 +ml−2 + ...+ 1),
hence
lk = q(ml−1 +ml−2 + ...+ 1) (4.3)
with q = l(m− 1)− p > 0.
But for D we know that k =
lλ(m− 1)
r
and k(k − 1) = λ(v − 1) so
k(k − 1) = kr
l(m− 1)
(ml − 1)
and we obtain a generalisation of (3.2):
l(k − 1) = r(ml−1 +ml−2 + ...+ 1). (4.4)
If we substitute (4.3) in (4.4) then
q(ml−1 +ml−2 + ...+ 1)− l = r(ml−1 +ml−2 + ...+ 1)
so (q − r)(ml−1 +ml−2 + ...+ 1) = l,
and therefore ml−1 +ml−2 + ... + 1 ≤ l if m > 1 and l ≥ ml−1 +ml−2 + ... + 1 > l,
which is impossible.
We conclude m ≤ 1, but this is a contradiction since m ≥ 5.
ORCID iDs
Eugenia O’Reilly-Regueiro https://orcid.org/0000-0001-5867-7258
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.05 / 261–280
https://doi.org/10.26493/1855-3974.2692.86d
(Also available at http://amc-journal.eu)
Mutually orthogonal cycle systems*
Andrea C. Burgess †
Department of Mathematics and Statistics, University of New Brunswick,
Saint John, NB, E2L 4L5, Canada
Nicholas J. Cavenagh
Department of Mathematics, The University of Waikato, Private Bag 3105,
Hamilton 3240, New Zealand
David A. Pike
Department of Mathematics and Statistics, Memorial University of Newfoundland,
St. John’s, NL, A1C 5S7, Canada
Received 8 September 2021, accepted 23 June 2022, published online 17 November 2022
Abstract
An ℓ-cycle system F of a graph Γ is a set of ℓ-cycles which partition the edge set of
Γ. Two such cycle systems F and F ′ are said to be orthogonal if no two distinct cycles
from F ∪F ′ share more than one edge. Orthogonal cycle systems naturally arise from face
2-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A
set of pairwise orthogonal ℓ-cycle systems of Γ is said to be a set of mutually orthogonal
cycle systems of Γ.
Let µ(ℓ, n) (respectively, µ′(ℓ, n)) be the maximum integer µ such that there exists a
set of µ mutually orthogonal (cyclic) ℓ-cycle systems of the complete graph Kn. We show
that if ℓ ≥ 4 is even and n ≡ 1 (mod 2ℓ), then µ′(ℓ, n), and hence µ(ℓ, n), is bounded
below by a constant multiple of n/ℓ2. In contrast, we obtain the following upper bounds:
µ(ℓ, n) ≤ n − 2; µ(ℓ, n) ≤ (n − 2)(n − 3)/(2(ℓ − 3)) when ℓ ≥ 4; µ(ℓ, n) ≤ 1 when
ℓ > n/
√
2; and µ′(ℓ, n) ≤ n − 3 when n ≥ 4. We also obtain computational results for
small values of n and ℓ.
Keywords: Orthogonal cycle decompositions, cyclic cycle systems, Heffter arrays, completely-redu-
cible, super-simple.
Math. Subj. Class. (2020): 05B30
*Authors A.C. Burgess and D.A. Pike acknowledge research support from NSERC Discovery Grants RGPIN-
2019-04328 and RGPIN-2016-04456, respectively. Thanks are given to the Centre for Health Informatics and
Analytics of the Faculty of Medicine at Memorial University of Newfoundland for access to computational re-
sources.
†Corresponding author.
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
262 Ars Math. Contemp. 23 (2023) #P2.05 / 261–280
1 Introduction
We say that a graph Γ decomposes into subgraphs Γ1,Γ2, . . . ,Γt, if the edge sets of the Γi
partition the edges of Γ. If F = {Γi | 1 ≤ i ≤ t} where Γi ∼= H for each 1 ≤ i ≤ t,
then we say that F is an H-decomposition of Γ. An ℓ-cycle system of a graph Γ is a
decomposition of Γ into ℓ-cycles. In the case where Γ is the complete graph Kn we say
that there is an ℓ-cycle system of order n. Necessary and sufficient conditions for the
existence of an ℓ-cycle system of order n were given in [1, 26]; see also [6]. Namely, at
least one ℓ-cycle system of order n > 1 exists if and only if 3 ≤ ℓ ≤ n, n(n − 1) ≡ 0
(mod 2ℓ) and n is odd.
Two ℓ-cycle systems F and F ′ of the same graph Γ are said to be orthogonal if, for all
cycles C ∈ F and C ′ ∈ F ′, C and C ′ share at most one edge. A set of pairwise orthogonal
ℓ-cycle systems of Γ is said to be a set of mutually orthogonal cycle systems of Γ. In
this paper we are interested in the maximum µ such that there exists a set of µ mutually
orthogonal ℓ-cycle systems of order n; we denote this value by µ(ℓ, n).
In the array below we exhibit a set of four mutually orthogonal cycle systems of order
9. We have determined computationally that µ(4, 9) = 4; i.e., this set is maximum.
{(1, 2, 3, 4), (1, 3, 6, 5), (1, 6, 2, 7), (1, 8, 2, 9), (2, 4, 7, 5), (3, 5, 8, 7), (3, 8, 6, 9),
(4, 5, 9, 8), (4, 6, 7, 9)},
{(1, 2, 6, 8), (1, 3, 5, 7), (1, 4, 8, 5), (1, 6, 5, 9), (2, 3, 6, 4), (2, 5, 4, 9), (2, 7, 3, 8),
(3, 4, 7, 9), (6, 7, 8, 9)},
{(1, 2, 8, 7), (1, 3, 4, 6), (1, 4, 9, 8), (1, 5, 3, 9), (2, 3, 8, 5), (2, 4, 5, 6), (2, 7, 5, 9),
(3, 6, 9, 7), (4, 7, 6, 8)},
{(1, 2, 9, 3), (1, 4, 6, 9), (1, 5, 4, 7), (1, 6, 3, 8), (2, 3, 7, 6), (2, 4, 8, 7), (2, 5, 6, 8),
(3, 4, 9, 5), (5, 7, 9, 8)}.
Orthogonal cycle systems arise from face 2-colourable embeddings of graphs on sur-
faces, which satisfy two conditions natural to polyhedra and similar phenomena: each pair
of faces share at most one edge and each edge belongs to exactly two faces.
Let µKn be the multigraph in which each edge of Kn is replaced by µ parallel edges. A
decomposition F of µKn into a subgraph H is said to be super-simple if no two copies of H
share more than one edge, and completely-reducible if F partitions into µ decompositions
of Kn. It follows that a set of µ mutually orthogonal cycle systems of Kn is equivalent to
a completely-reducible super-simple decomposition of µKn into cycles; see [12] for more
details.
In the case ℓ = 3, observe that a pair of ℓ-cycle systems is orthogonal if and only if
the cycle systems are disjoint. It is not hard to see that there are at most n − 2 pairwise
disjoint triple systems of order n; a set of systems which meets this bound is called a large
set of disjoint Steiner triple systems, or LTS(n). An LTS(7) does not exist [13]; however
E-mail addresses: andrea.burgess@unb.ca (Andrea C. Burgess), nickc@waikato.ac.nz (Nicholas
J. Cavenagh), dapike@mun.ca (David A. Pike)
A. C. Burgess et al.: Mutually orthogonal cycle systems 263
in [23, 24], it is shown that an LTS(n) exists if and only if n > 7 and n ≡ 1 or 3 (mod 6),
except for a finite list of possible exceptions. The exceptional cases are all solved in [27].
In this paper, we are often interested in cyclic cycle systems of the complete graph
Kn. Let G be an additive group of order n and suppose Kn has vertex set G. Given a
cycle C = (c0, c1, . . . , cℓ−1) in Kn, for each element g ∈ G, define the cycle C + g =
(c0+ g, c1+ g, . . . , cℓ−1+ g). We say that a cycle system F of Kn is G-regular if, for any
C ∈ F and g ∈ G, we have that C + g ∈ F . In the case that G is a cyclic group, we refer
to a Zn-regular cycle system as cyclic. In a cyclic cycle system F , the orbit of the cycle
C ∈ F is the set of cycles {C + g | g ∈ Zn}; a cyclic cycle system can be completely
specified by listing a set of starter cycles, that is, a set of representatives for the orbits of
the cycles under the action of Zn.
The existence problem for cyclic cycle systems has attracted much attention. Clearly,
in order for a cyclic ℓ-cycle system of odd order n to exist, we must have that 3 ≤ ℓ ≤ n
and ℓ divides n(n−1)/2. However, additional conditions for existence also come into play.
There is no cyclic ℓ-cycle system of order n when (ℓ, n) ∈ {(3, 9), (15, 15)}; ℓ = n = pm
for some prime p and integer m ≥ 2; or ℓ < n < 2ℓ and gcd(ℓ, n) is a prime power [7, 9].
Buratti [7] has conjectured that a cyclic ℓ-cycle system of order n exists for any other
admissible pair (ℓ, n); this conjecture is still open. The existence problem for cyclic cycle
systems of the complete graph has been solved in a number of cases, including when n ≡ 1
or ℓ (mod 2ℓ) [8, 9, 22, 25, 28] (see also [4, 5, 18]), ℓ ≤ 32 [31, 32], ℓ is twice or thrice a
prime power [30, 31, 32], or ℓ is even and m > 2ℓ [29].
We explore the maximum µ′ such that there exists a set of µ′ mutually orthogonal cyclic
ℓ-cycle systems of order n; this value is denoted by µ′(ℓ, n). Pairs of orthogonal cyclic
cycle systems of the complete graph arise from Heffter arrays with certain orderings. A
Heffter array H(n; k) is an n × n matrix such that each row and column contains k filled
cells, each row and column sum is divisible by 2nk + 1 and either x or −x appears in the
array for each integer 1 ≤ x ≤ nk. A Heffter array is said to have a simple ordering if,
for each row and column, the entries may be cyclically ordered so that all partial sums are
distinct modulo 2nk + 1. The following was first shown by Archdeacon [2] as part of a
more general result; consult [11] to see this result stated more explicitly.
Theorem 1.1. If H(n; k) is a Heffter array with a simple ordering, then there exists a pair
of orthogonal cyclic decompositions of K2nk+1 into k-cycles. In particular,
µ′(k, 2nk + 1) ≥ 2.
Thus the following is implied by existing literature on Heffter arrays.
Theorem 1.2 ([3, 11, 14, 17]). Let n ≥ k. Then µ′(k, 2nk + 1) ≥ 2 whenever:
• k ∈ {3, 5, 7, 9} and nk ≡ 3 (mod 4);
• k ≡ 0 (mod 4);
• n ≡ 1 (mod 4) and k ≡ 3 (mod 4);
• n ≡ 0 (mod 4) and k ≡ 3 (mod 4) (for large enough n).
With an extra condition on the orderings of the entries of a Heffter array, these orthogo-
nal cycle systems in turn biembed to yield a face 2-colourable embedding on an orientable
surface. Face 2-colourable embeddings on orientable surfaces have been studied for a va-
riety of combinatorial structures [16, 19, 20, 21]. Recently, Costa, Morini, Pasotti and
264 Ars Math. Contemp. 23 (2023) #P2.05 / 261–280
Pellegrini [15] employed a generalization of Heffter arrays to construct pairs of orthogonal
ℓ-cycle systems of the complete multipartite graph in certain cases.
In [12], it is shown that for every graph H and fixed integer k ≥ 1, for sufficiently
large n (satisfying some elementary necessary divisibility conditions), there exists a set of
k pairwise orthogonal decompositions of Kn into H (i.e., no two copies of H share more
than one edge). Aside from this quite general asymptotic result, to our knowledge, sets of
mutually orthogonal ℓ-cycle systems of size greater than 2 have not been studied for ℓ ≥ 4.
In this paper, our focus for cyclic cycle systems is in the case n ≡ 1 (mod 2ℓ), for
which it is possible to construct a cyclic ℓ-cycle system with no short orbit. In particular,
we will find lower bounds on µ(ℓ, n) by constructing sets of mutually orthogonal cyclic
even cycle systems. Specifically, we show that if ℓ is even and n ≡ 1 (mod 2ℓ), then
µ′(ℓ, n) is bounded below by a constant multiple of n/ℓ2, i.e., µ′(ℓ, n) = Ω(n/ℓ2). Our
main result is as follows.
Theorem 1.3. If ℓ ≥ 4 is even, n ≡ 1 (mod 2ℓ) and N = (n− 1)/(2ℓ), then
µ(ℓ, n) ≥ µ′(ℓ, n) ≥ N
aℓ+ b
− 1,
where
(a, b) =
{
(4,−2), if ℓ ≡ 0 (mod 4),
(24,−18), if ℓ ≡ 2 (mod 4).
In Section 2, when ℓ = 4, we improve the bound of Theorem 1.3 to µ(ℓ, n) ≥
µ′(ℓ, n) ≥ 4N (Lemma 2.1). Section 3 establishes some notation and preliminary results.
The general result for ℓ ≡ 0 (mod 4) is proved in Section 4 (Theorem 4.3), while the
bound for ℓ ≡ 2 (mod 4) is proved in Section 5 (Theorem 5.5). In contrast, in Section 6
we establish upper bounds, namely µ(ℓ, n) ≤ n− 2; µ(ℓ, n) ≤ (n− 2)(n− 3)/(2(ℓ− 3))
for ℓ ≥ 4; µ(ℓ, n) ≤ 1 for ℓ >
√
n(n− 1)/2; and µ′(ℓ, n) ≤ n − 3 for n ≥ 4. Finally,
computational results for small values are given in the appendix.
2 Mutually orthogonal 4-cycle systems
Clearly n ≡ 1 (mod 8) is a necessary condition for a decomposition of Kn into 4-cycles,
cyclic or otherwise. Let [a, b, c, d]n denote the Zn-orbit of the 4-cycle (0, a, a+b, a+b+c),
where a+b+c+d is divisible by n. Observe that [a, b, c, d]n = [−d,−c,−b,−a]n. Where
the context is clear, we write [a, b, c, d]n = [a, b, c, d]. Let Dn = {1, 2, . . . , (n − 1)/2};
that is, Dn is the set of differences in Zn. We consider Zn as the set ±Dn ∪ {0}.
By observation, the maximum size of a set of mutually orthogonal cyclic 4-cycle sys-
tems of K9 is µ′(4, 9) = 2. Two such systems are [1,−2, 4,−3]9 and [1,−3, 4,−2]9. In
the non-cyclic case, an exhaustive computational search indicates that the maximum size of
a set of mutually orthogonal 4-cycle systems of K9 is µ(4, 9) = 4; see the example given
in Section 1.
Lemma 2.1. If n ≡ 1 (mod 8) and n ≥ 17, then there exists a set of (n − 1)/2 mutually
orthogonal cyclic 4-cycle systems of order n. In particular, µ′(4, n) ≥ (n− 1)/2.
Proof. We first describe how to construct a set of (n − 5)/2 mutually orthogonal cyclic
4-cycle systems; then we add two more by making some adjustments.
A. C. Burgess et al.: Mutually orthogonal cycle systems 265
Let N = (n − 1)/8. For each i, j with 1 ≤ i < j ≤ 2N , let Ci,j and C ′i,j be the pair
of orbits of 4-cycles:
Ci,j := {[2i− 1, 2j,−2i,−(2j − 1)]}, C ′i,j := {[2i− 1,−(2j − 1),−2i, 2j]}.
Next, let F1, F2, . . . F2N−1 be a set of 1-factors which decompose the complete graph on
vertex set {1, 2, . . . , 2N}.
For each 1-factor Fk, the sets
Fk :=
⋃
{i,j}∈Fk
i<j
Ci,j and F ′k :=
⋃
{i,j}∈Fk
i<j
C ′i,j
each describe a cyclic decomposition of Kn into 4-cycles. Observe that the set of such
decompositions constitutes a mutually orthogonal set of size 4N − 2 = (n− 5)/2.
We next make an adjustment to extend this set. Without loss of generality, let F1 =
{{1, 2}, {3, 4}, . . . , {2N − 1, 2N}}. Replace F1 and F ′1 with:
F∗ = {[4i− 3,−(4i− 2),−(4i− 1), 4i] | 1 ≤ i ≤ N},
F ′∗ = {[4i− 3, 4i,−(4i− 1),−(4i− 2)] | 1 ≤ i ≤ N}.
Then, we can add another pair of cyclic decompositions, orthogonal to each decomposition
in {F∗,F ′∗,F2, . . . ,F2N−1,F ′2, . . . ,F ′2N−1}, given by:
F2N := {[1,−3, 4N,−(4N − 2)]} ∪ {[4i+ 1,−(4i+ 3), 4i,−(4i− 2)] | 1 ≤ i < N}
and
F ′2N := {[1,−(4N − 2), 4N,−3]} ∪ {[4i+ 1,−(4i− 2), 4i,−(4i+ 3)] | 1 ≤ i < N}.
(Note that orthogonality requires N ≥ 2 at this final step.)
In the case n = 17, we have computationally determined that µ′(4, 17) = 10, which
improves on the bound given in Lemma 2.1. A set of ten mutually orthogonal cyclic 4-cycle
systems of order 17 is given in the appendix.
We exhibit the methods of the previous proof in the case n = 25. We start with a
1-factorization of K6:
F1 = {{1, 2}, {3, 4}, {5, 6}},
F2 = {{1, 3}, {2, 6}, {4, 5}},
F3 = {{1, 4}, {2, 5}, {3, 6}},
F4 = {{1, 5}, {2, 3}, {4, 6}},
F5 = {{1, 6}, {2, 4}, {3, 5}}.
The resulting 12 mutually orthogonal cyclic 4-cycle systems of order 25 are given by:
266 Ars Math. Contemp. 23 (2023) #P2.05 / 261–280
F∗ = {[1,−2,−3, 4], [5,−6,−7, 8], [9,−10,−11, 12]},
F ′∗ = {[1, 4,−3,−2], [5, 8,−7,−6], [9, 12,−11,−10]},
F2 = {[1, 6,−2,−5], [3, 12,−4,−11], [7, 10,−8,−9]},
F ′2 = {[1,−5,−2, 6], [3,−11,−4, 12], [7,−9,−8, 10]},
F3 = {[1, 8,−2,−7], [3, 10,−4,−9], [5, 12,−6,−11]},
F ′3 = {[1,−7,−2, 8], [3,−9,−4, 10], [5,−11,−6, 12]},
F4 = {[1, 10,−2,−9], [3, 6,−4,−5], [7, 12,−8,−11]},
F ′4 = {[1,−9,−2, 10], [3,−5,−4, 6], [7,−11,−8, 12]},
F5 = {[1, 12,−2,−11], [3, 8,−4,−7], [5, 10,−6,−9]},
F ′5 = {[1,−11,−2, 12], [3,−7,−4, 8], [5,−9,−6, 10]},
F6 = {[1,−3, 12,−10], [5,−7, 4,−2], [9,−11, 8,−6]},
F ′6 = {[1,−10, 12,−3], [5,−2, 4,−7], [9,−6, 8,−11]}.
Through computational means we determined that this collection of 12 mutually or-
thogonal cyclic 4-cycle systems of order 25 is maximal. However, it is not maximum, as
we also established computationally that µ′(4, 25) ≥ 17.
3 Preliminary lemmas for cycle length greater than 4
In this section, we introduce notation and basic results which will be needed later to con-
struct mutually orthogonal cycle systems with even cycle length ℓ ≥ 6.
Henceforth, for any integers a and b with a ≤ b, [a, b] is the set of integers {a, a +
1, . . . , b}. For a, b ∈ R with a < b, we also use the notation (a, b) to denote the set of
integers strictly between a and b.
Let the vertices of the complete graph Kn be labelled with [0, n − 1], where n is odd.
Then the difference associated with an edge {a, b} is defined to be the minimum value in the
set {|a− b|, n− |a− b|}. Let e1 and e2 be two edges of differences d and e, respectively.
Then we may write e1 = {a, a + d (mod n)} and e2 = {b, b + e (mod n)}, where
a, b ∈ [0, n− 1] are uniquely determined. We define the distance between e1 and e2 to be
the minimum value in the set {|a − b|, n − |a − b|}. Given a cycle C with vertices in Zn,
the set ∆C is defined to be the multiset of differences of the edges of C.
The idea is to construct cyclic systems using so-called balanced sets of differences.
The following definitions and lemma appear in [10].
Definition 3.1. If D = {d1, d2, . . . , d2k} is a set of positive integers, with di < di+1 for
i ∈ [1, 2k − 1], the alternating difference pattern of D is the sequence (s1, s2, . . . , sk)
where si = d2i − d2i−1 for every i ∈ [1, k]. Furthermore, D is said to be balanced if there
exists an integer τ ∈ [1, k] such that
∑τ
i=1 si =
∑k
i=τ+1 si.
Definition 3.2. Let D = {d1, d2, . . . , d2k} be a balanced set of positive integers. Let δ1,
δ2, . . . , δ2k be the sequence obtained by reordering the integers in D as follows:
δi =

di if 1 ≤ i ≤ 2τ − 1,
di+1 if 2τ ≤ i ≤ 2k − 1,
d2τ if i = 2k.
A. C. Burgess et al.: Mutually orthogonal cycle systems 267
Set c0 = 0 and ci =
∑i
h=1(−1)hδh for 1 ≤ i ≤ 2k − 1. We then define C(D) :=
(c0, c1, . . . , c2k−1).
Lemma 3.3 (Lemma 3.2 of [10])). Let k ≥ 2. If D is a balanced set of 2k positive integers,
then C(D) is a 2k-cycle satisfying ∆C(D) = D and vertex set V (C(D)) ⊂ [−d, d′],
where d = maxD and d′ = max(D \ {d}).
Corollary 3.4. Let k ≥ 2 and n ≡ 1 (mod 4k). Suppose that the set [1, (n − 1)/2]
partitions into sets D1, D2, . . . , D(n−1)/(4k), each of which is balanced and of size 2k.
Then cycles C(Di), i ∈ [1, (n− 1)/(4k)], form a set of starter cycles for a cyclic 2k-cycle
decomposition of Kn; in particular, the set
{C(Di) + j | i ∈ [1, (n− 1)/4k], j ∈ [0, n− 1]}
is a cyclic decomposition of Kn into 2k-cycles.
Proof. Let i ∈ [1, (n − 1)/2k]. Since Di ⊂ [1, (n − 1)/2], Lemma 3.3 implies that
V (C(Di)) ⊂ [−(n− 1)/2, (n− 1)/2]. Thus the vertices of V (C(Di)) are distinct in Zn.
The result follows.
Our general strategy will be to show that a pair of cyclically generated cycle systems is
orthogonal by showing that the sets of differences from any two cycles in different orbits
share at most one element. To this end, the following lemma will be used in Sections 4
and 5.
Lemma 3.5. Let δ,N > 0 and suppose there exist integers d and d′ such that d, d′ ∈
(N/2 − δN,N/2 + δN). If α and α′ are integers such that 1 ≤ α < α′ ≤ (1 − 2δ)/4δ,
then αd < α′d′.
Proof. Note that if δ > 110 , then the result is vacuously true since (1− 2δ)/4δ < 2. So we
assume δ ≤ 110 . For each positive integer s, define
Is = {si | N/2− δN < i < N/2 + δN ; i ∈ R}.
Let m = ⌊ 1−2δ4δ ⌋, and let S = [1,m]. Observe that α, α
′ ∈ S. Now, δ ≤ 1/(4m + 2)
implies that:
m(1 + 2δ) ≤ (m+ 1)(1− 2δ)
⇒ m(N/2 + δN) ≤ (m+ 1)(N/2− δN).
It follows that for each s ∈ S, every element of Is is strictly less than every element of
Is+1. Since αd ∈ Iα and α′d′ ∈ Iα′ , it follows that αd < α′d′.
The following variation of Lemma 3.5 will be used in Section 5.
Corollary 3.6. Let δ,N > 0 and suppose there exist integers d and d′ such that d, d′ ∈
(N/3 − δN,N/3 + δN). If α and α′ are integers such that 1 ≤ α < α′ ≤ (1 − 3δ)/6δ,
then αd < α′d′.
Proof. If m is a positive integer, m ≤ (1− 3δ)/6δ implies that
m(N/3 + δN) ≤ (m+ 1)(N/3− δN).
The remaining argument is similar to the previous lemma.
268 Ars Math. Contemp. 23 (2023) #P2.05 / 261–280
4 Orthogonal sets of 4k-cycle systems with k ≥ 2
Our aim in this section is to prove Theorem 4.3. In particular, for each k ≥ 2 and n ≡ 1
(mod 8k), we will show that µ′(n, 4k) = Ω(n/k2). That is, we construct a set of mutually
orthogonal 4k-cycle decompositions of Kn of size at least cn/k2 where c is a constant. In
particular, the number of mutually orthogonal decompositions constructed is⌈
n− 1
8k(16k − 2)
− 1
⌉
;
thus we have at least two orthogonal decompositions whenever
n− 1
8k(16k − 2)
> 2,
or equivalently n− 1 > 32k(8k − 1).
Let N and k be positive integers and let n = 8kN + 1. For each integer d ∈ (N/2 −
N/(16k−2), N/2) (there is at least one such integer whenever N > 16k−2), we construct
a cyclic 4k-cycle decomposition of Kn which we will denote by F(d).
The first d starter cycles in F(d) use the set of differences [1, 4kd]. For i ∈ [1, d], let
Sd,i = {i, d+ i, 2d+ i, . . . , (4k − 1)d+ i}.
Observe that the set Sd,i is balanced, with τ = k, for each i ∈ [1, d].
Henceforth in this section, let e := N − d. (In effect, e is a function of d.) Observe that
e ∈ (N/2, N/2 + N/(16k − 2)). The remaining e starter cycles in F(d) use differences
[4kd+ 1, 4kN ]. For i ∈ [1, e], take
Te,i = {4kd+ i, 4kd+ e+ i, 4kd+ 2e+ i, . . . , 4kd+ (4k − 1)e+ i}.
Observe that the set Te,i is balanced for each i ∈ [1, e], where τ = k. Moreover, since
4kd+ 4ke = 4kN , we have that(
d⋃
i=1
Sd,i
)
∪
(
e⋃
i=1
Te,i
)
= [1, 4kN ],
so by Corollary 3.4, the set of cycles
F(d) := {C(Sd,i) | i ∈ [1, d]} ∪ {C(Te,i) | i ∈ [1, e]}
is a set of starter cycles for a cyclic 4k-cycle system of order n = 8kN + 1.
In order to show that we have constructed an orthogonal set of decompositions, we will
make use of the following, which is a direct consequence of Lemma 3.5.
Lemma 4.1. Suppose d, d′ ∈ (N/2 − N/(16k − 2), N/2) such that d ̸= d′, and let
e = N − d and e′ = N − d′. Let α, α′ ∈ [1, 4k− 1]. Then no two of αd, α′d′, αe and α′e′
are equal. Moreover, if α < α′ then αd < α′d′ and αe < α′e′.
Lemma 4.2. Suppose d, d′ ∈ (N/2 − N/(16k − 2), N/2) such that d ̸= d′. Then the
decompositions F(d) and F(d′), as defined above, are orthogonal.
A. C. Burgess et al.: Mutually orthogonal cycle systems 269
Proof. In what follows, d ̸= d′, e = N − d and e′ = N − d′. Observe that e, e′ ∈
(N/2, N/2 +N/(16k − 2)).
It suffices to show that if C is a cycle from F(d) and C ′ is a cycle from F(d′), then C
and C ′ share at most one difference. Equivalently, we will show that:
(i) For any i ∈ [1, d] and i′ ∈ [1, d′], |Sd,i ∩ Sd′,i′ | ≤ 1;
(ii) For any i ∈ [1, e] and i′ ∈ [1, e′], |Te,i ∩ Te′,i′ | ≤ 1; and
(iii) For any i ∈ [1, d] and i′ ∈ [1, e′], |Sd,i ∩ Te′,i′ | ≤ 1.
To show (i), suppose to the contrary that {x, y} ⊆ Sd,i ∩ Sd′,i′ with x < y. Thus
y−x = αd = α′d′ for some α, α′ ∈ [1, 4k−1], contradicting Lemma 4.1. The justification
of (ii) is similar. For (iii), if x, y ∈ Sd,i ∩ Te′,i′ with x < y, then y − x = αd for some
α ∈ [1, 4k − 1] (since x, y ∈ Sd,i) and y − x = α′e′ for some α′ ∈ [1, 4k − 1] (since
x, y ∈ Te′,i′ ), so αd = α′e′, which again contradicts Lemma 4.1.
We note that the existence of two distinct integers in (N/2 − N/(16k − 2), N/2) is
guaranteed when N > 4(8k − 1), i.e. n− 1 > 32k(8k − 1).
Since n = 8Nk + 1, we have the following theorem.
Theorem 4.3. Let k ≥ 2 and n = 8Nk + 1. There is a set of mutually orthogonal cyclic
4k-cycle systems of order n of size at least
N
16k − 2
− 1 = n− 1
8k(16k − 2)
− 1.
Thus, if n ≡ 1 (mod 8k),
µ(n, 4k) ≥ µ′(n, 4k) ≥ n− 1
8k(16k − 2)
− 1.
5 Orthogonal sets of (4k + 2)-cycles
In this section, we show that for positive integers k and n ≡ 1 (mod 8k+4), µ′(n, 4k+2) =
Ω(n/k2). Specifically, we construct⌈
n− 1
(8k + 4)(96k + 30)
− 1
⌉
mutually orthogonal cyclic (4k + 2)-cycle decompositions of Kn. Thus we have at least
two orthogonal decompositions whenever
n− 1
(8k + 4)(96k + 30)
> 2,
or equivalently n− 1 > 48(2k + 1)(16k + 5).
270 Ars Math. Contemp. 23 (2023) #P2.05 / 261–280
Let N and k be positive integers and let n = 2(4k+2)N+1. For each d ≡ N (mod 2)
with d ∈ (N/3 − N/(48k + 15), N/3) (there is at least one such integer whenever N >
48k+15), we form a cyclic (4k+2)-cycle decomposition F(d) of Kn. Let e = (N−d)/2,
and observe that N/3 < e < N/3 + N/(2(48k + 15)) < N/3 + N/(48k + 15). Thus
e ∈ (N/3, N/3 +N/(48k + 15)).
For i ∈ [1, d], let
Sd,i,1 = {i, d+i, 2d+i, . . . , (4k−1)d+i} and Sd,i,2 = {4kN+4e+i, (4k+2)N−i+1},
and let Sd,i = Sd,i,1 ∪ Sd,i,2.
Now, when constructing the cycles containing differences in Sd,i, instead of
(4k + 2)N − i + 1, we will use the negative of this difference modulo n, that is, the
value
(8k + 4)N + 1− ((4k + 2)N − i+ 1) = (4k + 2)N + i.
We construct a starter cycle C ′(Sd,i) using the set of differences Sd,i but in a slightly
different way to Lemma 3.3.
C ′(Sd,i) =(0,−i, d,−d− i, . . . , kd,−kd− i, (k + 2)d,
− (k + 1)d− i, (k + 3)d,−(k + 2)d− i, . . . , 2kd,−(2k − 1)d− i,
(4k + 2)N − (2k + 1)d,−(2k + 1)d− i).
(Note that in the case k = 1, C ′(Sd,i) = (0,−i, d,−d− i, 4N + e− d,−3d− i).)
Lemma 5.1. Let i ∈ [1, d]. Working modulo n, the ordered sequence C ′(Sd,i) is a (4k+2)-
cycle with difference set Sd,i.
Proof. To see that no vertices are repeated (modulo n) within the sequence C ′(Sd,i), it
suffices to observe that:
− (4k + 2)N < −(2k + 1)d− i < −(2k − 1)d− i < −(2k − 2)d− i < · · ·
< −d− i < −i < 0 < d < 2d < · · · < kd < (k + 2)d < (k + 3)d < · · · < 2kd
< (4k + 2)N − (2k + 1)d < (4k + 2)N.
By inspection, and since (4k + 2)N − (2k + 1)d = 4kN + 4e − (2k − 1)d and
n− ((4k+ 2)N − i+ 1) = (4k+ 2)N + i, the set of differences of the edges of the cycle
C ′(Sd,i) is Sd,i.
Note that
d⋃
i=1
Sd,i = [1, 4kd] ∪ [4kN + 4e+ 1, 4kN + 4e+ d] ∪ [(4k + 2)N − d+ 1, (4k + 2)N ];
since 4kN + 4e+ d = (4k + 2)N − d, we have that
d⋃
i=1
Sd,i = [1, 4kd] ∪ [4kN + 4e+ 1, (4k + 2)N ].
A. C. Burgess et al.: Mutually orthogonal cycle systems 271
For j, ℓ ∈ [1, e], let
Te,j,1 = {4kd+ j, 4kd+ e+ j, . . . , 4kd+ (4k − 1)e+ j},
Te,j,2 = {4kN + j, 4kN + 2e+ j},
Ue,ℓ,1 = {4kd+ 4ke+ ℓ, 4kd+ (4k + 1)e+ ℓ, . . . , 4kd+ (8k − 1)e+ ℓ},
Ue,ℓ,2 = {4kN + e+ ℓ, 4kN + 3e+ ℓ},
and set Te,j = Te,j,1 ∪ Te,j,2 and Ue,ℓ = Ue,ℓ,1 ∪ Ue,ℓ,2.
The sets Te,j and Ue,ℓ are each balanced with τ = k + 1. We have that e⋃
j=1
Te,j
 ∪( e⋃
ℓ=1
Ue,ℓ
)
= [4kd+ 1, 4kd+ 8ke] ∪ [4kN + 1, 4kN + 4e]
= [4kd+ 1, 4kN + 4e],
since 4kd+ 8ke = 4kN . Observe that for fixed d,(
d⋃
i=1
Sd,i
)
∪
 e⋃
j=1
Te,j
 ∪( e⋃
ℓ=1
Ue,ℓ
)
= [1, (4k + 2)N ],
and thus by Corollary 3.4 and Lemma 5.1, the set of cycles
F(d) = {C ′(Sd,i) | i ∈ [1, d]} ∪ {C(Te,j) | j ∈ [1, e]} ∪ {C(Ue,ℓ) | ℓ ∈ [1, e]}
is a set of starter cycles for a (4k + 2)-cycle decomposition of Kn.
In order to show that the decompositions F(d), d ∈ (N/3 − N/(48k + 15), N/3),
are orthogonal, we will make use of the following lemma which is directly implied by
Corollary 3.6.
Lemma 5.2. Suppose there exist integers
d, d′, e, e′ ∈
(
N
3
− N
48k + 15
,
N
3
+
N
48k + 15
)
such that d ̸= d′ and e ̸= e′. Let α, α′ ∈ [1, 8k + 2]. Then αd ̸= α′d′ and αe ̸= α′e′.
Moreover, if α < α′, then αd < α′d′ and αe < α′e′.
Lemma 5.3. Suppose that βd + i = β′d′ + i′, where β, β′ ∈ [0, 4k − 1], i ∈ [1, d],
i′ ∈ [1, d′] and d′ < d. Then either β′ = β or β′ = β + 1.
Proof. From Lemma 5.2, (β + 1)d < (β + 2)d′. Now,
(β − 1)d′ + i′ ≤ βd′ ≤ βd < βd+ i
and
βd+ i ≤ (β + 1)d < (β + 2)d′ < (β + 2)d′ + i′;
hence
(β − 1)d′ + i′ < βd+ i < (β + 2)d′ + i′.
272 Ars Math. Contemp. 23 (2023) #P2.05 / 261–280
Lemma 5.4. Let d ̸= d′ such that d, d′ ≡ N (mod 2) and
d, d′ ∈
(
N
3
− N
48k + 15
,
N
3
+
N
48k + 15
)
.
Let e = (N − d)/2 and e′ = (N − d′)/2. Let i ∈ [1, d], i′ ∈ [1, d′], j, ℓ ∈ [1, e] and
j′, ℓ′ ∈ [1, e′]. Then for each X ∈ {Sd,i, Te,j , Ue,ℓ} and each Y ∈ {Sd′,i′ , Te′,j′ , Ue′,ℓ′},
|X ∩ Y | ≤ 1 with the exception Sd,i ∩ Sd′,i = {i, (4k + 2)N + i}.
Proof. Recall from the start of this section that e, e′ ∈ (N/3, N/3 + N/(48k + 15)). In
what follows, we frequently apply Lemma 5.2 to d, d′, e and e′. To prove the lemma, it
suffices to show the following:
(i) Sd,i ∩ Sd′,i = {i, (4k + 2)N − i+ 1} and if i ̸= i′ then |Sd,i ∩ Sd′,i′ | ≤ 1;
(ii) |Te,j ∩ Te′,j′ | ≤ 1, |Ue,ℓ ∩ Ue′,ℓ′ | ≤ 1 and |Te,j ∩ Ue′,ℓ′ | ≤ 1;
(iii) |Sd,i ∩ Te′,j′ | ≤ 1 and |Sd,i ∩ Ue′,ℓ′ | ≤ 1.
Proof of (i). In this case, we may assume without loss of generality that d′ < d. We note
that
4kN + 4e′ + i′ > 4kN > 4kd ≥ (4k − 1)d+ i and
4kN + 4e+ i > 4kN > 4kd′ ≥ (4k − 1)d′ + i′,
so Sd,i,1 ∩ Sd′,i′,2 = Sd′,i′,1 ∩ Sd,i,2 = ∅.
Now, supposing that |Sd,i,1 ∩ Sd′,i′,1| ≥ 2, it follows that for some x, (x, x + αd) =
(x, x + α′d′) where α, α′ ∈ [1, 4k − 1]; thus αd = α′d′, in contradiction to Lemma 5.2.
Next, supposing that |Sd,i,2 ∩ Sd′,i′,2| ≥ 2, then either
(a) 4kN + 4e+ i = 4kN + 4e′ + i′ and (4k + 2)N − i+ 1 = (4k + 2)N − i′ + 1, or
(b) 4kN + 4e+ i = (4k + 2)N − i′ + 1 and 4kN + 4e′ + i′ = (4k + 2)N − i+ 1.
In both cases, it is straightforward to check that e = e′, a contradiction.
Thus if |Sd,i∩Sd′,i′ | ≥ 2, it must be that |Sd,i,1∩Sd′,i′,1| = 1 and |Sd,i,2∩Sd′,i′,2| = 1.
If i = i′ then {i, (4k + 2)N − i + 1} ⊆ Sd,i ∩ Sd′,i′ . Moreover, recalling that Sd,i,1 ∩
Sd′,i′,2 = Sd′,i′,1 ∩ Sd,i,2 = ∅, it follows that |Sd,i ∩ Sd′,i′ | = 2. Hence if i = i′, then
Sd,i ∩ Sd′,i = {i, (4k + 2)N − i + 1}. We now assume that i ̸= i′. From Lemma 5.3,
|Sd,i,1 ∩ Sd′,i′,1| = 1 implies that either
(a) βd+ i = βd′ + i′, or
(b) βd+ i = (β + 1)d′ + i′
for some β, β′ ∈ [0, 4k − 1]. Now suppose that also |Sd,i,2 ∩ Sd′,i′,2| = 1. Since i ̸= i′,
we note that (4k + 2)N − i + 1 ̸= (4k + 2)N − i′ + 1. Also, it cannot be the case that
4kN + 4e+ i = (4k + 2)N − i′ + 1, since
4kN + 4e+ i = (4k + 2)N − 2d+ i ≤ (4k + 2)N − d < (4k + 2)N − d′
≤ (4k + 2)N − i′ < (4k + 2)N − i′ + 1.
Now suppose that 4kN + 4e + i = 4kN + 4e′ + i′. Then 2d − i = 2d′ − i′. If (a)
is true, then (β + 2)d = (β + 2)d′; since β + 2 > 0, we have d = d′, a contradiction.
A. C. Burgess et al.: Mutually orthogonal cycle systems 273
On the other hand, if (b) is true, then (β + 2)d = (β + 3)d′, contradicting Lemma 5.2.
Thus the only remaining possibility is that 4kN + 4e′ + i′ = (4k + 2)N − i + 1, so that
i+ i′ = 2N−4e′+1 = 2d′+1 is odd. Since d and d′ have the same parity, this contradicts
(a), so it must be that (b) is true. It follows that
(β + 3)d′ − βd+ 1 = 2i ≤ 2d.
Thus (β + 3)d′ ≤ (β + 2)d− 1 < (β + 2)d, contradicting Lemma 5.2.
Proof of (ii). We first note that the largest element in Te,j,1∪Ue,ℓ,1 is 4kd+(8k−1)e+ ℓ,
while the smallest element of Te,j,2 ∪ Ue,ℓ,2 is 4kN + j. Since
4kd+ (8k − 1)e+ ℓ ≤ 4kd+ 8ke = 4kN < 4kN + j,
it follows that Te,j,1 ∩ Te′,j′,2 = ∅, Ue,ℓ,1 ∩ Ue′,ℓ′,2 = ∅ and Te,j,1 ∩ Ue′,ℓ′,2 = ∅.
Now, if |Te,j,1 ∩ Te′,j′,2| ≥ 2, |Ue,ℓ,1 ∩ Ue′,ℓ′,1| ≥ 2 or |Te,j,1 ∩ Ue′,j′,1| ≥ 2, then
for some x, (x, x + αe) = (x, x + α′e′), where α, α′ ∈ [1, 8k − 1]. Thus αe = α′e′,
contradicting Lemma 5.2. If |Te,j,2 ∩ Te′,j′,2| ≥ 2, |Ue,ℓ,2 ∩ Ue′,ℓ′,2| ≥ 2 or |Te,ℓ,2 ∩
Ue′,ℓ′,2| ≥ 2, then it follows that e = e′, a contradiction.
Thus, if |Te,j∩Te′,j′ | ≥ 2, it must be that |Te,j,1∩Te′,j′,1| = 1 and |Te,j,2∩Te′,j′,2| = 1.
Since |Te,j,1∩Te′,j′,1| = 1, we have that for some α, α′ ∈ [0, 4k−1], 4kd+αe+j = 4kd′+
α′e′+j′, which implies that (8k−α)e−j = (8k−α′)e′−j′. Since |Te,j,2∩Te′,j′,2| = 1,
then 4kN+βe+j = 4kN+β′e′+j′ where β, β′ ∈ {0, 2}. Hence (8k−α+β)e = (8k−
α′+β′)e′, which contradicts Lemma 5.2 since (8k−α+β), (8k−α′+β′) ∈ [4k+1, 8k+2].
We conclude that |Te,j ∩ Te′,j′ | ≤ 1.
In a similar way, the assumption that |Ue,ℓ,1∩Ue,ℓ′,1| = 1 and |Ue,ℓ,2∩Ue,ℓ′,2| = 1 leads
to a contradiction, as does the assumption that |Te,j,1∩Ue′,ℓ′,1| = 1 and |Te,j,2∩Ue′,ℓ′,2| =
1. We conclude that |Ue,ℓ ∩ Ue′,ℓ′ | ≤ 1 and |Te,j ∩ Ue′,ℓ′ | ≤ 1.
Next, suppose that |Ue,ℓ,1 ∩ Ue′,ℓ′,1| = 1 and |Ue,ℓ,2 ∩ Ue′,ℓ′,2| = 1. Since |Ue,ℓ,1 ∩
Ue′,ℓ′,1| = 1, we have that for some α, α′ ∈ [4k, 8k− 1], 4kd+αe+ ℓ = 4kd′+α′e′+ ℓ′,
which implies that (8k − α)e − ℓ = (8k − α′)e′ − ℓ. Since |Ue,ℓ,2 ∩ Ue′,ℓ′,2| = 1, then
4kN + βe + ℓ = 4kN + β′e′ + ℓ′ where β, β′ ∈ {1, 3}. Hence (8k − α + β)e = (8k −
α′+β′)e′, which contradicts Lemma 5.2 since (8k−α+β), (8k−α′+β′) ∈ [2, 4k+3].
Finally, suppose that |Te,j,1 ∩ Ue′,ℓ′,1| = 1 and |Te,j,2 ∩ Ue′,ℓ′,2| = 1. Since |Te,j,1 ∩
Ue′,ℓ′,1| = 1, we have that for some α ∈ [0, 4k − 1], α′ ∈ [4k, 8k − 1], 4kd + αe +
j = 4kd′ + α′e′ + ℓ′, which implies that (8k − α)e − j = (8k − α′)e′ − ℓ′. Since
|Te,j,2 ∩ Ue′,ℓ′,2| = 1, then 4kN + βe + j = 4kN + β′e′ + ℓ′, where β ∈ {0, 2} and
β′ ∈ {1, 3}. Hence (8k−α+β)e = (8k−α′+β′)e′, which contradicts Lemma 5.2 since
4k + 1 ≤ 8k − α+ β ≤ 8k + 2 and 2 ≤ 8k − α′ + β′ ≤ 4k + 3.
Proof of (iii). Note that since
(4k − 1)d+ i ≤ 4kd < 4kN < 4kN + j′ < 4kN + e′ + ℓ′,
then Sd,i,1 ∩ Te′,j′,2 = ∅ and Sd,i,1 ∩ Ue′,ℓ′,2 = ∅. Moreover,
4kd′ + (4k − 1)e′ + j′ ≤ 4kd′ + 4ke′ < 4kd′ + (8k − 1)e′ + ℓ′ ≤ 4kd′ + 8ke′ = 4kN
< 4kN + 4e+ i,
and so Sd,i,2 ∩ Te′,j′,1 = ∅ and Sd,i,2 ∩ Ue′,ℓ′,1 = ∅.
274 Ars Math. Contemp. 23 (2023) #P2.05 / 261–280
By Lemma 5.2,
(4k − 2)d+ i ≤ (4k − 1)d < 4kd′ < 4kd′ + j′.
It follows that |Sd,i,1 ∩ Te′,j′,1| ≤ 1. Also, since d < N/3 < e′,
(4k − 1)d+ i ≤ 4kd < 4ke′ < 4kd′ + 4ke′ + ℓ′,
and thus Sd,i,1 ∩ Ue′,ℓ′,1 = ∅.
Now, using Lemma 5.2, we also have that
4kN+e′+ℓ′ ≤ 4kN+2e′ < 4kN+2e′+j′ ≤ 4kN+3e′ < 4kN+4e < 4kN+4e+ i,
and so Sd,i,2 ∩ Te′,j′,2 = ∅ and |Sd,i,2 ∩ Ue′,ℓ′,2| ≤ 1. It follows that |Sd,i ∩ Te′,j′ | ≤ 1
and |Sd,i ∩ Ue′,ℓ′ | ≤ 1.
End of Proof of Lemma 5.4.
Theorem 5.5. Let k ≥ 1 and n = (8k + 4)N + 1. There is a set of mutually orthogonal
cyclic (4k + 2)-cycle systems of order n of size at least⌈
N
96k + 30
− 1
⌉
=
⌈
n− 1
(8k + 4)(96k + 30)
− 1
⌉
.
Thus, if n ≡ 1 (mod 2(4k + 2)), then
µ(n, 4k + 2) ≥ µ′(n, 4k + 2) ≥
⌈
n− 1
(8k + 4)(96k + 30)
− 1
⌉
.
Proof. The number of integers strictly between N/3 − N/(48k + 15) and N/3 with the
same parity as N is at least ⌈N/(96k + 30) − 1⌉. Note that there are at least two distinct
integers of the same parity as N in this interval whenever
N
96k + 30
> 2,
or equivalently n − 1 > 48(2k + 1)(16k + 5). It thus suffices to show that for distinct
integers d and d′ with the same parity such that
d, d′ ∈
(
N
3
− N
48k + 15
,
N
3
)
,
the decompositions F(d) and F(d′) are orthogonal.
In turn, it suffices to deal with the exceptional case from Lemma 5.4. From Lemma 5.1,
the edges of differences i and (4k + 2)N − i + 1 within C ′(Sd,i) are {0,−i} and
{(4k+2)N−(2k+1)d,−(2k+1)d− i}, which are at distance (4k+2)N−(2k+1)d+ i.
Similarly, the edges of differences i and (4k+2)N− i+1 within C ′(Sd′,i) are {0,−i} and
{(4k+2)N−(2k+1)d′,−(2k+1)d′−i}, which are at distance (4k+2)N−(2k+1)d′+i.
If the pairs of edges within cycles generated from the starters C ′(Sd,i) and C ′(Sd′,i) co-
incide, then we must have that (2k + 1)d ≡ (2k + 1)d′ (mod n). But n and 2k + 1 are
coprime, so d = d′.
A. C. Burgess et al.: Mutually orthogonal cycle systems 275
6 Concluding remarks
The main results of this paper have been to establish lower bounds on the number of mu-
tually orthogonal cyclic ℓ-cycle systems of order n. For upper bounds on the number of
systems (not necessarily cyclic in nature) we have the following lemmata.
Lemma 6.1. If there exists a set of µ mutually orthogonal ℓ-cycle systems of order n, then
µ ≤ n− 2. That is, µ(ℓ, n) ≤ n− 2.
Proof. Consider a vertex w in Kn. The vertex w belongs to precisely (n − 1)(n − 2)/2
paths of length 2 in Kn where w is the center vertex of the path. Moreover, each such path
belongs to at most one ℓ-cycle from any set of µ mutually orthogonal ℓ-cycle systems. The
number of cycles in one ℓ-cycle system which contain vertex w is equal to (n−1)/2. Thus
µ(n− 1)/2 ≤ (n− 1)(n− 2)/2. The result follows.
Lemma 6.2. Let ℓ ≥ 4. Then
µ(ℓ, n) ≤ (n− 2)(n− 3)
2(ℓ− 3)
.
Proof. Suppose there exists a set {F1,F2, . . . ,Fµ} of mutually orthogonal ℓ-cycle systems
of Kn. Consider an edge {v, w} in Kn. Then for each i ∈ [1, µ], there is an ℓ-cycle
Ci ∈ Fi containing the edge {v, w}. Let H be the clique of size n− 2 in Kn not including
vertices v and w. Then the intersection of Ci with H is a path Pi with ℓ − 3 edges.
Moreover, orthogonality implies that the paths in the set {Pi | i ∈ [1, µ]} are pairwise
edge-disjoint. Thus, (ℓ− 3)µ is bounded by the number of edges in H; that is, (ℓ− 3)µ ≤
(n− 2)(n− 3)/2.
Observe that Lemma 6.2 improves Lemma 6.1 only if ℓ > (n+ 3)/2. If ℓ > n/
√
2, it
is not even possible to find a pair of orthogonal cycle systems, as shown in the following
lemma.
Lemma 6.3. If 2ℓ2 > n(n− 1) then µ(ℓ, n) ≤ 1.
Proof. Suppose there exists a pair {F1,F2} of mutually orthogonal ℓ-cycle systems of Kn.
Then F1 and F2 each contain n(n − 1)/(2ℓ) cycles of length ℓ. Let C be a cycle in F1.
By the definition of orthogonality, each edge of C intersects a unique cycle in F2. Thus
ℓ ≤ n(n− 1)/(2ℓ), contradiction.
When the systems are required to be cyclic, Lemma 6.1 can be slightly improved.
Lemma 6.4. Let n ≥ 4. If there exists a set of µ′ mutually orthogonal cyclic ℓ-cycle
systems of order n, then µ′ ≤ n− 3. That is, µ′(ℓ, n) ≤ n− 3.
Proof. Since µ′(ℓ, n) ≤ µ(ℓ, n), Lemma 6.1 implies that µ′(ℓ, n) ≤ n − 2. Suppose, for
the sake of contradiction that µ′(ℓ, n) = n− 2. Thus there exists a set of n− 2 orthogonal
cyclic decompositions of Kn where the vertices are labelled with elements of Zn. Let
a ∈ [1, (n − 1)/2]. Suppose that the path (−a, 0, a) of length 2 does not occur in a cycle
from one of these decompositions. Then the total number of paths of length 2 containing
0 which appear in one of the cycles is less than (n − 1)(n − 2)/2. However, there are
(n− 2)(n− 1)/2 cycles containing vertex 0, contradicting the condition of orthogonality.
276 Ars Math. Contemp. 23 (2023) #P2.05 / 261–280
Let Ca be the cycle containing the path (−a, 0, a) and let F be the decomposition
of Kn containing Ca. Since our decomposition is cyclic, there is also a cycle C ′ ∈ F
containing (0, a, 2a); since C ′ and Ca share an edge we must have C ′ = Ca. Inductively,
Ca = (0, a, 2a, . . . ). In particular C1 = (0, 1, 2, . . . , n − 1) and thus ℓ = n. But since
µ′(ℓ, n) = n − 2 ≥ 2 and n > (n − 1)/2, there is a cycle C ′′ ̸= Ca in a decomposition
F ′ ̸= F containing a repeated difference a ∈ [1, (n − 1)/2]. The cycle C ′′ shares two
edges with Ca, contradicting the condition of orthogonality.
It is worth noting that for certain congruencies the upper bound in Lemma 6.4 can be
made significantly smaller. For example, if n ≡ 3 (mod 6) then µ′(3, n) = 1, because in
this case any cyclic decomposition necessarily contains the cycle (0, n/3, 2n/3).
In the appendix we give computational results for µ′(ℓ, n) when ℓ and n are small. As
yet we are unaware of any instances for which the bound of Lemma 6.4 is tight, and so we
ask if equality ever occurs.
Question 1. For which values of ℓ and n, if any, is µ′(ℓ, n) = n− 3?
ORCID iDs
Nicholas J. Cavenagh https://orcid.org/0000-0002-9151-3842
David A. Pike https://orcid.org/0000-0001-8952-3016
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Appendix
We computed sets of mutually orthogonal cyclic ℓ-cycle systems of order n = 2ℓ + 1 for
small values of ℓ, and in so doing we empirically determined or bounded µ′(ℓ, 2ℓ + 1) in
these cases. Recall from Lemma 6.4 that µ′(ℓ, 2ℓ+ 1) ≤ 2ℓ− 2.
Note that for any cyclic ℓ-cycle system of order 2ℓ + 1, the cycles of the system com-
prise a single Z2ℓ+1-orbit. To find sets of mutually orthogonal cyclic ℓ-cycle systems of
order 2ℓ+ 1, we first determined the orbit for each possible system and then constructed a
graph in which each system is represented as a vertex and adjacency denotes orthogonality.
Maximum cliques were then sought. The results for 3 ≤ ℓ ≤ 11 are summarised in Table 1.
For 9 ≤ ℓ ≤ 11, we found cliques of order 8 but we do not yet know whether larger cliques
exist (the computational task becomes increasingly challenging as the number of systems
grows).
We now present examples of the Z2ℓ+1-orbits for the sets of mutually orthogonal cyclic
ℓ-cycle systems of order 2ℓ+ 1 that we found. Each orbit is represented by the differences
that occur on the edges of its cycles, using notation from Section 2.
ℓ = 3, n = 7
[1, 2,−3], [1,−3, 2]
ℓ = 4, n = 9
[1,−2,−3, 4], [1, 4,−3,−2]
ℓ = 5, n = 11
[1,−2, 4, 3, 5], [1, 3,−2, 5, 4], [1, 4, 5,−2, 3], [1, 5, 3, 4,−2]
A. C. Burgess et al.: Mutually orthogonal cycle systems 279
ℓ n = 2ℓ+ 1 No. of Cyclic Systems µ′(ℓ, 2ℓ+ 1)
3 7 2 2
4 9 6 2
5 11 24 4
6 13 168 5
7 15 1344 8
8 17 11136 8
9 19 128304 ≥ 8
10 21 1504248 ≥ 8
11 23 19665040 ≥ 8
Table 1: Number of mutually orthogonal cyclic ℓ-cycle systems of order 2ℓ+ 1.
ℓ = 6, n = 13
[1, 2, 3,−4, 5, 6], [1,−4,−2, 3,−5,−6], [1, 5, 3, 6,−4, 2], [1,−5,−4, 3,−6,−2], [1, 6, 3, 2, 5,−4]
ℓ = 7, n = 15
[1, 2, 6,−4,−7,−3, 5], [1,−2,−3,−5,−4, 7, 6], [1, 3, 4,−2, 6,−5,−7], [1,−3, 4, 2,−5,−6, 7],
[1, 5,−3,−7,−4, 6, 2], [1, 6, 7,−4,−5,−3,−2], [1, 7,−6,−5, 2, 4,−3], [1,−7,−5, 6,−2, 4, 3]
ℓ = 8, n = 17
[1, 2, 3, 4,−6,−7,−5, 8], [1,−2,−3, 8, 5,−6, 4,−7], [1, 3, 7,−8,−6, 2,−4, 5],
[1,−3,−5, 6, 4,−8, 7,−2], [1, 4, 5, 2,−3, 6,−7,−8], [1, 5,−7, 6,−8,−3, 2, 4],
[1,−6,−8, 2, 5,−4, 7, 3], [1,−8,−7, 5,−3, 4, 2, 6]
ℓ = 9, n = 19
[1, 2, 3, 4, 5,−6,−7, 9, 8], [1,−2, 3,−4,−7, 6, 8, 9, 5], [1, 5, 8,−3, 7,−6,−4, 9, 2],
[1,−5,−7,−6,−8,−2,−4, 3, 9], [1, 6,−3,−9, 2,−7, 4,−5,−8],
[1,−6, 8,−7,−3,−5,−2, 4,−9], [1, 7, 3, 6,−8, 9,−5, 2, 4], [1, 9,−7, 8, 4, 3, 5, 2,−6]
ℓ = 10, n = 21
[1, 2, 3, 4, 5,−7, 6, 9,−10, 8], [1,−2, 3,−4, 5,−6, 8,−9,−7,−10],
[1, 3,−2,−5,−10, 9,−6,−8, 4,−7], [1,−6,−10, 7,−3, 9,−5,−2,−8,−4],
[1, 7,−9,−8, 5, 6, 4, 3, 2, 10], [1,−8,−5, 3,−6,−9, 7,−10, 2, 4],
[1, 10, 3, 6,−7, 5,−8,−2, 4, 9], [1,−10, 3, 5,−4,−9,−2,−7, 8,−6]
ℓ = 11, n = 23
[1, 2, 3, 4, 5, 6, 7, 8, 9,−10, 11], [1,−2, 3,−4, 5,−6, 7,−11,−10, 9, 8],
[1, 3, 2,−4,−5,−11,−6, 10, 8,−7, 9], [1,−3, 10, 6, 4, 7, 9, 11, 8,−2,−5],
[1,−4, 8, 6, 3, 5,−9,−2, 10,−11,−7], [1, 5,−11,−8,−3, 9,−7, 2, 6,−4, 10],
[1, 10,−7,−8, 3,−5, 9, 4, 6,−11,−2], [1,−11, 5,−7, 4, 2,−9,−6, 8, 10, 3]
280 Ars Math. Contemp. 23 (2023) #P2.05 / 261–280
Below we present examples of mutually orthogonal cyclic 4-cycle systems of orders 17 and
25; these are mentioned in Section 2.
ℓ = 4, n = 17
{[1, 2, 3,−6], [4,−5,−7, 8]}, {[1,−2,−3, 4], [5, 8,−7,−6]}, {[1,−3,−8,−7], [2, 4, 5, 6]},
{[1, 4,−7, 2], [3,−5, 8,−6]}, {[1,−4, 8,−5], [2, 7,−3,−6]}, {[1, 5, 2,−8], [3,−4,−6, 7]},
{[1,−5,−3, 7], [2,−6, 8,−4]}, {[1,−6,−3, 8], [2,−4, 7,−5]}, {[1,−7,−8,−3], [2, 6, 5, 4]},
{[1,−8,−6,−4], [2, 5, 3, 7]}
ℓ = 4, n = 25
{[1, 2, 3,−6], [4,−5, 12,−11], [7,−8,−9, 10]},
{[1,−2,−3, 4], [5,−6,−7, 8], [9,−10,−11, 12]},
{[1, 3, 4,−8], [2, 5, 7, 11], [6,−9,−10,−12]}, {[1,−3,−4, 6], [2,−5,−7, 10], [8,−11,−9, 12]},
{[1, 4, 2,−7], [3,−5,−9, 11], [6,−8,−10, 12]}, {[1,−4,−2, 5], [3, 6, 7, 9], [8,−12,−11,−10]},
{[1, 5, 3,−9], [2,−4, 10,−8], [6, 12,−7,−11]}, {[1,−5,−3, 7], [2, 6, 4,−12], [8, 11,−10,−9]},
{[1, 7,−10, 2], [3,−11,−4, 12], [5, 9,−8,−6]}, {[1,−7,−8,−11], [2, 12, 5, 6], [3, 10,−4,−9]},
{[1, 8, 4, 12], [2,−10,−6,−11], [3,−7, 9,−5]}, {[1,−8, 11,−4], [2,−12, 3, 7], [5, 10,−6,−9]},
{[1,−9,−3, 11], [2,−6,−4, 8], [5, 12,−10,−7]},
{[1,−10, 11,−2], [3, 8,−5,−6], [4,−7, 12,−9]}, {[1,−11,−12,−3], [2, 8, 10, 5], [4,−6, 9,−7]},
{[1, 12,−3,−10], [2, 7,−5,−4], [6, 11,−9,−8]}, {[1,−12, 8, 3], [2, 9,−4,−7], [5, 11,−6,−10]}
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.06 / 281–296
https://doi.org/10.26493/1855-3974.2621.26f
(Also available at http://amc-journal.eu)
Some remarks on the square graph of the
hypercube
Seyed Morteza Mirafzal *
Department of Mathematics, Lorestan University, Khoramabad, Iran
Received 6 May 2021, accepted 26 June 2022, published online 18 November 2022
Abstract
Let Γ = (V,E) be a graph. The square graph Γ2 of the graph Γ is the graph with the
vertex set V (Γ2) = V in which two vertices are adjacent if and only if their distance in
Γ is at most two. The square graph of the hypercube Qn has some interesting properties.
For instance, it is highly symmetric and panconnected. In this paper, we investigate some
algebraic properties of the graph Q2n. In particular, we show that the graph Q
2
n is distance-
transitive. We show that the graph Q2n is an imprimitive distance-transitive graph if and
only if n is an odd integer. Also, we determine the spectrum of the graph Q2n. Finally, we
show that when n > 2 is an even integer, then Q2n is an automorphic graph, that is, Q
2
n is a
distance-transitive primitive graph which is not a complete or a line graph.
Keywords: Square of a graph, distance-transitive graph, hypercube, automorphism group, Johnson
graph, automorphic graph.
Math. Subj. Class. (2020): Primary 05C25, 94C15
1 Introduction
In this paper, a graph Γ = (V,E) is considered as an undirected simple graph where
V = V (Γ) is the vertex-set and E = E(Γ) is the edge-set. For all the terminology and
notation not defined here, we follow [1, 3, 5, 6, 9].
Let Γ = (V,E) be a graph. The square graph Γ2 of the graph Γ is the (simple)
graph with vertex set V in which two vertices are adjacent if and only if their distance in
Γ is at most two. It is easy to see that Aut(Γ) ≤ Aut(Γ2), where Aut(Γ) denotes the
automorphism group of the graph Γ. Thus, if the graph Γ is a vertex-transitive graph, then
Γ2 is a vertex-transitive graph. A graph Γ of order n > 2 is Hamilton-connected if for any
pair of distinct vertices u and v, there is a Hamilton u-v path, namely, there is a u-v path
*The author is thankful to the anonymous reviewers for their valuable comments and suggestions.
E-mail address: smortezamirafzal@yahoo.com (Seyed Morteza Mirafzal)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
282 Ars Math. Contemp. 23 (2023) #P2.06 / 281–296
of length n − 1. It is clear that if a graph Γ is Hamilton-connected then it is Hamiltonian.
A graph Γ of order n > 2 is panconnected if for every two vertices u and v, there is a
u-v path of length l for every integer l with d(u, v) ≤ l ≤ n − 1. Note that if a graph
Γ is panconnected, then it is Hamilton-connected. It is a well known fact that when a
graph Γ is 2-connected, then its square Γ2 is panconnected [4, 7]. Using this fact, and an
algebraic property of Johnson graphs, recently it has been proved that the Johnson graphs
are panconnected [10].
Let n ≥ 2 be an integer. The hypercube Qn is the graph whose vertex-set is {0, 1}n,
where two n-tuples are adjacent if they differ in precisely one coordinate. This graph has
been studied from various aspects by many authors. Some recent works concerning some
algebraic aspects of this graph include [14, 17, 24, 28]. It is a well known fact that the
graph Qn is a distance-transitive graph [1, 3], and hence it is edge-transitive. Now, using a
well known result due to Watkins [27], it follows that the connectivity of Qn is maximal,
that is, n. Like the hypercube Qn, its square, namely, the graph Q2n has some interesting
properties. For instance, when n ≥ 2, then Qn is 2-connected. Now using a known result
due to Chartrand and Fleischner [4, 7], it follows that Q2n is a panconnected graph. Also,
since Qn is vertex-transitive, the graph Q2n is vertex-transitive, as well. Hence Q
2
n is a
regular graph and it is easy to check that its valency is n+
(
n
2
)
=
(
n+1
2
)
. If n = 2, then Q2n is
the complete graph K4. When n = 3, then Q2n is a 6-regular graph with 8 vertices. This
graph is isomorphic with a graph known as the coktail-party graph CP (4) [1]. It can be
shown that when n = 4, then the graph Q2n is a 10-regular graph with 16 vertices, which is
isomorphic to the complement of the graph known as the Clebsch graph [9].
In this paper, we determine the automorphism group of the graph Q2n. Then we show
that Q2n is a distance-transitive graph. This implies that the connectivity of the graph Q
2
n
is maximal, namely, its valency
(
n+1
2
)
. Also, we will see that the graph Q2n is an imprim-
itive distance-transitive graph if and only if n is an odd integer. A graph Γ is called an
automorphic graph, when it is a distance-transitive primitive graph which is not a com-
plete or a line graph [1]. In the last step of the paper, we show that the graph Q2n is an
automorphic graph if and only if n is an even integer.
2 Preliminaries
The graphs Γ1 = (V1, E1) and Γ2 = (V2, E2) are called isomorphic, if there is a bijection
α : V1 −→ V2 such that {a, b} ∈ E1 if and only if {α(a), α(b)} ∈ E2 for all a, b ∈ V1.
In such a case the bijection α is called an isomorphism. An automorphism of a graph Γ
is an isomorphism of Γ with itself. The set of automorphisms of Γ with the operation of
composition of functions is a group called the automorphism group of Γ and denoted by
Aut(Γ).
The group of all permutations of a set V is denoted by Sym(V ) or just Sym(n) when
|V | = n. A permutation group G on V is a subgroup of Sym(V ). In this case we say that
G acts on V . If G acts on V we say that G is transitive on V (or G acts transitively on V )
if given any two elements u and v of V , there is an element β of G such that β(u) = v. If
Γ is a graph with vertex-set V then we can view each automorphism of Γ as a permutation
on V and so Aut(Γ) = G is a permutation group on V .
A graph Γ is called vertex-transitive if Aut(Γ) acts transitively on V (Γ). We say that Γ
is edge-transitive if the group Aut(Γ) acts transitively on the edge set E, namely, for any
{x, y}, {v, w} ∈ E(Γ), there is some π in Aut(Γ), such that π({x, y}) = {v, w}. We say
S. M. Mirafzal: Some remarks on the square graph of the hypercube 283
that Γ is symmetric (or arc-transitive if for all vertices u, v, x, y of Γ such that u and v are
adjacent, and also, x and y are adjacent, there is an automorphism π in Aut(Γ) such that
π(u) = x and π(v) = y. We say that Γ is distance-transitive if for all vertices u, v, x, y of
Γ such that d(u, v) = d(x, y), where d(u, v) denotes the distance between the vertices u
and v in Γ, there is an automorphism π in Aut(Γ) such that π(u) = x and π(v) = y.
A vertex cut of the graph Γ is a subset U of V such that the subgraph Γ − U induced
by the set V − U is either trivial or not connected. The connectivity κ(Γ) of a nontrivial
connected graph Γ is the minimum cardinality of all vertex cuts of Γ. If we denote by δ(Γ)
the minimum degree of Γ, then κ(Γ) ≤ δ(Γ). A graph Γ is called k-connected (for k ∈ N)
if |V (Γ)| > k and Γ − X is connected for every subset X ⊂ V (Γ) with |X| < k. It is
trivial that if a positive integer m is such that m ≤ κ(Γ), then Γ is an m-connected graph.
We have the following fact.
Theorem 2.1 ([27]). If a connected graph Γ is edge-transitive, then κ(Γ) = δ(Γ), where
δ(Γ) is the minimum degree of vertices of Γ.
Let n, k ∈ N with k < n, and let [n] = {1, ..., n}. The Johnson graph J(n, k) is
defined as the graph whose vertex set is V = {v | v ⊆ [n], |v| = k} and two vertices
v,w are adjacent if and only if |v ∩ w| = k − 1. The class of Johnson graphs is a well
known class of distance-transitive graphs [3]. It is an easy task to show that the set of
mappings H = {fθ | θ ∈ Sym([n])}, fθ({x1, ..., xk}) = {θ(x1), ..., θ(xk)}, is a subgroup
of Aut(J(n, k)) [9]. It has been shown that Aut(J(n, k)) ∼= Sym([n]) if n ̸= 2k, and
Aut(J(n, k)) ∼= Sym([n]) × Z2, if n = 2k, where Z2 is the cyclic group of order 2
[3, 13, 18].
Although in most situations it is difficult to determine the automorphism group of a
graph Γ and how it acts on its vertex and edge sets, there are various papers in the literature,
and some of the recent works include [8, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 26, 28].
Let G be any abstract finite group with identity 1, and suppose Ω is a subset of G, with
the properties:
(i) x ∈ Ω =⇒ x−1 ∈ Ω,
(ii) 1 /∈ Ω.
The Cayley graph Γ = Cay(G; Ω) is the (simple) graph whose vertex-set and edge-set are
defined as follows:
V (Γ) = G, E(Γ) = {{g, h} | g−1h ∈ Ω}.
It can be shown that the Cayley graph Γ = Cay(G; Ω) is connected if and only if the set Ω
is a generating set in the group G [1].
The group G is called a semidirect product of N by Q, denoted by G = N ⋊ Q, if G
contains subgroups N and Q such that:
(i) N ⊴G (N is a normal subgroup of G)
(ii) NQ = G; and
(iii) N ∩Q = 1.
284 Ars Math. Contemp. 23 (2023) #P2.06 / 281–296
3 Main results
The hypercube Qn is the graph whose vertex set is {0, 1}n, where two n-tuples are adja-
cent if they differ in precisely one coordinate. It is easy to show that Qn = Cay(Zn2 ;S),
where Z2 is the cyclic group of order 2, and S = {ei | 1 ≤ i ≤ n}, where ei =
(0, ..., 0, 1, 0, ..., 0), with 1 at the ith position. It is easy to show that the set H = {fθ|θ ∈
Sym([n])}, fθ(x1, ..., xn) = (xθ(1), ..., xθ(n)) is a subgroup of the group Aut(Qn). It is
clear that H ∼= Sym([n]). We know that in every Cayley graph Γ = Cay(G;S), the group
Aut(Γ) contains a subgroup isomorphic with the group G. In fact, if x ∈ Zn2 , and we
define the mapping fx(v) = x + v, for every v ∈ V (Qn), then fx is an automorphism of
the hypercube Qn. Hence Zn2 is (isomorphic with) a subgroup of Aut(Qn). It has been
proved that Aut(Qn) = ⟨Zn2 ,Sym([n])⟩ ∼= Zn2 ⋊ Sym([n]) [14]. It is clear that when Γ is
a graph then Aut(Γ) is a subgroup of Aut(Γ2). Thus we have Aut(Qn) ≤ Aut(Q2n). In
the sequel, we wish to show that the graph Q2n is a distance-transitive graph, and for doing
this we need the automorphism group of Q2n. When n = 3, then Q
2
n is isomorphic with the
coktail-party graph CP (4). The complement of this graph is a disjoint union of 4 copies
of K2. Thus Aut(Q23) ∼= Sym([2]) wrI Sym([4]), where I = {1, 2, 3, 4} [3, 22] (for an
acquaintance with the notion of wreath product of groups see [6]). Now it can be checked
that this graph is a distance-transitive graph. Hence, in the sequel we assume that n > 4.
It is easy to see that for the graph Q2n we have, Q
2
n = Cay(Zn2 ;T ), T = S ∪ S1, where
S1 = {ei + ej | i, j ∈ [n], i ̸= j}. Let A = Aut(Q2n) and A0 be the stabilizer subgroup of
the vertex v = 0 in A. Since Q2n is a vertex-transitive graph, then from the orbit-stabilizer
theorem we have |A| = |A0||V (Q2n)| = 2n|A0|. The following lemma determines an
upper bound for |A0|.
Lemma 3.1. Let n > 4 and A = Aut(Q2n). Let A0 be the stabilizer subgroup of the vertex
v = 0. Then |A0| ≤ (n+ 1)!.
Proof. Let Γ = Q2n. We know that Γ = Cay(Zn2 ;T ), T = S ∪ S1, where S = {ei | 1 ≤
i ≤ n} and S1 = {ei + ej | i, j ∈ [n], i ̸= j}. Let f ∈ A0. Then f(T ) = T . Let G
be the subgraph of Γ which is induced by the subset T . Let h = f |T be the restriction
of the mapping f to the subset T . It is clear that h is an automorphism of the graph
G. It is easy to see that the mapping Φ: A0 → Aut(G), which is defined by the rule
Φ(g) = g|T , is a group homomorphism. Thus we have A0ker(Φ) ∼= im(Φ), and hence we
have |A0| = | ker(Φ)|| im(Φ)|. Since im(Φ) is a subgroup of Aut(G), then we have
|A0| ≤ | ker(Φ)||Aut(G)|. If we show that |Aut(G)| ≤ (n+ 1)! and ker(Φ) = {1}, then
the lemma is proved. Hence in the rest of the proof we show that:
(i) |Aut(G)| ≤ (n+ 1)!,
(ii) ker(Φ) = {1}.
(i) We give two proofs for proving this claim. The first is more elementary than the second,
but we need some parts of it in the proof of (ii). The second is based on the automorphism
group of the Johnson graph J(n, k).
Proof 1 of (i). Consider the graph G. In T = V (G), consider the subgraphs induced by
the subsets C0 = S = {ei| 1 ≤ i ≤ n}, Ci = {ei, ei + ej | 1 ≤ j ≤ n, i ̸= j}, 1 ≤ i ≤ n
(we also denote by Ci the subgraph induced by the set Ci ). It is clear that C0 is an n-
clique in the graph G. Note that if ei+ er and ei+ es are two elements of Ci, then we have
S. M. Mirafzal: Some remarks on the square graph of the hypercube 285
(ei + er)− (ei + es) = er + es ∈ T . Hence each Ci is also an n-clique in the graph G. It
can be shown that each Ci, 0 ≤ i ≤ n is a maximal n-clique in G. It is clear that if i ̸= 0,
then C0 ∩ Ci = {ei}. Moreover, if i, j ∈ {1, ..., n} and i ̸= j, then Ci ∩ Cj = {ei + ej}.
Let M be a maximal n-clique in the graph G. It is not hard to show that M = Cj for
some j ∈ {0, 1, ..., n}. If a is an automorphism of the graph G, then a(Cj) is a maximal
n-clique in the graph G. Hence the natural action of a on the set X = {C0, C1, ..., Cn} is
a permutation on X . Let G1 be the graph with the vertex set X in which two vertices v and
w are adjacent if and only if v ∩ w ̸= ∅. Now, it is clear that G1 ∼= Kn+1, the complete
graph on n + 1 vertices, and hence Aut(G1) ∼= Sym(X). Let a ∈ Aut(G) be such that
a(Cj) = Cj , for each j ∈ {0, 1, ..., n}. Noting that C0 ∩Ci = {ei}, i ̸= 0, we deduce that
a(x) = x for every x ∈ C0. Note that the vertex ei + ej is the unique common neighbor of
vertices ei and ej in the graph G which is not in C0. This implies that a(ei+ej) = ei+ej .
Therefore we have a(v) = v for every v ∈ T . Now it is easy to see that the mapping
π : Aut(G) → Aut(G1) defined by the rule π(a) = fa, where fa(Ci) = a(Ci) for every
Ci ∈ X , is an injection and therefore we have (n+ 1)! ≥ |Aut(G)|.
Proof 2 of (i). Consider the graph G. We show that this graph is isomorphic with the
Johnson graph J(n+ 1, 2). We define the mapping
f : V (G) → V (J(n+ 1, 2)),
by the rule:
f(v) =
{
{i, n+ 1}, if v = ei
{i, j}, if v = ei + ej
It is clear that f is a bijection. Let {v, w} be an edge in the graph G. Then we have three
possibilities:
(1) {v, w} = {ei, ej}, (2) {v, w} = {ei, ei + ek}, (3) {v, w} = {ei + ek, ei + ej}.
Now, we have (1) f({v, w}) = {{i, n + 1}, {j, n + 1}}, (2) f({v, w}) = {{i, n +
1}, {i, k}}, (3) f({v, w}) = {{i, k}, {j, k}}. It follows that f is a graph isomorphism.
Hence, Aut(G) ∼= Aut(J(n+1, 2)). Since Aut(J(n+1, 2)) ∼= Sym([n+1]) [3, 13, 18],
then we have Aut(G) ∼= Sym([n+ 1]).
(ii) we now show that ker(Φ) = {1}. Let f ∈ ker(Φ). Then f(0) = 0 and h = f |T
is the identity automorphism of the graph G. Hence f(x) = x for every x ∈ T . Note
that when x ∈ T , then w(x) ∈ {1, 2}, where w(x) is the weight of x, that is, the number
of 1s in the n-tuple x. Let x ∈ V (Γ) and w(x) = m. We show by induction on m, that
f(x) = x. It is clear that when m = 0, 1, 2, then the claim is true. Let the claim be
true when w(x) ≤ m, m ≥ 2. We show that if w(x) = m + 1, then f(x) = x. Let
y = ei1 + ... + eim + eim+1 be a vertex of weight m + 1. Let v = y + eim + eim+1 .
Since W (v) = m − 1, thus f(v) = v. Let N be the subgraph of Γ which is induced by
the set N(v). Since Γ is vertex-transitive, then G ∼= N . Also, since f(v) = v, then the
restriction of f to N(v) is an automorphism of the graph N . In N(v) we define the subsets
M0 = {v + ei| 1 ≤ i ≤ n}, Mi = {v + ei, v + ei + ej | 1 ≤ j ≤ n, j ̸= i}, 1 ≤ i ≤ n. It
can be check that the subgraph induced by each Mi is a maximal n-clique in the graph N .
Also, M0 ∩Mi = {v + ei}. Moreover, v + ei + ej is the unique common neighbor of the
vertices v + ei and v + ej in the graph N which is not in M0. If x ∈ M0, then f(x) = x,
because w(x) ≤ m. This implies that f(Mi) = Mi. Now, by an argument similar to what
286 Ars Math. Contemp. 23 (2023) #P2.06 / 281–296
is done in Proof 1, we can see that f(x) = x for every x ∈ N(v). Since y ∈ N(v), we
have f(y) = y. We now conclude that f is the identity automorphism of Γ. Hence we have
ker(Φ) = {1}.
Theorem 3.2. Let n > 4 and Γ = Q2n be the square of the hypercube Qn. Then we have
Aut(Γ) ∼= Zn2 ⋊ Sym([n+ 1]).
Proof. Let A0 be the stabilizer subgroup of the vertex v = 0 in the group Aut(Γ). We know
from Lemma 3.1, that |A0| ≤ (n+ 1)!. Let T and X = {C0, ..., Cn} be the sets which are
defined in the proof of Lemma 3.1. Note that Zn2 is a vector space over the field Z2 and Ci,
0 ≤ i ≤ n, is a basis for this vector space. Let fi : C0 → Ci be a bijection. We can linearly
extend fi to an automorphism e(fi) of the group Zn2 . It is clear that e(fi) ∈ A0. We
know that every automorphism of the group Zn2 which fixes the set T is an automorphism
of the graph Γ. We can see that when x, y ∈ Ci and x ̸= y then x + y ∈ T . Thus we
have e(fi)(er + es) = e(fi)(er) + e(fi)(es) ∈ T . Hence we have e(fi)(T ) = T . Since
the number of permutations fi is n!, hence the number of automorphisms of e(fi) is n!.
Note that when i ̸= j, then e(fi) ̸= e(fj). Now since 0 ≤ i ≤ n, then we have at least
(n+ 1)(n!) = (n+ 1)! distinct automorphisms in the group A0. Thus by Lemma 3.1, we
have |A0| = (n + 1)!. We saw, in the proof of Lemma 3.1, that A0 is isomorphic with a
subgroup of Sym([n+ 1]). Hence we deduce that A0 ∼= Sym([n+ 1]).
We know, by the orbit-stabilizer theorem, that |V (Γ)||A0| = |Aut(Γ)|. Thus we have
|Aut(Γ)| = 2n[(n+1)!]. For every v ∈ Zn2 , the mapping fv(x) = v+x, for every x ∈ Zn2 ,
is an automorphism of the graph Γ. It is easy to check that L = {fv| v ∈ Zn2} is a subgroup
of Aut(Γ) which is isomorphic with Zn2 . Also it is easy to check that L∩A0 = {1}. Hence
we have |LA0| = |L||A0| = 2n[(n+ 1)!] = |Aut(Γ)|. This implies that Aut(Γ) = LA0.
Also we can see that for every v ∈ Zn2 and every a ∈ A0 we have a−1fva = fa−1(v). Thus
we deduce that L is a normal subgroup of Aut(Γ). We now conclude that
Aut(Γ) ∼= L⋊A0 ∼= Zn2 ⋊ Sym([n+ 1]).
The graph Q2n has some interesting properties. In the next theorem, we show that Q
2
n
is distance-transitive.
Theorem 3.3. Let n ≥ 4 be an integer. Then the graph Q2n is a distance-transitive graph.
Proof. Let v and w be vertices in Q2n. It is easy to check that d(x, y) = ⌈
w(x+y)
2 ⌉. Hence
we have d(x, 0) = ⌈w(x)2 ⌉. Let D be the diameter of Q
2
n. it follows from the first two
sentences that D = ⌈n2 ⌉. Let A0 be the stabilizer subgroup the vertex v = 0 in Aut(Q
2
n).
Since the graph Q2n is a vertex-transitive graph, it is sufficient to show that the action of
A0 on the set Γk is transitive, where Γk is the set of vertices at distance k from the vertex
v = 0. Let x and y be two vertices in Γk. There are two possible cases, that is,
(i) w(x) = w(y) or
(ii) w(x) ̸= w(y).
(i) Let w(x) = w(y). We know that w(x) ∈ {2k, 2k − 1}. Without loss of generality, we
can assume that w(x) = 2k. Let x = ei1 + ... + ei2k and y = ej1 + ... + ej2k . There are
vertices ex1 , ..., exn−2k and ey1 , ..., eyn−2k in Q
2
n such that
{ei1 , ..., ei2k , ex1 , ..., exn−2k} = C0 = {e1, e2, ..., en} = {ej1 , ..., ej2k , ey1 , ..., eyn−2k}.
S. M. Mirafzal: Some remarks on the square graph of the hypercube 287
Let f be the permutation on the set C0 which is defined by the rule, f(eir ) = ejr , 1 ≤ r ≤
2k, and f(exl) = eyl , 1 ≤ l ≤ n − 2k. We now can see that e(f)(x) = y, where e(f) is
the linear extension of f to Zn2 (see the proof of Theorem 3.2).
(ii) Let w(x) ̸= w(y). Without loss of generality we can assume that w(x) = 2k − 1
and w(y) = 2k. Let x = ei1 + ... + ei2k−1 and y = ej1 + ... + ej2k . Note that y =
(ej1 + ej2k) + (ej2 + ej2k) + ... + (ej2k−2 + ej2k) + (ej2k−1 + ej2k). There are vertices
ex1 , ..., exn−2k+1 and ey1 = ej2k , ey2 , ..., eyn−2k+1 in Q
2
n such that
{ei1 , ..., ei2k−1 , ex1 , ..., exn−2k+1} = C0,
{ej1 + ej2k , ej2 + ej2k , ...+ ej2k−2 + ej2k , ej2k−1 + ej2k}∪
{ey1 , ey2 + ej2k , ..., eyn−2k+1 + ej2k} = Cj2k
We now define the bijection g from C0 to Cj2k by the rule g(eir ) = ejr + ej2k , and
g(ex1) = ey1 , g(exi) = eyi + ej2k , i ̸= 1. Let e(g) be the linear extension of g to Zn2 . This
yields that e(g) is an automorphism of the graph Q2n such that e(g)(x) = y.
Theorem 3.3 implies many results. For instance, we now can deduce from it the fol-
lowing corollary, which is important in applied graph theory and interconnection networks.
Corollary 3.4. Let n ≥ 4 be an integer. Then the connectivity of the graph Q2n is maximal,
namely, n+
(
n
2
)
(its valency).
Proof. By Theorem 3.3 the graph Q2n is distance-transitive, then it is edge-transitive. Thus,
it follows from Theorem 2.1, that the connectivity of the graph Q2n is its valency, namely,
n+
(
n
2
)
.
A block B, in the action of a group G on a set X , is a subset of X such that B∩g(B) ∈
{B, ∅}, for each g in G. If G is transitive on X , then we say that the permutation group
(X,G) is primitive if the only blocks are the trivial blocks, that is, those with cardinality 0, 1
or |X|. In the case of an imprimitive permutation group (X,G), the set X is partitioned
into a disjoint union of non-trivial blocks, which are permuted by G. We refer to this
partition as a block system. A graph Γ is said to be primitive or imprimitive according to
the group Aut(Γ) acting on V (Γ) has the corresponding property. In the sequel, we need
the following definition.
Definition 3.5. A graph Γ = (V,E) of diameter D is said to be antipodal if for any
u, v, w ∈ V such that d(u, v) = d(u,w) = D, then we have d(v, w) = D or v = w.
Let Γi(x) denote the set of vertices of Γ at distance i from the vertex x. Let Γ be a
distance-transitive graph. From Definition 3.5 it follows that if ΓD(x) is a singleton set,
then the graph Γ is antipodal. It is easy to see that the hypercube Qn is antipodal, since
every vertex u has a unique vertex at maximum distance from it. Note that this graph is at
the same time bipartite. We have the following fact [1].
Proposition 3.6. A distance-transitive graph Γ of diameter D has a block X = {u} ∪
ΓD(u) if and only if Γ is antipodal, where ΓD(u) is the set of vertices of Γ at distance D
from the vertex u.
Also, we have the following important fact [1].
288 Ars Math. Contemp. 23 (2023) #P2.06 / 281–296
Theorem 3.7. An imprimitive distance-transitive graph is either bipartite or antipodal.
(Both possibilities can occur in the same graph.)
We now can state and prove the following fact concerning the square of the hypercube
Qn.
Corollary 3.8. Let n ≥ 4 be an integer. Then, the square of the hypercube Qn, namely, the
graph Q2n, is an imprimitive distance-transitive graph if and only if n is an odd integer.
Proof. We know from Theorem 3.3, that the graph Γ = Q2n is a distance-transitive graph.
Let n = 2k be an even integer. If D denotes the diameter of Q2n, then D = k. Let
C0 = {e1, ..., en} be the standard basis of the hypercube Qn. Let w = e1 + e2 + ...+ en
and B1 = {w + ei | 1 ≤ i ≤ n}. Consider the vertex u = 0. It is easy to show that
ΓD(u) = {w} ∪B1. Two vertices w and w + e1 are in ΓD(u), but they are not at distance
k = D from each other, since they are adjacent and k > 1. Thus, when n is an even
integer, then the graph Q2n is not antipodal. Since the girth of Q
2
n is 3, then this graph is
not bipartite. Now, Theorem 3.7 implies that the graph Γ = Q2n is not imprimitive.
Now assume that n = 2k + 1 is an odd integer. It is easy to see that D = k + 1 and
ΓD(0) = {w}. Therefore by Proposition 3.6, Γ is antipodal, and hence has the set {0, w}
as a block. We now conclude that, when n is an odd integer, then Q2n is an imprimitive
graph.
Let Γ = (V,E) be a simple connected graph with diameter D. A distance-regular
graph Γ = (V,E), with diameter D, is a regular connected graph of valency k with the
following property. There are positive integers
b0 = k, b1, ..., bD−1; c1 = 1, c2, ..., cD,
such that for each pair (u, v) of vertices satisfying u ∈ Γi(v), we have
(1) the number of vertices in Γi−1(v) adjacent to u is ci, 1 ≤ i ≤ D.
(2) the number of vertices in Γi+1(v) adjacent to u is bi, 0 ≤ i ≤ D − 1.
The intersection array of Γ is i(Γ) = {k, b1, ..., bD−1; 1, c2, ..., cd}.
It is easy to show that if Γ is a distance-transitive graph, then it is distance-regular [1].
Hence, the hypercube Qn, n > 2 is a distance-regular graph. We can verify by an easy
argument that the intersection array of Qn is
{n, n− 1, n− 2, ..., 1; 1, 2, 3, ..., n}.
In other words, for hypercube Qn, we have bi = n− i, ci = i, 1 ≤ i ≤ n− 1, and b0 = n,
cn = n. In the following theorem, we determine the intersection array of the square of the
hypercube Qn [1].
Theorem 3.9. Let n > 3 be an integer and Γ = Q2n be the square of the hypercube Qn.
Let D denote the diameter of Q2n. Then for the intersection array of this graph we have
b0=
(
n+1
2
)
, bi=
(
n−2i+1
2
)
, ci=
(
2i
2
)
, 1 ≤ i ≤ D − 1. Also, cD=
(
n+1
2
)
, when n is an odd
integer and cD=
(
n
2
)
when n is an even integer.
S. M. Mirafzal: Some remarks on the square graph of the hypercube 289
Proof. Since Q2n is a regular graph of valency
(
n+1
2
)
, thus we have b0=
(
n+1
2
)
. Let u be a
vertex in Q2n at distance i from the vertex v = 0. It is easy to check that w(u) = 2i or
w(u) = 2i− 1. This implies that that the diameter of the graph Q2n is D = ⌈n2 ⌉.
Let u be a vertex in Q2n at distance i ≥ 1 from the vertex v = 0, such that i ̸= D.
There are two cases, that is, w(u) = 2i, or w(u) = 2i − 1. Without lose of generality we
can assume that w(u) = 2i. Hence u is of the form u = ej1 + ej2 + ... + ej2i . Now it is
easy to show that if x is a vertex of Q2n adjacent to u and at distance i− 1 from the vertex
v = 0, then x must be of the form x = u+ ek + el, where ek, el ∈ {ej1 , ej2 , ..., ej2i}. It is
clear that the number of such xs is equal to
(
2i
2
)
. Moreover, If x is a vertex of Q2n adjacent
to u and at distance i + 1 from the vertex v = 0, then x must be of the forms x = u + ek
or x = u + ek + el, where ek, el ∈ {e1, e2, ..., en} − {ej1 , ej2 , ..., ej2i}. It is clear that
the number of such xs is equal to
(
n−2i
1
)
+
(
n−2i
2
)
=
(
n−2i+1
2
)
. We now deduce that when
1 ≤ i ≤ D − 1, then ci=
(
2i
2
)
, and bi=
(
n−2i+1
2
)
.
When n is an odd integer, then the vertex u = e1 + e2 + ... + en is the unique vertex
of Q2n at distance D from the vertex v = 0. Thus cD=
(
n+1
2
)
, namely, the valency of u. If
n is an even integer, then ΓD(0) = {u, u+ ei| 1 ≤ i ≤ n} is the set of vertices of Γ = Q2n
at distance D from the vertex v = 0. Now, by a similar argument which is done in the first
section of the proof, it can be shown that cD=
(
n
2
)
.
Remark 3.10. There are distance-regular graphs Γ = (V,E), with the property that
their squares are not distance-regular. For instance, consider the cycle Cn with vertex
set {0, 1, 2, ..., n− 1}. It is well known that Cn is a distance-regular graph of diameter [n2 ]
with the intersection array:
{2, 1, 1, ..., 1, 1; 1, 1, 1, ..., 1, 2} when n is an even integer and,
{2, 1, 1, ..., 1, 1; 1, 1, 1, ..., 1, 1} when n is an odd integer [1].
Now, assume that n ≥ 7. It can be shown by an easy argument that Γ = C2n is not a
distance-regular graph. To see this fact, let v be a vertex in Cn at distance i from the vertex
0, and ci(v) = |Γi−1(0)∩N(v)|. It is easy to show that Γi(0) = {2i,−2i, 2i−1,−2i+1},
and ci(2i) = 1, but ci(2i− 1) = 2.
Remark 3.11. Let n, k ∈ N with k < n, and let [n] = {1, ..., n}. Consider the Johnson
graph J(n, k). It is clear that the order of this graph is
(
n
k
)
. It is easy to check that J(n, k) ∼=
J(n, n − k), hence we assume that 1 ≤ k ≤ n2 . The class of Johnson graphs is one of the
most well known and interesting subclass of distance-regular graphs [3]. It is easy to show
that if v and w are vertices in the Johnson graph J(n, k), then d(v, w) = k−|v∩w|. Thus,
the diameter of the Johnson graph J(n, k) is k. Note that the graph J(n, 1) is the complete
graph Kn and hence it is distance-regular. The diameter of the graph J(n, 2) is 2, hence the
diameter of its square is 1. Thus the graph J2(n, 2) is the complete graph Km, and hence
it is a distance-regular graph (m=
(
n
2
)
). We can show that when k = 3, then the square of
Johnson graph Γ = J(n, k) is a distance-regular graph if and only if n = 6. For checking
this, let v = {1, 2, 3}. Note that the diameter of the graph Γ2 is 2 and a vertex w in Γ2
is at distance 2 from v if and only if |v ∩ w| = 0. Moreover w is at distance 1 from v if
and only if |v ∩ w| ∈ {1, 2}. Hence Γ21(v) = V (Γ) − {v, vc} and Γ22(v) = {vc}, where
vc is the complement of the set v in the set {1, 2, ..., 6}. Thus vc = {4, 5, 6}. Now, it is
clear that b0(v)=
(
3
2
)(
3
1
)
+
(
3
1
)(
3
2
)
=18. Also, for every w ∈ Γ21(v), c1(w) = 1 and b1(w) = 1,
and c2(vc) = |Γ21(v)| = 18. Thus the graph Γ2 = J2(6, 3) is a distance-regular graph
290 Ars Math. Contemp. 23 (2023) #P2.06 / 281–296
with intersection array {18, 1; 1, 18}. But, if n > 6, then the graph Γ2 = J2(n, 3) is not
distance-regular. In fact if n > 6, then for the vertex v = {1, 2, 3}, each of the vertices
u = {1, 2, 4} and w = {1, 4, 5} is in Γ21(v). If x ∈ Γ22(v) is adjacent to u, then 4 ∈ x, and
hence x = {4}∪y, where y ⊂ vc−{4} and |y| = 2. We now can deduce that b1(u)=
(
n−4
2
)
.
On the other hand, if x ∈ Γ22(v) is adjacent to w, then 4 ∈ x and 5 /∈ x, or 5 ∈ x and 4 /∈ x
or 4, 5 ∈ x. Thus, b1(w)=2
(
n−5
2
)
+
(
n−5
1
)
=
(
n−4
2
)
+
(
n−5
2
)
. This implies that when n ≥ 7
then the graph J2(n, 3) cannot be distance-regular.
By a similar argument we we can show that the graph J2(8, 4) is distance-regular, but
if n > 8, then the graph J2(n, 4) is not distance-regular.
Remark 3.12. Let Γ = (V,E) be a graph. Γ is said to be a strongly regular graph with
parameters (n, k, λ, µ), whenever |V | = n, Γ is a regular graph of valency k, every pair of
adjacent vertices of Γ have λ common neighbor(s), and every pair of non-adjacent vertices
of Γ have µ common neighbor(s). It is clear that the diameter of every strongly regular
graph is 2. It is easy to show that if a graph Γ is a distance-regular graph of diameter
2 and order n, with intersection array (b0, b1; c1, c2), then Γ is a strongly regular graph
with parameters (n, b0, b0 − b1 − 1, c2). We know that the diameter of the graph Q2n is
⌈n2 ⌉. Now, it follows from Theorem 3.3, that Q
2
3 is a strongly regular graph with parameter
(8, 6, 4, 6). This graph is known as the coktail-party graph CP (4) [1]. Also, the graph Q24
is a strongly regular graph with parameter (16, 10, 6, 6). We know that when a graph Γ is a
strongly regular graph with parameters (n, k, λ, µ), then its complement is again a strongly
regular graph with parameter (n, n− k − 1, n− 2− 2k + µ, n− 2k + λ) [9]. Hence, the
complement of the graph Q24 is a strongly regular graph with parameter (16, 5, 0, 2). This
graph is known as the Clebsch graph [9] and it is the unique strongly regular graph with
parameters (16, 5, 0, 2). Figure 1 displays a version of the Clebsch graph (the complement
of the graph Q24) in the plane [9].
Figure 1: The Clebsch graph.
S. M. Mirafzal: Some remarks on the square graph of the hypercube 291
4 The spectrum of the square of the hypercube
The square of the hypercube Qn has some further interesting algebraic properties. For
obtaining some of those properties, we need the spectrum of this graph. The spectrum of
Qn is known [1], however we are not aware of a paper showing the spectrum of Q2n. Here
we compute by means of an algebraic and self-contained method the spectrum of Q2n.
Let Γ = (V,E) be a graph with the vertex set {v1, · · · , vn}. Then the adjacency matrix
of Γ is an n× n matrix A = (aij), in which columns and rows are labeled by V and aij is
defined as follow:
aij = A(vi, vj) =
{
1 if vi is adjacent to vj
0 otherwise.
If Ax = λx, x ̸= 0, then λ is an eigenvalue of A, and x is an eigenvector of A corre-
sponding to λ [9]. Let λ1, · · · , λr be eigenvalues of A with multiplicities m1, · · · ,mr,
respectively. The spectrum of the graph Γ is defined as
Spec(Γ) =
{
λ1, λ2, · · · , λr
m1 m2 · · · mr
}
.
When we work with graphs there is an additional refinement. We can suppose that
an eigenvector is a real function f on the vertices. Then if at any vertex v you sum up
the values of f on its neighboring vertices, you should get λ times the values of f at v.
Formally, ∑
w∈N(v)
f(w) = λf(v).
Let G be a finite abelian group (written additively) of order |G| with identity element
0=0G. A character χ of G is a homomorphism from G into the multiplicative group U
of complex numbers of absolute value 1, that is, a mapping from G into U with χ(g1 +
g2) = χ(g1)χ(g2) for all g1, g2 ∈ G. If G is a finite abelian group, then there are integers
n1, · · · , nk, such that G = Zn1 × · · · × Znk . Let S = {s1, · · · , sn} be a non-empty
subset of G such that 0 ̸∈ S and S = −S. Let Γ = Cay(G;S). Assume f : G −→ C∗
is a character where C∗ is the multiplicative group of the complex numbers. If ωij =
e
2πij
ni , 0 ≤ i ≤ k, 1 ≤ j ≤ ni, is an nith root of unitary, then f is of the form f =
f(ω1,··· ,ωk), where f(ω1,··· ,ωk)(x1, · · · , xk) = ω
x1
1 ω
x2
2 · · ·ω
xk
k , for each (x1, x2, ..., xk) ∈
G [12].
If v is a vertex of Γ, then we know that N(v) = {v + s1, · · · , v + sn} is the set of
vertices that are adjacent to v. We now have
∑
w∈N(v)
f(w) =
n∑
i=1
f(v + si) =
n∑
i=1
f(v)f(si) = f(v)(
n∑
i=1
f(si)).
Therefore, if we let λ = λf =
∑
s∈S f(s) then we have
∑
w∈N(v) f(w) = λff(v),
and hence the mapping f is an eigenvector for the Cayley graph Γ with corresponding
eigenvalue λ = λf =
∑
s∈S f(s).
292 Ars Math. Contemp. 23 (2023) #P2.06 / 281–296
Theorem 4.1. Let n > 3 be an integer and Q2n be the square of the hypercube Qn. Then
each of the eigenvalues of Q2n is of the form,
λi =
1
2
n(n+ 1)− 2i(n+ 1) + 2i2,
for 0 ≤ i ≤ ⌊n+12 ⌋. Moreover, the multiplicity of λ0 is 1, the multiplicity of λi is m(λi)=(
n
i
)
+
(
n
n+1−i
)
, for 1 ≤ i ≤ ⌊n+12 ⌋, when n is an even integer, and m(λi)=
(
n
i
)
+
(
n
n+1−i
)
for 1 ≤ i < ⌊n+12 ⌋, when n is an odd integer, with m(λj)=
(
n
j
)
for j = ⌊n+12 ⌋.
Proof. According to what is stated before this theorem, every eigenvector of the graph
Γ = Q2n = Cay(Zn2 ;S) is of the form f = f(ω1,··· ,ωn), where each ωi, 1 ≤ i ≤ n, is a
complex number such that ω2i = 1, namely, ωi ∈ {1,−1}. We now have
λf =
∑
w∈S
f(w) =
n∑
i=1
f(ei) +
n∑
i,j=1, i ̸=j
f(ei + ej)
=
n∑
i=1
f(ei) +
n∑
i,j=1, i ̸=j
f(ei)f(ej).
Note that for every vertex v = (x1, . . . , xn), xi ∈ {0, 1} in Q2n, we have
f(x1, . . . , xn) = f(w1,...,wn)(x1 . . . , xn) = w
x1
1 . . . w
xn
n .
Note that in the computing of the value of wx11 . . . w
xn
n we can ignore wi when wi = 1.
Thus, for ek = (0, . . . , 0, 1, 0 . . . , 0), where 1 is the kth entry, we have;
f(ek) = f(w1,...,wn)(0, . . . , 0, 1, 0, . . . , 0)
= w01 . . . w
1
kw
0
k+1 . . . w
0
n =
{
−1 if wk = −1
1 if wk = 1
Hence, if in the n-tuple (w1, . . . , wn) the number of −1s is i (and therefore the number of
ls is (n− i)), then in the sum
n∑
k=1
f(ek) =
n∑
k=1
f(w1,...,wn)(0, . . . , xk, 0, . . . , 0), xk = 1,
the contribution of −1 is i and the contribution of 1 is n− i. Therefore, we have
n∑
k=1
f(ek) = −i+ (n− i) = n− 2i.
On the other hand, since
(
n∑
k=1
f(ek))
2 =
n∑
k=1
f(ek)
2
+ 2
n∑
i,j=1, i ̸=j
f(ei)f(ej),
therefore, we have
S. M. Mirafzal: Some remarks on the square graph of the hypercube 293
n∑
i,j=1, i ̸=j
f(ei)f(ej) =
1
2
((n− 2i)2 −
n∑
k=1
f(ek)
2
).
Now since
∑n
k=1 f(ek)
2
= n, thus we have
λf =
n∑
i=1
f(ei) +
n∑
i,j=1, i ̸=j
f(ei)f(ej) = (n− 2i) +
1
2
((n− 2i)2 − n)
=
1
2
n+
1
2
n2 − 2ni+ 2i2 − 2i = 1
2
n(n+ 1)− 2i(n+ 1) + 2i2.
Note that f = f(w1,w2,...,wn), and the number of sequences (w1 . . . , wn) in which i entries
are −1 is
(
n
i
)
. If we denote λf by λi, then we deduce that every eigenvalue of the graph
Q2n is of the form
λi =
1
2
n(n+ 1)− 2i(n+ 1) + 2i2, 0 ≤ i ≤ n. (∗∗)
Consider the real function f(x) = 12n(n + 1) − 2x(n + 1) + 2x
2. Then λi = f(i),
i ∈ {0, 1, ..., n}. This function reaches its minimum at x = n+12 . Now by using some
calculus, we can see that f(x) = f(n + 1 − x). Thus, we have λi = f(i) = f(n + 1 −
i) = λn+1−i, 1 ≤ i ≤ n. Now it follows that if n = 2k, then the multiplicity of λi is(
n
i
)
+
(
n
n+1−i
)
, 1 ≤ i ≤ k. Note that when n = 2k + 1, then n+ 1− (k + 1) = k + 1, thus
λn+1−(k+1) = λk+1. Hence if n = 2k + 1, then the multiplicity of λi is
(
n
i
)
+
(
n
n+1−i
)
,
1 ≤ i ≤ k, and the multiplicity of λk+1 is
(
n
k+1
)
. Note that since the graph Q2n is a(
n+1
2
)
-regular graph, hence the multiplicity of λ0=
(
n+1
2
)
= 12 (n+ 1)n is 1.
Let Γ = (V,E) be a graph. The line graph L(Γ) of the graph Γ is constructed by
taking the edges of Γ as vertices of L(Γ), and joining two vertices in L(Γ) whenever the
corresponding edges in Γ have a common vertex. Note that if e = {v, w} is an edge of Γ,
then its degree in the graph L(Γ) is deg(v) + deg(w) − 2. Concerning the eigenvalues of
the line graphs, we have the following fact [1, 9].
Proposition 4.2. If λ is an eigenvalue of a line graph L(Γ), then λ ≥ −2.
Therefore, if λ < −2 is an eigenvalue of a graph graph Γ, then Γ is not a line graph.
A (c, d)-biregular graph is a bipartite graph in which each vertex in one part has degree
c and each vertex in the other part has degree d [25]. It is known and easy to prove that
if the line graph of the graph Γ is regular, then Γ is a regular or a (c, d)-biregular bipartite
graph.
Theorem 4.3. Let n ≥ 4 be an integer and Q2n be the square of the hypercube Qn. Then
Q2n cannot be a line graph.
Proof. Let k = ⌊n2 ⌋. Hence, if n is an even integer, then n = 2k and if n is an odd integer
then n = 2k + 1. It follows from Theorem 4.1, that the smallest eigenvalue of the graph
Q2n is λk, when n is an even integer and λk+1, when n is an odd integer. Now consider the
eigenvalue λk of the graph Q2n in (**) (in the proof of Theorem 4.1). Therefore if n is an
even integer, then we have
λk = k(2k + 1)− 2k(2k + 1) + 2k2 = k(2k + 1− 4k − 2 + 2k) = −k.
294 Ars Math. Contemp. 23 (2023) #P2.06 / 281–296
Moreover if n = 2k + 1, then we have,
λk+1 = (2k + 1)(k + 1)− 2(k + 1)(2k + 2) + 2(k + 1)2
= (k + 1)(2k + 1− 4k − 4 + 2k + 2) = −k − 1.
We now deduce that when n ≥ 5, then λk ≤ −3. Now, it follows from Proposition 4.2,
when n ≥ 5, then the graph Q2n can not be a line graph.
Our argument shows that if λ is an eigenvalue of the graph Q24, then λ ≥ −2, and hence
in this way we can not say anything about our claim.
We now show that Q24 is not a line graph. On the contrary, assume that Q
2
4 is a line graph.
Thus, there is a graph ∆ such that Q24 = L(∆). Since Q
2
4 is a regular graph, hence
(i) ∆ is a regular graph, or
(ii) ∆ is a biregular bipartite graph.
(i) Let ∆ = (V,E) be a t-regular graph of order h. Since Q24 is 10-regular, thus, L(∆) =
Q24 is a 2t−2 = 10-regular graph, and hence t = 6. Therefore we have 16 = |E| = 126h =
3h, which is impossible.
(ii) Let ∆ = (A ∪ B,E) be a (c, d)-biregular bipartite graph such that every vertex in
A (B) is of degree c (d). Hence we have 16 = |E| = c|A| = d|B|. Thus c and d
must divide 16. On the other hand, if e = {a, b} is an edge of ∆, then we must have
deg(a)+deg(b)−2 = 10 = c+d−2. Hence we have c+d = 12. We now can check that
{c, d} = {4, 8}. Without loss of generality, we can assume that d = 8 and c = 4. Hence
we must have |A| ≥ 8. Now since each vertex in A is of degree c = 4, then we must have,
16 = |E| = c|A| = 4|A| ≥ 4× 8 = 32, which is impossible.
Our argument shows that the graph Q24 is also not a line graph.
An automorphic graph is a distance-transitive graph whose automorphism group acts
primitively on its vertices, and not a complete graph or a line graph.
Automorphic graphs are apparently very rare. For instance, there are exactly three cubic
automorphic graphs [1, 2]. It is clear that for n ≥ 3, the graph Q2n is not a complete graph.
We now derive from Corollary 3.8, and Theorem 4.3, the following important result.
Corollary 4.4. Let n ≥ 4 be an integer. Then the square of the hypercube Qn, that is, the
graph Q2n, is an automorphic graph if and only if n is an even integer.
5 Conclusion
In this paper, we proved that the square of the distance-transitive graph Qn, that is, the
graph Q2n, is again a distance-transitive graph (Theorem 3.3). We showed that there are im-
portant classes of distance-transitive graphs (including the cycle Cn, n ≥ 7), such that their
squares are not even distance-regular (and hence are not distance-transitive) (Remark 3.11).
Also, we determined the spectrum of the graph Q2n (Theorem 4.1). Moreover, we showed
that when n > 3 is an even integer, then the graph Q2n is an automorphic graph, that is, a
distance-transitive primitive graph which is not a complete or a line graph (Corollary 4.4).
ORCID iDs
Seyed Morteza Mirafzal https://orcid.org/0000-0002-3545-7022
S. M. Mirafzal: Some remarks on the square graph of the hypercube 295
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.07 / 297–303
https://doi.org/10.26493/1855-3974.2707.29c
(Also available at http://amc-journal.eu)
A new family of additive designs
Andrea Caggegi
Dipartimento di Ingegneria, Università degli Studi di Palermo, Viale delle Scienze,
90128 Palermo, Italy
Received 10 October 2021, accepted 1 October 2022, published online 18 November 2022
Abstract
In this paper we construct a family of 2-(qn, sp2, λ) additive designs D = (P,B),
where q is a power of a prime p and P is a n-dimensional vector space over GF(q), and we
compute their parameters explicitly. These designs, except for some special cases, had not
been considered in the previous literature on additive block designs.
Keywords: Block designs, additive designs.
Math. Subj. Class. (2020): 05B05, 05B25, 05B07
1 Additive designs
Point-flat designs D = (P,B) of an affine geometry AG(d, p) over GF(p), as well as of
a projective geometry PG(d, 2) over GF(2), are basic examples of 2-(v, k, λ) designs for
which, if P is taken to be GF(pd), respectively GF(2d+1)∗ = GF(2d+1) \ {0}, then the
blocks have the property that the sum of their points in P is zero.
As soon as k > 4, the family B of blocks of any of these designs is strictly contained in
the family Bk (respectively, B∗k) of all the k-subsets of GF(pd) (respectively, GF(2d+1)∗)
whose elements sum up to zero. In [19], and in [13] for the case p = 2, it is shown that
the incidence structure Dk = (P,Bk) is a 2-(pd, k, λ) design if and only if k = mp for
some integer m, and that, in such a case, the automorphism group of Dk is the group of
invertible affine mappings ϕ(x) = ϕ0(x) + ϕ(0) over GF(p), with ϕ0 ∈ GL(d, p). In this
case, by applying a well-known result of Li and Wan [15] (see also [14, Theorem 2.4] and
[20]), one finds that
λ =
1
pd
(
pd − 2
k − 2
)
+ ck
k − 1
pd
(
pd−1 − 1
m− 1
)
,
where ck = (−1)m if p = 2 and ck = 1 otherwise.
E-mail address: andreacaggegibibd@gmail.com (Andrea Caggegi)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
298 Ars Math. Contemp. 23 (2023) #P2.07 / 297–303
Moreover, for p = 2, the incidence structure D∗k = (GF(2d+1)∗,B∗k) is a 2-(2d+1 − 1,
k, λ) design for any integer k, and, again, the parameter λ is given by an explicit formula
[13, Proposition 2.6], whereas the automorphism group of D∗k is the group GL(d+1, 2) of
invertible linear mappings on GF(2d+1) over GF(2). Among the subdesigns of the latter
designs one finds the only known Steiner 2-design over a finite field, found by Braun et al.
[2] and revisited in [6], when seen as a 2-(8191, 7, 1) design (note that 8191 = 213 − 1), as
well as the 2-(2v − 1, 7, 7) designs over GF(2) considered in [4], [21].
More generally, in [8] and [9] a 2-(v, k, λ) design D = (P,B) is said to be additive
if P can be embedded in a finite commutative group G in such a way that the sum of the
elements in every block is zero. Moreover, it is shown that symmetric and affine resolvable
2-designs are additive and that, for these designs, and for a suitable choice of G, the blocks
are exactly the (unordered) k-tuples of elements in P which sum up to zero, so that the
automorphism group of D coincides with the stabilizer of P in the automorphism group
of G. On the contrary, it is shown that the only additive Steiner triple systems are the
point-line designs of AG(d, 3) and PG(d, 2) (cf. also [11] and [12]).
With a similar construction to that considered in the present paper, in [18] an additive
2-design is provided, for which no embedding can be found in such a way that the blocks
are characterized as the k-sets of elements of P summing up to zero, thereby settling an
open question posed in [9].
Interestingly enough, the search for new additive designs occasionally produces new
designs which, in addition to being additive, turn out to be also the first known examples of
designs with a certain set of parameters. For instance, in [16] an additive 2-(81, 6, 2) design
is constructed, which is also the first known example of a simple 2-design (that is, with no
repeated blocks) with these parameters, whereas in [17] an additive Steiner 2-(124, 4, 1)
design is presented. More generally, some infinite classes of additive Steiner 2-designs are
presented in [5] and [3], in the latter case as a notable application of the method of partial
differences.
The goal of this paper is to introduce a class of (additive) block designs that are subde-
signs of D = (GF(pd),Bk) and which seem not to have appeared so far in the literature.
2 Some new designs
In [7] we considered the 2-(n2, 2n, 2n − 1) design obtained by taking the points and the
(unordered) pairs of distinct parallel lines of a finite affine plane of order n > 2. Similarly,
in this paper we consider an incidence structure whose blocks are unions of suitable parallel
lines in an affine geometry over GF(p). We obtain an additive subdesign of the design
D = (GF(pd),Bk) considered here in Section 1, for which we are able to compute the
parameters.
Note that one finds, among these designs, the classical point-flat designs AG2(n, 3),
n ≥ 2, and AG3(n, 2), n ≥ 3. Interestingly enough, in some special cases the 2-(v, k, λ)
designs that we construct have a smaller λ than that of the corresponding point-flat designs
of AG(d, p) with the same parameters v and k.
As usual, we say that m vectors x1, x2, . . . , xm are affinely independent if the m − 1
vectors x2 − x1, . . . , xm − x1 are linearly independent.
Theorem 2.1. Let q be a power of a prime p, and let P be a n-dimensional vector space
over GF(q). Let m be divisible by p, with 3 ≤ m ≤ n + 1, and let B consist of all
A. Caggegi: A new family of additive designs 299
subsets b(x1, x2, . . . , xm) of P of the form
b(x1, x2, . . . , xm) = {xj + s(x1 + x2 + · · ·+ xm)| 1 ≤ j ≤ m and s ∈ GF(p)},
where x1, x2, . . . , xm ∈ P are affinely independent vectors over GF(q), and GF(p) is the
fundamental subfield of GF(q). Then D = (P,B) is a 2 − (qn,mp, λ) additive design,
with
λ =

(qn−q)···(qn−qm−2)
(m−1)! pm−2 (p−1) (mp− 1) if m > 4,
(qn−q)(qn−q2)
24 if m = 4,
qn−q
6 if m = 3.
(2.1)
Proof. Suppose b(y1, y2, . . . , ym) ∈ B. Since the vectors y1, y2, . . . , ym ∈ P are affinely
independent, the sum (y2− y1)+ · · ·+(ym− y1) is not zero and, since m is divisible by p
and my1 = 0, we deduce that y1+y2+ · · ·+ym is not zero, as well. Since the case m = 2
is excluded by hypothesis, the sets {yi+s(y1+y2+· · ·+ym)| s ∈ GF(p)} and {yj+s(y1+
y2+ · · ·+ ym)| s ∈ GF(p)} are disjoint, for i ̸= j, thus b(y1, y2, . . . , ym) contains exactly
mp elements (note that in the excluded case where m = 2 the two sets are coincident).
Because m ≤ n + 1 and because G = Aff(P) (the affine group of P over GF(q)) acts
2-homogeneously on P and permutes the subsets {w1, w2, . . . , wm} of P consisting of m
affinely independent vectors, the block-set B may be written as B = bG0 (the G-orbit of a
fixed block b0 = b(x1, x2, . . . , xm)), and it follows from [1, Proposition 4.6, page 175] (or
from [10, Remark 4.29, page 82]) that D is a 2− (v, k, λ) design with parameters v = qn,
k = mp and
b = |B| = |G|
|Sb0 |
,
where Sb0 = {f ∈ Aff(P)| f(b0) = b0} is the setwise stabilizer of the base block b0.
Since, for every block b = b(y1, y2, . . . , ym) of D,∑
y∈b
y =
{(
p+m
(
p
2
))
(y1 + y2 + · · ·+ ym) for p > 2
m(y1 + y2 + · · ·+ ym) for p = 2
,
which is the zero vector in either case, the design D is additive by [8, Proposition 2.7,
page 277].
In order to determine the number b of blocks of D, we claim that, if b = b(y1, y2, . . . , ym)
is any block of the 2-design D and if we denote by Rb the number of (unordered) sets
{z1, z2, . . . , zm} ⊂ b consisting of affinely independent vectors z1, z2, . . . , zm having the
property that b(z1, z2, . . . , zm) = b(y1, y2, . . . , ym), then Rb does not depend on b and we
have
Rb =

pm−1(p− 1), if m > 4,
56, if m = 4,
72, if m = 3.
Indeed, if t1, t2, . . . , tm ∈ GF(p) are chosen in such a way that t1+ t2+ · · ·+ tm ̸= −1 ∈
GF(p), then the m (distinct) vectors zi = yi + ti(y1 + y2 + · · · + ym) of P (belonging
300 Ars Math. Contemp. 23 (2023) #P2.07 / 297–303
to b) are affinely independent and have the property that b(z1, z2, . . . , zm) = b. Hence
Rb ≥ pm − pm−1 = pm−1(p− 1).
On the other hand, since
lj = {yj + τ(y1 + y2 + · · ·+ ym)| τ ∈ GF(q)} (j = 1, 2, . . . ,m)
are m distinct parallel lines of P such that b ⊆ l1∪ l2∪· · ·∪ lm, we infer: if b(w1, w2, . . . ,
wm) = b for suitable affinely independent vectors w1, w2, . . . , wm ∈ b, and if m > 4,
then the block b is strictly contained in the affine subspace over GF(p) through the m
given affinely independent points and defines uniquely the direction y1 + y2 + · · · + ym
of the parallel lines, thus the m-set {w1, w2, . . . , wm} meets each of the m lines lj (j =
1, 2, . . . ,m) in just one point (vector), otherwise some of the yj would not belong to b =
b(y1, y2, . . . , ym) = b(w1, w2, . . . , wm). Hence there are c1, c2, . . . , cm ∈ GF(p) such
that wj = yj + cj(y1 + y2 + · · ·+ ym) for j = 1, 2, . . . ,m. Therefore we must have Rb ≤
pm−1(p − 1), if m > 4. Thus we proved that, if m > 4, then Rb ≤ pm−1(p − 1) ≤ Rb,
that is, Rb = pm−1(p− 1).
Suppose now m = 4. Thus p = 2 and the four lines yi + ⟨y1 + y2 + y3 + y4⟩ (with
i = 1, 2, 3, 4), whose union is b, fill a whole 3-dimensional space over GF(2). Then four
vectors (points) z1, z2, z3, z4 ∈ b have the property that b(z1, z2, z3, z4) = b if and only if
z1, z2, z3, z4 are non-coplanar points of (the affine space) b: choosing 3 points out of the 8,
and a further point not in the plane through them, we obtain 4 non-coplanar points, in
(
4
3
)
different ways, hence Rb = 4×
(
8
3
)
/
(
4
3
)
= 56, if m = 4.
Finally, suppose m = 3. Then p = 3 and the three lines yi + ⟨y1 + y2 + y3⟩ (with
i = 1, 2, 3), whose union is b, are coplanar, hence b is a finite affine plane of order 3. Then
three vectors (points) z1, z2, z3 ∈ b are affinely independent (and have the property that
b(z1, z2, z3) = b) if and only if z1, z2, z3 are non-collinear points of (the affine plane) b.
Therefore Rb =
(
9
3
)
− 12 = 72, if m = 3, and the claim is proved.
Since q
n(qn−1)(qn−q)···(qn−qm−2)
1·2·3···m is the number of all the m-subsets of P consisting
of affinely independent vectors, counting in two ways the number of flags (W, b), where
W = {w1, w2, . . . , wm} is an m-subset of P consisting of affinely independent vectors
and b = b(y1, y2, . . . , ym) is a block of D through W , we obtain by the above argument
qn(qn−1)(qn−q)···(qn−qm−2)
1·2·3···m = p
m−1(p− 1)b , if m > 4,
qn(qn−1)(qn−q)(qn−q2)
24 = 56b , if m = 4,
qn(qn−1)(qn−q)
6 = 72b , if m = 3,
and this gives the number b of blocks. The parameter λ follows consequently.
Remark 2.2. It is worth noting that the cases where m = 3, 4 are sensibly different from
those where m > 4.
Let us first point out that, since the 2−(qn,mp, λ) designs D considered in Theorem 2.1
have v = qn points, it is natural to ask in what cases such designs arise just as classical
point-flat designs AGµ(n, q) of the affine geometries AG(n, q). It turns out that this is the
case only for AG2(n, 3), n ≥ 2, and AG3(n, 2), n ≥ 3. Indeed, the µ-flat through m
affinely independent points has k = qm−1 points, and this equals k = mp only in the cases
A. Caggegi: A new family of additive designs 301
where m = 2 and q = 4 (which is excluded), m = 3 and q = 3, and m = 4 and q = 2.
The fact that in these two cases the blocks turn out to be affine subspaces has already been
pointed out in the above proof.
In all the remaining cases, the designs D in Theorem 2.1 are not point-flat designs
AGµ(n, q). For q = pc,m = ph, such designs D are 2 − (pcn, ph+1, λ) designs, hence
they have the same parameters v and k as the point-flat designs AGh+1(cn, p) of the affine
geometries AG(cn, p), thus it is appropriate to compare the value of the parameter λ in
(2.1) for D with the value of λ for AGh+1(cn, p). As we will now see, for m = p = 3,
q = 3c (resp., for m = 4, p = 2, q = 2c), with c > 1, the value of λ in (2.1) is smaller
than the corresponding value of λ for the point-plane design AG2(cn, 3) (resp., for the
point-flat design AG3(cn, 2)). In either case, the design D has a GF(p)-structure, but not
a GF(q)-structure.
(i) For m = 3 and q = 3c, c > 1, D is a 2 −
(
3cn, 9, 3
cn−3c
6
)
design, whereas the
point-plane design AG2(cn, 3) has a larger λ = 3
cn−3
6 , whose difference with the
parameter λ of D is 3
c−3
6 , which increases exponentially with c. The smallest exam-
ple is the case n = c = 2: in this case, D is a 2 − (81, 9, 12) design, whereas the
point-plane design AG2(4, 3) is a 2− (81, 9, 13) design.
(ii) For m = 4 and q = 2c, c > 1,D is a 2−
(
2cn, 8, (2
cn−2c)(2cn−22c)
24
)
design, whereas
the point-flat design AG3(cn, 2) has a larger value of λ =
(2cn−2)(2cn−4)
24 .
On the contrary, for m = p = q > 3 the parameter λ for D becomes much larger than
that for the point-plane design AG2(n, p). For instance, for the smallest case m = p = q =
5, n = 4, D is a 2− (625, 25, 372000) design, whereas the point-plane design AG2(4, 5) is
a 2− (625, 25, 31) design. And the situation in the cases that do not have a corresponding
AGµ(n, q) to be compared with is not different: for q = 3, n = 5, and m = 6, D is a
2− (243, 18, λ) design, with λ = 1718496.
Remark 2.3. As the affine group Aff(P) has order |Aff(P)| = qn(qn − 1)(qn − q) · · ·
(qn − qn−1) and b = |Aff(P)||Sb0 | , we may conclude that the stabilizer Sb0 is a group of order
|Sb0 | =

(1 · 2 · 3 · · ·m)pm−1(p− 1)(qn − qn−1)(qn − qn−2)· · ·(qn − qm−1), if m > 4,
1344(qn − q3) · · · (qn − qn−1), if m = 4,
432(qn − qn−1)(qn − qn−2) · · · (qn − q2), if m = 3.
.
Remark 2.4. The design Dk = (GF(2n),Bk), considered in [13, Proposition 2.5], is a
3-design for any even k. Similarly, for p = 2, the 2-design D = (P,B) considered in
Theorem 2.1 is a 3-design if and only if q = 2. Indeed, let q = 2, and let {P1, P2, P3} and
{Q1, Q2, Q3} be two 3-subsets of P. Since the group of affinities of P acts 3-transitively
on P, there exists an (invertible) affinity ρ such that ρ(Pi) = Qi, i = 1, 2, 3. Moreover,
ρ(b(y1, y2, . . . , ym)) = b(ρ(y1), ρ(y2), . . . , ρ(ym)) for any subset {y1, y2, . . . , ym} of P
consisting of m affinely independent vectors, hence P1, P2, P3 belong to a block b if and
only if Q1, Q2, Q3 belong to the block ρ(b). Therefore D is a 3-design.
Now let q ̸= 2. If D = (P,B) were a 3-design, then the corresponding derived design
at the point 0 would be a 2-design. By definition, every block of the latter design is of the
form b(x1 = 0, x2, . . . , xm) \ {0} = {xj + s(x1 + x2 + · · · + xm)| 1 ≤ j ≤ m and
302 Ars Math. Contemp. 23 (2023) #P2.07 / 297–303
s ∈ GF(2)} \ {0}, where x2, . . . , xm are linearly independent vectors over GF(q), hence
one can prove that, for any nonzero x in P, and for any scalar c in GF(q) \GF(2), the two
vectors x and cx cannot lie in a common block. Therefore D is not a 3-design for p = 2
and q ̸= 2.
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.08 / 305–313
https://doi.org/10.26493/1855-3974.2568.55c
(Also available at http://amc-journal.eu)
On metric dimensions of hypercubes
Aleksander Kelenc *
University of Maribor, FERI, Koroška cesta 46, 2000 Maribor, Slovenia and
Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
Aoden Teo Masa Toshi
Independent researcher, Singapore
Riste Škrekovski †
University of Ljubljana, FMF, Jadranska 19, 1000 Ljubljana, Slovenia and
Faculty of Information Studies, Ljubljanska cesta 31a, 8000 Novo Mesto, Slovenia
Ismael G. Yero ‡
Universidad de Cádiz, Departamento de Matemáticas, Av. Ramón Puyol, s/n,
11202 Algeciras, Spain
Received 24 February 2021, accepted 25 July 2022, published online 2 December 2022
Abstract
In this note we show two unexpected results concerning the metric, the edge metric
and the mixed metric dimensions of hypercube graphs. First, we show that the metric and
the edge metric dimensions of Qd differ by at most one for every integer d. In particular,
if d is odd, then the metric and the edge metric dimensions of Qd are equal. Second, we
prove that the metric and the mixed metric dimensions of the hypercube Qd are equal for
every d ≥ 3. We conclude the paper by conjecturing that all these three types of metric
dimensions of Qd are equal when d is large enough.
Keywords: Edge metric dimension, mixed metric dimension, metric dimension, hypercubes.
Math. Subj. Class. (2020): 05C12, 05C76
*Corresponding author. Partially supported by the Slovenian Research Agency ARRS via grants J1-1693 and
J1-2452.
†Acknowledges the Slovenian research agency ARRS, program No. P1–0383 and project No. J1-3002.
‡Partially supported by the Spanish Ministry of Science and Innovation through the grant PID2019-105824GB-
I00.
E-mail addresses: aleksander.kelenc@um.si (Aleksander Kelenc), aodenteo@gmail.com (Aoden Teo Masa
Toshi), skrekovski@gmail.com (Riste Škrekovski), ismael.gonzalez@uca.es (Ismael G. Yero)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
306 Ars Math. Contemp. 23 (2023) #P2.08 / 305–313
1 Introduction
The metric dimension of connected graphs was introduced about 50 years ago in [6, 22],
in connection with modeling navigation systems in networks, although this notion was
already known by then for general metric spaces from [1]. Given a connected graph G
and two vertices u, v ∈ V (G), the distance dG(u, v) between these two vertices is the
length of a shortest path connecting v and u. The vertices u, v are distinguished or resolved
by a vertex x ∈ V (G) if dG(u, x) ̸= dG(v, x). A given set of vertices S is a metric
generator for the graph G, if every two vertices of G are distinguished by a vertex of S.
The cardinality of the smallest possible metric generator for G is the metric dimension of
G, which is denoted by dim(G). The terminology of metric generators was introduced in
[11], and the previous two works referred to such sets as resolving sets and locating sets,
respectively. We herewith follow the terminology of [11]. A metric generator for G of
cardinality dim(G) is called a metric basis. Although the classical metric dimension is an
old topic in graph theory, there are still several open problems that remain unsolved. Recent
investigations on this concern are [3, 4, 8, 16]. More results and open questions concerning
metric dimension and related variants can be found in the recent surveys [15] and [23].
In order to uniquely identify the edges of a graph, by using vertices, the edge metric
dimension of connected graphs was introduced in [10] as follows. Let G be a connected
graph and let uv be an edge of G such that u, v ∈ V (G). The distance between a vertex
x ∈ V (G) and the edge uv is defined as, dG(uv, x) = min{dG(u, x), dG(v, x)}. It is said
that two distinct edges e1, e2 ∈ E(G) are distinguished or resolved by a vertex v ∈ V (G)
if dG(e1, v) ̸= dG(e2, v). A set S ⊂ V (G) is called an edge metric generator for G if
and only if for every pair of edges e1, e2 ∈ E(G), there exists an element of S which
distinguishes the edges. The cardinality of a smallest possible edge metric generator of
a graph is known as the edge metric dimension, and is denoted by edim(G). After the
seminal paper [10], a significant number of researches on such parameter have appeared.
Among them, some of the most recent ones are [3, 12, 13, 14, 19]. See also the survey
[15] for some other contributions. It is natural to consider comparing the metric and edge
metric dimensions of graphs. However, as first proved in [10], and continued in [13, 14],
both parameters are not in general comparable since there exist connected graphs G for
which edim(G) < dim(G), edim(G) = dim(G) or edim(G) > dim(G).
In order to combine the unique identification of vertices and of edges in only one
scheme, the mixed metric dimension of graphs was introduced in [9]. For a connected
graph G, a vertex w ∈ V (G) and an edge uv ∈ E(G) are distinguished or resolved by
a vertex x ∈ V (G) if dG(w, x) ̸= dG(uv, x). A set S ⊂ V (G) is called a mixed metric
generator for G if and only if for every pair of elements of the graphs (vertices or edges)
e, f ∈ E(G)∪V (G), there exists a vertex of S which distinguishes them. The cardinality of
a smallest possible mixed metric generator of G is known as the mixed metric dimension of
G, and is denoted by mdim(G). Some recent studies on mixed metric dimension of graphs
are [20, 21]. Clearly, every mixed metric generator must be a metric generator as well
as an edge metric generator, and so, mdim(G) ≥ max{dim(G), edim(G)}, for any con-
nected graph G. Moreover, since dim(G) and edim(G) are in general not comparable (see
[13, 14] for more information on this fact), several situations relating these three parame-
ters can be found. That is, there are graphs G with mdim(G) ≫ max{dim(G), edim(G)},
mdim(G) = dim(G) ≫ edim(G), mdim(G) = edim(G) ≫ dim(G), or mdim(G) =
dim(G) = edim(G).
A. Kelenc et al.: On metric dimensions of hypercubes 307
The metric dimension of hypercube graphs has attracted the attention of several re-
searchers from long ago. For instance, the work of Lindström [17] is probably one of the
oldest ones, and for some recent ones we suggest the works [7, 18, 24]. Surprisingly, for
other related invariants there has been comparatively little research on hypercube graphs,
although one can find some interesting recent results on this topic such as those that ap-
peared in [5, 7]. It is our goal to present some results on the close connections that exist
among the metric, the edge metric and the mixed metric dimensions of hypercube graphs.
The d-dimensional hypercube, denoted by Qd, with d ∈ N, is a graph whose vertices
are represented by d-dimensional binary vectors, i.e., u = (u1, . . . , u2) ∈ V (Qd) where
ui ∈ {0, 1} for every i ∈ {1, . . . , d}. Two vertices are adjacent in Qd if their vectors differ
in exactly one coordinate. Hypercubes can be also seen as the d times Cartesian product
of the graph P2, that is, Qd ∼= P2□P2□ · · ·□P2, or recursively, Qd ∼= Qd−1□P2. The
distance between two vertices in Qd represents the total number of coordinates in which
their vectors differ. The hypercube Qd is bipartite, and has 2d vertices and d · 2d−1 edges.
We remark that Q2 is the cycle C4 and that Q4 can be also seen as the torus graphs C4□C4.
2 Results
Our first contribution is to relate the metric generators with the edge metric generators of
bipartite graphs.
Lemma 2.1. Let G be a connected bipartite graph. Then, every metric generator for G is
also an edge metric generator.
Proof. Let S be an arbitrary metric generator for G. We will show that S is an edge metric
generator as well.
Let e1 = x1y1 and e2 = x2y2 be two arbitrary distinct edges of G. Since G is bipartite
and e1, e2 are distinct, one can w.l.o.g. assume that x1, x2 (with x1 ̸= x2) belong to one
of the bipartition sets and y1, y2 to the other one. Hence the distance between u = x1 and
v = x2 is even.
Now, as u and v are distinct, there must be a vertex s ∈ S that distinguishes them,
i.e. d(s, u) ̸= d(s, v). We may assume that d(s, u) + 1 ≤ d(s, v). Since u and v are on
even distance, it follows that distances d(s, u) and d(s, v) are of same parity, otherwise we
encounter a closed walk of odd length in G, which is not possible in a bipartite graph. This
implies d(s, u) + 2 ≤ d(s, v), and now we easily derive
d(e1, s) ≤ d(u, s) < d(v, s)− 1 ≤ d(e2, s).
In particular, d(e1, s) < d(e2, s) implies that e1, e2 are distinguished by s ∈ S. Since
the choice of these two edges was arbitrary, we conclude that S is also an edge metric
generator.
It is then natural to think in the opposite direction with regard to the result above. In
particular, we ask if an edge metric generator for a bipartite graph is also a metric generator.
In contrast with the result above, achieving this seems to be a challenging task. However,
we have at least managed to show a weaker result for an infinite family of bipartite graphs,
namely the hypercubes Qd. That is, when d is odd, every edge metric generator for Qd is
indeed a metric generator, and when d is even, every edge metric generator is “almost” a
metric generator.
308 Ars Math. Contemp. 23 (2023) #P2.08 / 305–313
From now on we denote by αi the vector of dimension d whose ith-coordinate is 1,
and the remaining coordinates are 0. Also, by “⊕” we represent the standard (binary)
XOR operation. Notice that, for any vertex u ∈ V (Qd), u ⊕ αi means switching the
ith-coordinate of u from 0 to 1, or vice versa.
Lemma 2.2. Let S be an edge metric generator of Qd. If there exist two distinct vertices u
and v not distinguished by S, then they must be antipodal in Qd and d is even. If d is odd,
then S is also a metric generator of Qd.
Proof. Suppose that u = (u1, u2, . . . , ud) and v = (v1, v2, . . . , vd) are not antipodal.
Then, ui = vi for some i. Let Q0d−1 and Q
1
d−1 be the half-cubes regarding the dimension
i. Notice that u and v belongs to a same half-cube, say Q0d−1. Let eu and ev be the edges
corresponding to the component i (in Qd) incident with u and v, respectively. In other
words, as u⊕αi and v⊕αi are the neighbours of u and v in Q1d−1, we have eu = (u, u⊕αi)
and ev = (v, v⊕αi). We claim that the edges eu and ev are not distinguished by S. To see
this, observe that if s ∈ S belongs to Q0d−1, then
d(s, eu) = d(s, u) = d(s, v) = d(s, ev).
Also, if s ∈ S belongs to Q1d−1, then
d(s, eu) = d(s, u⊕ αi) = d(s, v ⊕ αi) = d(s, ev).
We hence derive that the edges eu and ev are not distinguished by S, which is a contradic-
tion.
Based on the above arguments we conclude that u and v are antipodal, i.e. d(u, v) =
d. Hence, every vertex x of S satisfies d(u, x) + d(x, v) = d. As every vertex s ∈ S
must be equally distanced from u and v, we conclude that d(u, s) = d(s, v) = d/2, and
consequently, d must be even. This establishes the main claim.
Finally, observe that if d is odd, then no vertex is equally distanced from two antipodal
vertices of Qd, and therefore, S is a metric generator of Qd.
Next lemma will ensure that enlarging an edge metric generator of Qd with one chosen
element, we get a metric generator of Qd.
Lemma 2.3. Let S be an edge metric generator of Qd and let s be an arbitrary element of
S. Then, S ∪ {s⊕ α1} is a metric generator of Qd.
Proof. If S is a metric generator of Qd, then S∪{s⊕α1} is so too, and we are done. Thus,
we assume that S is not a metric generator of Qd. Then, by Lemma 2.2, d is even and there
must exist antipodal vertices u and v such that d(u, x) = d(v, x) = d/2 for every x ∈ S.
This will not be the case for s ⊕ α1, as |d(u, s ⊕ α1) − d(v, s ⊕ α1)| = 2. Therefore, we
conclude that S ∪ {s⊕ α1} is a metric generator of Qd.
Since Qd is a bipartite graph, the two previous lemmas give us the following conse-
quence.
Theorem 2.4. Let d ≥ 1. Then
edim(Qd) ≤ dim(Qd) ≤ edim(Qd) + 1,
with the second inequality being tight only if d is even.
A. Kelenc et al.: On metric dimensions of hypercubes 309
Proof. The lower bound holds by Lemma 2.1. The upper bound and its possible tightness
(for more than one case) follows by Lemmas 2.2 and 2.3.
Notice that the upper bound dim(Qd) ≤ edim(Qd) + 1 is indeed tight for the case Q4,
since 4 = dim(Q4) = edim(Q4) + 1, as proved in [10].
We now turn our attention to relating the metric dimension with the mixed metric di-
mension of hypercubes. To this end, we will need the following two results. We must
remark that the first of next two lemmas already appeared in [18]. We include its proof for
completeness.
Lemma 2.5. If S is a metric generator (in particular, a metric basis) of Qd and s ∈ S,
then (S \{s})∪{s′} is also a metric generator (in particular, a metric basis) of Qd, where
s′ ∈ V (Qd) is the antipodal vertex of s.
Proof. If s ∈ S distinguishes some pair of vertices x and y of Qd, then s′ distinguishes
such pair as well, since d(x, s′) = d− d(x, s) and d(y, s′) = d− d(y, s). This also means
that no metric basis of Qd contains two antipodal vertices. Thus, if S is a metric generator
(or a metric basis) of Qd, then S \ {s} ∪ {s′} is a metric generator (or a metric basis) as
well.
Lemma 2.6. If S is a metric generator of Qd, then there is at most one index i ∈ {1, . . . , d}
such that all the vertices from S have the same value at the ith coordinate.
Proof. Suppose that there exist two different indices i and j such that all vertices from S
have the same value at the ith and jth coordinates. First, let us consider the case when
there are zeroes at such coordinates. Other cases can be shown by using similar arguments.
Now, let x ∈ V (Qd) be a vertex having zeroes at all coordinates, except at the ith, and let
y be a vertex having zeroes at all positions except at the jth. Then, d(x, s) = d(y, s) for
any vertex s ∈ S, a contradiction.
The mixed metric dimension of hypercubes Q1 and Q2 are 2 and 3, respectively. This
can be derived from results for paths and cycles from [9]. This gives us that dim(Qd) <
mdim(Qd), for d ∈ {1, 2}. For all higher dimensions the mixed metric dimension is equal
to the metric dimension as we next show.
Theorem 2.7. Let d ≥ 3. Then
dim(Qd) = mdim(Qd).
Proof. First, {(1, 1, 1), (0, 1, 0), (0, 0, 1)} and {(1, 1, 1, 1), (0, 1, 0, 0), (0, 0, 1, 0),
(0, 0, 0, 1)} are mixed metric bases for Q3 and Q4, respectively. Thus, the equality follows
for these cases since dim(Q3) = 3 and dim(Q4) = 4. It remains to check the equality for
d ≥ 5.
Let S be a metric basis for Qd with d ≥ 5. By Lemma 2.1, S is an edge metric
generator of Qd. In this sense, in Qd we only need to distinguish those pairs of elements,
one of them being a vertex and the other one, an edge. For this, let u be an arbitrary vertex
and let e = xy be an arbitrary edge of Qd.
Suppose first that u is not a vertex of e. As d(u, x) and d(u, y) are of different parity,
we may assume that u and x are on even distance. Now, let si be a vertex from S that
310 Ars Math. Contemp. 23 (2023) #P2.08 / 305–313
distinguishes u and x. Similarly, as in Lemma 2.1, notice that d(si, u) and d(si, x) are of
the same parity, and as they are different, we have that |d(si, u) − d(si, x)| ≥ 2. So, if
d(si, u) < d(si, x), then we derive
d(si, u) < d(si, u) + 1 ≤ d(si, x)− 1 ≤ d(si, e),
and if d(si, x) < d(si, u), then we have
d(si, e) ≤ d(si, x) < d(si, u).
Thus, in both cases e and u are distinguished by a vertex from S.
So all the pairs of elements (vertices and edges) considered in the upper part are distin-
guished by an arbitrary metric basis. To conclude the proof, we need to construct a metric
basis of cardinality |S| that will also distinguish incident vertices and edges.
Suppose now that u is an endpoint of e, say u = x. To distinguish u and e there needs
to be a vertex s ∈ S which is from the half-cube Qd−1 that contains vertex y and does not
contain vertex x. To distinguish all such pairs there must be at least one vertex from the
mixed metric generator in every half-cube Qd−1. For any index i ∈ {1, . . . , d}, there exists
a vertex from a mixed metric generator having 0 on the ith coordinate, and a vertex from
a mixed metric generator having 1 on the ith coordinate. In other words, a mixed metric
basis does not have a column of zeroes or a column of ones at an arbitrary index i (if we
arrange all vectors of the mixed metric basis as a matrix with such vectors as the rows of
such matrix).
We have started from an arbitrary metric basis S. Since Qd is a vertex transitive graph,
we may assume that the vertex s1 = (0, 0, . . . , 0) (all coordinates equal to 0) is in S. If
S does not contain a column of zeroes, then S is also a mixed metric basis. Otherwise,
by Lemma 2.6, there exists only one such column, say at index i0. By Lemma 2.5, we
know that we can replace any of the vertices from the set S with its antipodal vertex and
the incurred set S′ = S \ {s} ∪ {s′} is a metric basis too, since the column at index i0 (all
zeroes) ensures that no two vertices in S are antipodal to each other. Moreover, in view of
Lemma 2.1, S is an edge metric generator as well.
There exist at least four different vertices s1 = (0, 0, . . . , 0), s2, s3 and s4 in the set S,
since dim(Qd) ≥ 4, for d ≥ 5. We construct four sets S′i in the following way:
S′1 = (S \ {s1}) ∪ {s′1}, S′2 = (S \ {s2, s3}) ∪ {s′2, s′3},
S′3 = (S \ {s2}) ∪ {s′2}, S′4 = (S \ {s1, s3}) ∪ {s′1, s′3},
and consider the next situations:
(I): If S′1 is not a mixed metric generator, then there is a column of ones in S′1 at some
index i1.
(II): If S′2 is not a mixed metric generator, then there is a column of zeroes in S′2 at some
index i2.
(III): If S′3 is not a mixed metric generator, then there is a column of zeroes in S′3 at some
index i3.
(IV): If S′4 is not a mixed metric generator, then there is a column of ones in S′4 at some
index i4.
Observe that all these indices i0, i1, i2, i3, and i4 are different. If none of the four sets
S′i defined above is a mixed metric generator, then the initial set S looks as follows.
A. Kelenc et al.: On metric dimensions of hypercubes 311
i0 i1 i2 i3 i4 . . .
s1 : 0 0 0 0 0 . . .
s2 : 0 1 1 1 1 . . .
s3 : 0 1 1 0 0 . . .
s4 : 0 1 0 0 1 . . .
...
...
...
...
...
...
s|S| : 0 1 0 0 1 . . .
We now take a look at the columns i1, i2, i3 and i4. Let v1 be a vertex having zeroes at
all positions except at i1 and i3 and let v2 be a vertex having zeroes at all positions except
at i2 and i4. Then, d(v1, s) = d(v2, s), for any vertex s ∈ S, a contradiction. Therefore,
at least one of the sets S′i has to be a mixed metric generator, and therefore, the equality
mdim(Qd) = dim(Qd) follows since any mixed metric basis is also a metric basis.
In view of the asymptotical result for the metric dimension of hypercubes from [2],
Theorems 2.4 and 2.7 give us the following consequences.
Corollary 2.8. Let d ≥ 3. Then
dim(Qd)− 1 ≤ edim(Qd) ≤ dim(Qd) = mdim(Qd).
Corollary 2.9. Let d ≥ 2. Then
mdim(Qd) ∼ edim(Qd) ∼ dim(Qd) ∼
2d
log2 d
.
We conclude this short paper with the following conjecture.
Conjecture 2.10. If d is large enough, then
edim(Qd) = dim(Qd).
As the above conjecture does not hold for d = 4, d must be at least 5.
ORCID iDs
Aleksander Kelenc https://orcid.org/0000-0003-1633-9845
Riste Škrekovski https://orcid.org/0000-0001-6851-3214
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 23 (2023) #P2.09 / 315–334
https://doi.org/10.26493/1855-3974.2706.3c8
(Also available at http://amc-journal.eu)
Complete forcing numbers of graphs*
Xin He , Heping Zhang †
School of Mathematics and Statistics, Lanzhou University, Lanzhou,
Gansu 730000, P.R. China
Received 10 October 2021, accepted 30 June 2022, published online 13 December 2022
Abstract
The complete forcing number of a graph G with a perfect matching is the minimum
cardinality of an edge set of G on which the restriction of each perfect matching M is a
forcing set of M . This concept can be view as a strengthening of the concept of global
forcing number of G. Došlić in 2007 obtained that the global forcing number of a con-
nected graph is at most its cyclomatic number. Motivated from this result, we obtain that
the complete forcing number of a graph is no more than 2 times its cyclomatic number
and characterize the matching covered graphs whose complete forcing numbers attain this
upper bound and minus one, respectively. Besides, we present a method of constructing a
complete forcing set of a graph. By using such method, we give closed formulas for the
complete forcing numbers of wheels and cylinders.
Keywords: Perfect matching, global forcing number, complete forcing number, cyclomatic number,
wheel, cylinder.
Math. Subj. Class. (2020): 05C70, 05C90, 92E10
1 Introduction
Let G be a graph with vertex set V (G) and edge set E(G). A matching of G is a set of
disjoint edges of G. A perfect matching M of G is a matching that covers all vertices of
G. An edge of G is termed allowed if it lies in some perfect matching of G and forbidden
otherwise. A forcing set of M is a subset of M contained in no other perfect matching of
G. The forcing number of M is the minimum possible cardinality of forcing sets of M . We
may refer to a survey [6] on this topic. A subset S ⊆ E(G) \M is called an anti-forcing
set of M [14] if G − S has a unique perfect matching M . The anti-forcing number of M
is the smallest cardinality of anti-forcing sets of M .
*The authors are grateful to anonymous reviewers for their careful reading and valuable suggestions to improve
this manuscript. This work is supported by National Natural Science Foundation of China (Grant No. 11871256
and 12271229).
†Corresponding author.
E-mail addresses: hex2015@lzu.edu.cn (Xin He), zhanghp@lzu.edu.cn (Heping Zhang)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
316 Ars Math. Contemp. 23 (2023) #P2.09 / 315–334
Let G be a graph with a perfect matching. Concerning all perfect matchings of G,
Vukičević et al. [21, 22] introduced the concept of global (or total) forcing set, which is
defined as a subset of E(G) on which there are no two distinct perfect matchings coincid-
ing. The minimum possible cardinality of global forcing sets is called the global forcing
number of G. For more about the global forcing number of a graph, the reader is referred
to [3, 7, 20, 27].
As a strengthening of the concept of global forcing set of G, Xu et al. [24] proposed the
concept of the complete forcing set of G, which is defined as a subset of E(G) on which the
restriction of each perfect matching M is a forcing set of M . A complete forcing set with
the minimum cardinality is called a minimum complete forcing set of G, and its cardinality
is called the complete forcing number of G, denoted by cf(G). If G has at least two perfect
matchings, then cf(G) is larger than the global forcing number of G [24].
A subgraph G0 of G is said to be nice if G−V (G0) has a perfect matching. Obviously,
an even cycle C of G is nice if and only if there is a perfect matching M of G such that
E(C) ∩M is a perfect matching of C. We call each of the two perfect matchings of C a
frame (or a typeset [24]) of C, which was ever used in [1] to obtain a min-max theorem for
the Clar problem on 2-connected plane bipartite graphs.
Xu et al. established the following equivalent condition for a subset of edges of a graph
to be a complete forcing set.
Theorem 1.1 ([24]). Let G be a graph with a perfect matching. Then S ⊆ E(G) is a
complete forcing set of G if and only if for any nice cycle C of G, the intersection of S and
each frame of C is nonempty.
Let S be a complete forcing set of G. For a perfect matching M of G, from Theo-
rem 1.1, S\M contains at least one edge of every M -alternating cycle of G. By Lemma 2.1
of [14], S \M is an anti-forcing set of M . So a complete forcing set of G both forces and
antiforces each perfect matching. Further, Chan et al. [4] obtained that the complete forc-
ing number of a catacondensed hexagonal system is equal to the number of hexagons plus
the Clar number and presented a linear-time algorithm for computing it. Besides, some
certain explicit formulas for the complete forcing numbers of rectangular polynominoes,
polyphenyl systems, spiro hexagonal systems and primitive coronoids have been derived
[5, 15, 16, 23]. In recent papers [11, 12], we established a sufficient condition for an edge
set of a hexagonal system (HS) to be a complete forcing set in terms of elementary edge-cut
cover, which yields a tight upper bound on the complete forcing numbers of HSs. For a
normal HS, we gave two lower bounds on its complete forcing number by the number of
hexagons and matching numbers of some subgraphs of its inner dual graph, respectively.
In addition, we showed that the complete forcing numbers of catacondensed HSs, normal
HSs without 2× 3 subsystems, parallelogram, regular hexagon- and rectangle-shaped HSs
attain one of the two above lower bounds.
Let c(G) = |E(G)|− |V (G)|+ω(G) denote the cyclomatic number of G, where ω(G)
is the number of components of G. In 2007, Došlić [7] obtained that the global forcing
number of a connected graph is at most its cyclomatic number and gave a characterization:
a connected (bipartite) graph has the global forcing number attaining its cyclomatic number
if and only if each cycle is nice (such graphs are called 1-cycle resonant graphs; see [9]).
As a corollary, the global forcing number of any catacondensed HS is equal to the number
of hexagons.
Motivated by Došlić’s result, in this paper we obtain that the complete forcing number
X. He and H. Zhang: Complete forcing numbers of graphs 317
of a graph is no more than 2 times its cyclomatic number by presenting a method of con-
structing a complete forcing set of a graph (see the next section). Moreover, in Section 3,
we show that the complete forcing number of a matching covered graph attains the above
upper bound if and only if such graph is either K2 (a complete graph with 2 vertices) or an
even cycle. Besides, we characterize the matching covered graphs whose complete forcing
numbers are equal to 2 times their cyclomatic numbers minus 1 in terms of ear decompo-
sition. In the last section, we present some lower bounds on the complete forcing numbers
of some types of graphs including plane elementary bipartite graphs and cylinders. Com-
bining such methods, we give closed formulas for the complete forcing numbers of wheels
and cylinders.
2 An upper bound on complete forcing number
All graphs considered in this paper are simple and all the bipartite graphs are given a proper
black and white coloring: any two adjacent vertices receive different colors.
Let G be a graph. Suppose that V ′ is a nonempty subset of V (G). The subgraph of
G whose vertex set is V ′ and whose edge set is the set of those edges of G that have both
end-vertices in V ′ is called the subgraph of G induced by V ′ and is denoted by G[V ′]. The
induced subgraph G[V \V ′] is denoted by G−V ′. For E′ ⊆ E(G), the spanning subgraph
(V (G), E(G) \ E′) is denoted by G− E′ [2]. For a nonempty proper subset V ′ of V (G),
the set of all edges of G having exactly one end-vertex in V ′ is called an edge cut of G and
denoted by ∂G(V ′) (or simply ∂(V ′)).
For v ∈ V (G) and e ∈ E(G), for simplicity we use G− v, G− e and ∂(v) to represent
G − {v}, G − {e} and ∂({v}) respectively. Further the cardinality of ∂G(v) is called the
degree of v in G and is denoted by dG(v) (or simply d(v)).
In this section, we will present a method of constructing a complete forcing set of G
in terms of elementary edge cut, which was introduced in [19, 26] to show the existence
of perfect matchings in HS and plays an important role in resonance theory of graphs
[13, 25, 28] especially in the computation of Clar number of HSs [10]. Elementary edge
cut was previously defined in bipartite graphs, we extend this concept to general graphs as
follows.
We call an edge cut D of G an elementary edge cut (e-cut for short) if it satisfies the
following three conditions:
(1) ω(G − D) = ω(G) + 1, that is, there are exactly two components O1 and O2 of
G−D that are different from of all components of G.
(2) At least one of O1 and O2 is a bipartite graph.
(3) All edges of D are incident with the vertices of the same color in one bipartite com-
ponent of O1 and O2 (for example, the bold edges of G1 in Figure 10 form an e-cut
of G1).
A bridge of G is an edge cut of G consisting of exactly one edge. A cut-vertex of G is
a vertex whose deletion increases the number of components. A block of G is a maximal
connected subgraph of G that has no cut-vertices. Each block with at least 3 vertices is
2-connected. The blocks of a loopless graph are its isolated vertices, bridges, and maximal
2-connected subgraph. A block of G that contains exactly one cut-vertex of G is called an
end-block of G.
318 Ars Math. Contemp. 23 (2023) #P2.09 / 315–334
Let D be an e-cut of a graph G with at least two edges. Then we define an e-cut deletion
operation (simply ED operation) of G in the following steps:
(1) Delete D from G,
(2) Delete the set B consisting of all bridges of G−D, and
(3) Delete the isolated vertices of G−D −B.
Let G′ be the subgraph obtained from G by an ED operation. Then G′ has neither
isolated vertices nor bridges. If G′ is not empty, then each block of G′ is 2-connected. Let
v′ be a non-cut-vertex of a block of G′ with at most one cut-vertex of G′. Then ω(G′ −
∂G′(v
′)) = ω(G′) + 1 and v′ is a component of G′ − ∂G′(v′), so ∂G′(v′) is an e-cut of G′
with at least 2 edges and we can do an ED operation on G′. If we can do l ED operations
from G and obtain the following subgraph sequence G = G1 ⊃ G2 ⊃ · · · ⊃ Gl+1,
where Gi is not empty graph and Gi+1 is obtained by doing an ED operation from Gi for
i = 1, 2, . . . , l, then we call this procedure an e-cut decomposition from G1 to Gl+1. Let
Di be the deleted e-cut from Gi. Then
c(Gi −Di) = |E(Gi)| − |Di| − |V (Gi)|+ ω(Gi) + 1 = c(Gi)− (|Di| − 1).
Let Bi be the set of bridges deleted from Gi −Di. Then we have
c((Gi −Di)−Bi) = c(Gi −Di) = c(Gi)− (|Di| − 1).
Since deleting the isolated vertices from Gi − Di − Bi keeps its cyclomatic number un-
changed,
c(Gi+1) = c((Gi −Di)−Bi) = c(Gi)− (|Di| − 1).
So, we have
c(Gi)− c(Gi+1) = |Di| − 1. (2.1)
Since |Di| ≥ 2, Equation (2.1) implies that an ED operation on Gi decrease the cyclomatic
number by at least 1.
From the above discussion, we have the following result.
Lemma 2.1. If a graph G has an e-cut with at least two edges, then there exists an e-cut
decomposition from G to empty graph.
Lemma 2.2. Let G be a graph without isolated vertices or bridges. If H is a 2-connected
induced subgraph of G, then there exists an e-cut decomposition from G to H .
Proof. If G is not 2-connected, then there is a block B of G such that H is not an induced
subgraph of B and B contains at most one cut-vertex of G. Since G has neither isolated
vertices nor bridges, B is 2-connected. Let v1 be a vertex of B that is not a cut-vertex of
G. Then ∂G(v1) is an e-cut of G with at least two edges. We can use ∂G(v1) to do an ED
operation on G. If G is 2-connected and G ̸= H , let v2 be a vertex of V (G) \ V (H). Then
∂G(v2) is an e-cut of G with at least two edges. We can use ∂G(v2) to do an ED operation
on G. In either of the above two cases, we can see that H is still an induced subgraph of
the resulting graph. Clearly, we can do ED operations repeatedly like the above until the
resulting subgraph is H .
X. He and H. Zhang: Complete forcing numbers of graphs 319
Lemma 2.3. Let G be a graph that admits a perfect matching and F be the set of all
forbidden edges of G. If there exists an e-cut decomposition from G − F = G1 to Gl+1
(l ≥ 1) such that Gl+1 is empty graph or each cycle of Gl+1 is not a nice cycle of G, then
D1 ∪D2 ∪ · · · ∪Dl is a complete forcing set of G, where Di (i = 1, 2, . . . , l) is the e-cut
deleted from Gi in the e-cut decomposition. Further, cf(G) ≤ c(G) + l − c(Gl+1).
Proof. For i = 1, 2, . . . , l, let Bi be the set of bridges deleted from Gi − Di in the e-cut
decomposition. Then E(G) can be partitioned into F ∪D1 ∪B1 ∪D2 ∪B2 ∪ · · · ∪Dl ∪
Bl ∪ E(Gl+1). Since every edge of a nice cycle C is allowed, E(C) ∩ F = ∅.
Claim. For i = 1, 2, . . . , l, if there is a nice cycle C of G that has an edge in Di ∪Bi, then
each frame of C intersects D1 ∪D2 ∪ · · · ∪Di.
Proof. We shall proceed by induction on i. For i = 1, if E(C)∩B1 ̸= ∅, then E(C)∩D1 ̸=
∅. Choose an edge e1 in E(C) ∩ D1. Since D1 is an e-cut of G1, there is a bipartite
component O1 of G1 − D1 such that all edges of D1 are incident with the same colored
vertices of O1 (say black). After C passes through e1 to black end-vertex in O1, C must
pass through another edge of D1 from black end-vertex in O1. Let e2 be the first such edge.
Then the path of C1 between black vertices of e1 and e2 in O1 has even length. This yields
that edges e1 and e2 in D1 belong to different frames of C and the claim holds for i = 1.
Suppose that the claim holds for i ≤ l − 1. We shall prove it for i + 1. If C has an edge
in E(G) \E(Gi+1), that is, C has some edge in D1 ∪B1 ∪D2 ∪B2 ∪ · · · ∪Di ∪Bi, by
the induction hypothesis, the intersection of each frame of C and D1 ∪ D2 ∪ · · · ∪ Di is
nonempty. So we may assume that E(C) ⊆ E(Gi+1). Since C has an edge in Di+1∪Bi+1,
similarly we have that each frame of C must have an edge in Di+1. Consequently, the claim
holds.
Since Gl+1 is empty graph or every cycle of Gl+1 is not a nice cycle of G, every nice
cycle of G contains an edge in D1∪B1∪D2∪B2∪· · ·∪Dl∪Bl. By the claim, each frame
of each nice cycle of G intersects D1∪D2∪· · ·∪Dl. So, by Theorem 1.1, D1∪D2∪· · ·∪Dl
is a complete forcing set of G.
From Equation (2.1), we have
c(G1)− c(Gl+1) =
l∑
i=1
(|Di| − 1), (2.2)
and then
l∑
i=1
|Di| = c(G1) + l − c(Gl+1).
Since G1 = G− F , c(G1) ≤ c(G) and
cf(G) ≤ |D1 ∪D2 ∪ · · · ∪Dl| = c(G1) + l − c(Gl+1) ≤ c(G) + l − c(Gl+1).
From Lemma 2.3, we have the following upper bound on the complete forcing number.
Theorem 2.4. Let G be a graph that admits a perfect matching. Then cf(G) ≤ 2c(G).
Proof. If G has a unique perfect matching, then cf(G) = 0, and the conclusion holds. If
G has at least two perfect matchings, let F be the set of all forbidden edges of G. Then
320 Ars Math. Contemp. 23 (2023) #P2.09 / 315–334
each K2 block of G − F is a component, so there is a 2-connected block B of G − F
with at most one cut-vertex of G − F . Let v be a vertex of B that is not a cut-vertex of
G− F . Then ∂G−F (v) is an e-cut of G− F with at least two edges. By Lemma 2.1 there
exists an e-cut decomposition from G − F to empty graph: G1 ⊃ G2 ⊃ · · · ⊃ Gl+1,
where G1 = G − F and Gl+1 = ∅. For i = 1, 2, . . . , l, let Di be the e-cut deleted from
Gi in this e-cut decomposition. From Equation (2.2), since |Di| ≥ 2 and c(G1) ≤ c(G),
we have l ≤
∑l
i=1(|Di| − 1) = c(G1) ≤ c(G). Combining with Lemma 2.3, we have
cf(G) ≤ c(G) + l − c(Gl+1) ≤ 2c(G).
3 Some extremal matching covered graphs
A connected graph G is said to be matching covered if it has at least two vertices and
each edge is allowed. Every matching covered graph with at least four vertices is 2-
connected [18].
In this section, we will characterize the matching covered graphs whose complete forc-
ing numbers attain the upper bound given in Theorem 2.4 and minus one, respectively.
Theorem 3.1. Let G be a matching covered graph. Then cf(G) = 2c(G) if and only if G
is either K2 or an even cycle.
Proof. The sufficiency is obvious. So we consider the necessity. If c(G) = 0, then G is
a tree. Since G is matching covered, G can only be K2. For c(G) ≥ 1, suppose to the
contrary that G is not an even cycle. Then G has a vertex v with degree at least 3. Let
D1 = ∂(v). Then since G is 2-connected, G − D1 has exactly two components and D1
is an e-cut with |D1| ≥ 3. We use D1 to do an ED operation on G1 = G and obtain
G2, and then we do ED operations from G2 repeatedly until the empty graph is obtained.
Consequently, we obtain an e-cut decomposition G = G1 ⊃ G2 ⊃ · · · ⊃ Gl+1 = ∅
(l ≥ 1). For i = 2, 3, . . . , l, let Di be the e-cut deleted from Gi in this e-cut decomposition.
By Equation (2.1), c(G2)−c(G1) = |D1|−1 ≥ 2 and c(Gi+1)−c(Gi) = |Di|−1 ≥ 1 (i =
2, 3, . . . , l). Combining with Equation (2.2), we have l + 1 ≤
∑l
i=1(|Di| − 1) = c(G)−
c(Gl+1), and thus l ≤ c(G)− 1. By Lemma 2.3, c(G) ≤ 2c(G)− 1, a contradiction.
Corollary 3.2. Let G be a graph with a perfect matching. Then cf(G) = 2c(G) if and
only if
(i) each forbidden edge of G is a bridge, and
(ii) each component of the graph obtained by deleting all forbidden edges from G is
either K2 or an even cycle.
Proof. Let G0 be the graph obtained from G by deleting all forbidden edges. Since each
forbidden edge of G does not appear in any minimum complete forcing set of G, cf(G) =
cf(G0). Let O1, O2, . . . , Ot (t ≥ 1) be the components of G0. By Theorem 2.4 we have
cf(G) = cf(G0) =
t∑
i=1
cf(Oi) ≤ 2
t∑
i=1
c(Oi) = 2c(G0) ≤ 2c(G).
In the above expression, Theorem 3.1 implies that the third equality holds if and only if
each Oi is either K2 or an even cycle, and the fifth equality holds if and only if G0 and G
have the same cyclomatic number, that is, each forbidden edge of G is a bridge.
X. He and H. Zhang: Complete forcing numbers of graphs 321
A connected graph G is said to be elementary if all its allowed edges form a connected
subgraph of G. A connected bipartite graph is elementary if and only if each edge is al-
lowed [17]. An elementary bipartite graph has the so-called “bipartite ear decomposition”.
Let x be an edge. Join the end vertices of x by a path P1 of odd length (the so-called
“first ear”). We proceed inductively to build a sequence of bipartite graphs as follows: If
Gr−1 = x + P1 + P2 + · · · + Pr−1 has already been constructed, add the r-th ear Pr (a
path of odd length) by joining any two vertices in different colors of Gr−1 such that Pr has
no other vertices in common with Gr−1. The decomposition Gr = x+P1+P2+ · · ·+Pr
will be called an (bipartite) ear decomposition of Gr. It is known that a bipartite graph G
is elementary if and only if G has a bipartite ear decomposition [17]. We can see that the
number r of ears is equal to the cyclomatic number of G.
2P
r
b
2b
3b
3w
r
w
2w
( )a ( )b
3P
r
P
1x P+
'
1P
'
2P
'
3P
a
b
Figure 1: Two examples for graphs G with cf(G) = 2c(G)− 1.
Theorem 3.3. Let G be a matching covered graph. Then cf(G) = 2c(G) − 1 if and only
if G is a bipartite graph and one of the following holds :
(i) c(G) = 2 (see Figure 1(a));
(ii) G has an ear decomposition G = x + P1 + P2 + · · · + Pr (r ≥ 3) such that one
frame of x+P1 contains at least r−1 edges w2b2, w3b3, . . . , wrbr and the two ends
of P2, P3, . . . , Pr are the two end-vertices of w2b2, w3b3, . . . , wrbr, respectively (see
Figure 1(b)).
Proof. Sufficiency. (i) If c(G) = 2, by the ear decomposition of G, G contains two 3-
degree vertices, denoted by a and b. Let P ′1, P
′
2, P
′
3 be the 3 internally disjoint paths from
a to b (see Figure 1(a)) and S be a complete forcing set of G. If |S| ≤ 2, then one of
P ′1, P
′
2 and P
′
3 has no edges in S, say P
′
1. We can see that one of the two nice cycles
P ′1 ∪ P ′2 and P ′1 ∪ P ′3 has a frame containing no edges of S, which contradicts that S is a
complete forcing set by Theorem 1.1. So cf(G) ≥ 3. Conversely, we can see that ∂G(a)
is a complete forcing set of G, which means that cf(G) ≤ 3. Consequently, we have
cf(G) = 3 = 2c(G)− 1.
322 Ars Math. Contemp. 23 (2023) #P2.09 / 315–334
(ii) For 2 ≤ i ≤ r, let Ci = Pi∪{wibi}. Then we can see that C2, C3, . . . , Cr are r−1
vertex-disjoint nice cycles of G. Let S be a complete forcing set of G. By Theorem 1.1,
each frame of Ci (2 ≤ i ≤ r) has at least one edge of S, so each Ci contains 2 edges of S.
Further the frame of the nice cycle x+P1 that does not contain {w2b2, w3b3, . . . , wrbr} has
an edge in S. So |S| ≥ 2r − 1. Conversely, let D1 = ∂G(b2) and Di (i = 2, 3, . . . , r − 1)
be any two adjacent edges of Ci+1. We use D1, D2, . . . , Dr−1 to do ED operations from
G in turn and obtain empty graph finally. By Lemma 2.3, D1 ∪ D2 ∪ · · · ∪ Dr−1 is a
complete forcing set of G and cf(G) ≤ |D1 ∪D2 ∪ · · · ∪Dr−1| = 3+2(r− 2) = 2r− 1.
Consequently, we have cf(G) = 2r − 1 = 2c(G)− 1.
Necessity. If c(G) = 0 or 1, then G is K2 or an even cycle. By Theorem 3.1, cf(G) =
2c(G), contradicting cf(G) = 2c(G) − 1. So c(G) ≥ 2 and |V (G)| ≥ 4. Since G is
matching covered, G is 2-connected.
Claim 1. For an e-cut decomposition from G = G1 to Gl+1 = ∅, if there is an integer k
(1 ≤ k ≤ l) such that |Dk| ≥ 4 or there are two integers m and n (1 ≤ m < n ≤ l) such
that |Dm| ≥ 3 and |Dn| ≥ 3, then cf(G) ≤ 2c(G)− 2.
Proof. If |Dk| ≥ 4 (1 ≤ k ≤ l), then since |Di| ≥ 2 for i = 1, 2, . . . , k − 1, k + 1, . . . , l,∑l
i=1(|Di| − 1) ≥ l + 2. From Equation (2.2), we have l ≤ c(G)− 2, and thus cf(G) ≤
2c(G)− 2 by Lemma 2.3.
If |Dm| ≥ 3 and |Dn| ≥ 3 (1 ≤ m < n ≤ l). Then since |Di| ≥ 2 (i = 1, 2, . . . ,
m− 1,m+1, . . . , n− 1, n+1, . . . , l),
∑l
i=1(|Di| − 1) ≥ l+2. From Equation (2.2), we
have l ≤ c(G)− 2, and thus cf(G) ≤ 2c(G)− 2 by Lemma 2.3.
If G has a vertex v0 with degree at least 4, let D1 = ∂(v0). Since G is 2-connected,
G − ∂(v0) has exactly two components and D1 is an e-cut of G. Then we can give an
e-cut decomposition from G to empty graph by taking D1 as the first e-cut. By Claim 1,
cf(G) ≤ 2c(G) − 2, a contradiction. Thus dG(v) ≤ 3 for each vertex v ∈ V (G). In
addition, since c(G) ≥ 2, G has a 3-degree vertex v1. Since G is 2-connected, G − v1 is
connected.
Claim 2. G is a bipartite graph.
Proof. Suppose to the contrary that G is not a bipartite graph. Let v be a vertex of G.
Then G − v is not a bipartite graph as well. Otherwise, G − v has a bipartition (W,B)
(|W | < |B|). If v is adjacent to a vertex w of W in G, then vw is a forbidden edge of G,
which contradicts that G is matching covered. So v can only be adjacent to vertices of B
in G, and thus G is a bipartite graph, a contradiction to the supposition. Hence, G− v1 has
an odd cycle C1.
Let D1 = ∂(v1). Since G is 2-connected, G−D1 has exactly two components and D1
is an e-cut of G with |D1| = 3. We obtain G2 by doing an ED operation on G1 = G via D1.
Since G[V (C1)] is 2-connected and G[V (C1)] is still a subgraph of G2, from Lemma 2.2,
there exists an e-cut decomposition from G2 to G[V (C1)] = Gm. For i = 2, 3, . . . ,
m − 1, we denote by Di the deleted e-cut from Gi in this e-cut decomposition. If Gm
has a 3-degree vertex vm, let Dm = ∂G[V (C1)](vm). We can give an e-cut decomposition
from Gm to empty graph by taking Dm as the first e-cut. Combining the above two e-cut
decompositions, we have an e-cut decomposition from G1 to empty graph with |D1| =
|Dm| = 3. By Claim 1, cf(G) ≤ 2c(G) − 2, a contradiction. If Gm is an odd cycle,
by Lemma 2.3, D1 ∪ D2 ∪ · · · ∪ Dm−1 is a complete forcing set of G and cf(G) ≤
X. He and H. Zhang: Complete forcing numbers of graphs 323
c(G) + (m − 1) − c(Gm). Since |D1| = 3 and |Di| ≥ 2 (i = 2, 3, . . . ,m − 1), c(G) −
c(Gm) =
∑m−1
i=1 (|Di| − 1) ≥ m, so we have m ≤ c(G) − c(Gm) = c(G) − 1. Hence,
cf(G) ≤ c(G) + (m− 1)− c(Gm) ≤ 2c(G)− 3, a contradiction.
Claim 3. Each block of G− v1 is either K2 or an even cycle.
Proof. Let B be a block of G − v1. Then dB(v) ≤ 2 for each v ∈ V (B). Otherwise, let
v′ be a vertex of B with dB(v′) = 3. By Lemma 2.2, there exists an e-cut decomposition
from G = G1 to B = Gm by taking D1 = ∂G(v1) as the first e-cut. Let Dm = ∂Gm(v
′).
Then Dm is an e-cut of B and we can give an ED decomposition from B to empty graph
by taking Dm as the first e-cut. Combining with the above two e-cut decompositions, we
have an e-cut decomposition from G to empty graph with |D1| = |Dm| = 3. By Claim 1,
cf(G) ≤ 2c(G) − 2, a contradiction. Since G is a bipartite graph by Claim 2, each block
of G− v1 is K2 or an even cycle.
In the following we may assume that v1 is a black vertex of G.
Claim 4. If each block of G− v1 is K2, then c(G) = 2 and (i) holds.
Proof. Obviously G− v1 is a tree. If G− v1 has no 3-degree vertices, then it is a path P .
Since G is 2-connected, the end-vertices of P are adjacent to v1 and receive white. Further,
since dG(v1) = 3, v1 has third white neighbor as an internal vertex of P (see Figure 2(a)).
So c(G) = 2.
1w 1w
1v
1v
1v
( )a ( )b ( )c
P
Figure 2: Illustration for Claim 4.
If G − v1 has a 3-degree vertex, then G − v1 has only one 3-degree vertex, denoted
by w1. Otherwise, G − v1 has at least four 1-degree vertices, but just three of them is
adjacent to v1 in G, so G has a 1-degree vertex, a contradiction. Thus G − v1 has three
1-degree vertices which are adjacent to v1 in G. It follows that w1 is a white vertex (see
Figure 2(c)); Otherwise, G has an odd number of vertices (see Figure 2(b)), a contradiction.
So c(G) = 2.
In what follows we suppose that G− v1 has a block that is an even cycle.
Claim 5. Each even cycle block of G− v1 has at most two 3-degree vertices in G.
324 Ars Math. Contemp. 23 (2023) #P2.09 / 315–334
C
1
v
3
w
2
w
P
Figure 3: An even cycle block C of G− v1 has exactly three 3-degree vertices of G.
Proof. If G− v1 has an even cycle block that has four 3-degree vertices in G, then it has at
least one end-block that has no vertices that are adjacent to v1 in G. This causes G to have
a cut-vertex, which contradicts that G is 2-connected. If there is an even cycle block C of
G−v1 that has exactly three 3-degree vertices of G, then two of such three vertices w2 and
w3 have the same color in G. Let P be a path contained in C with ends w2 and w3 (see
Figure 3). Then each internal vertex of P is still a 2-degree vertex of G. Further, since G
has an e-cut ∂(V (P )) of four edges, we can give an e-cut decomposition from G to empty
graph by taking ∂(V (P )) as the first e-cut. By Claim 1, we have cf(G) ≤ 2c(G) − 2, a
contradiction.
Claim 6. G− v1 is not 2-connected, and has no vertices contained in three K2 blocks.
P T
1v 1v
4w 4w5w 5w
( )a ( )b
Figure 4: (a) w4 and w5 have the same color; (b) w4 and w5 have different colors.
Proof. If G − v1 is 2-connected, then by Claim 3, G − v1 is an even cycle and v1 is
adjacent to three vertices of this cycle in G, which contradicts Claim 5. So, G − v1 is not
2-connected.
Suppose to the contrary that G − v1 has a vertex w4 incident with three K2 blocks.
Then G−v1 has at least 3 end-blocks. Since G is 2-connected and dG(v1) = 3, G−v1 has
exactly three end-blocks. Let P be a shortest path between w4 and a 3-degree vertex w5 of
G − v1 in an even cycle block so that each internal vertex of P is a 2-degree vertex in G.
If w4 and w5 have the same color, then ∂(V (P )) is an e-cut of G (see Figure 4(a)). There
exists an e-cut decomposition from G to empty graph by taking ∂(V (P )) as the first e-cut.
By Claim 1, we have cf(G) ≤ 2c(G) − 2, a contradiction. If w4 and w5 have different
colors, let T be the tree consisting of P and the remaining two K2 blocks of G − v1 that
X. He and H. Zhang: Complete forcing numbers of graphs 325
has an end-vertex w4. Then ∂(V (T )) is an e-cut of G (see Figure 4(b)). Similarly we have
cf(G) ≤ 2c(G)− 2, a contradiction.
By Claims 3, 5 and 6, G− v1 has exactly two end-blocks which each has a white non-
cut-vertex of G− v1 adjacent to v1 in G, and G− v1 can be constructed as follows: r − 1
disjoint paths P ′1, P
′
2, . . . , P
′
r−1 connect r−2 disjoint even cycles C1, C2, . . . , Cr−2 in turn
so that P ′i only connects Ci−1 and Ci for i = 2, 3, . . . , r − 2, where r ≥ 3, and P ′1 and
P ′r−1 connect only C1 and Cr−2 respectively (see Figure 5(a)). Let v2 be the third neighbor
of v1 in G− v1.
1v
2v
4v
3v
2rC -1C
'
2P
'
1P
'
i
P
'
1rP -
( )a ( )b
Figure 5: (a) The construction of G− v1; (b) Illustration for Claim 7.
Claim 7. v2 must be an internal vertex of paths P ′1 and P ′r−1.
Proof. If v2 belongs to some even cycle Ck (1 ≤ k ≤ r − 2) in G − v1, then Ck has
three 3-degree vertices of G, which contradicts Claim 5. If v2 is an internal vertex of P ′i
(2 ≤ i ≤ r−2) (see Figure 5(b)), let the ends of P ′i be v3 and v4. Then there exists an e-cut
decomposition from G to empty graph by taking ∂(v3) and ∂(v4) as the first two e-cuts.
Since |∂(v3)| = |∂(v4)| = 3, by Claim 1, cf(G) ≤ 2c(G) − 2, a contradiction. Hence v2
is an internal vertex of P ′1 or P
′
r−1 and the claim holds.
1C
i
C
1iC - -1i
C
i
C
+1,1iv
,2iv
,1iv
1C
( )c
'
i
P
'
i
P
,1iv
,2iv
( )a ( )b
2rC -
1rC -
1C
1v
5v
6v
2v
'
2P
'
1P
Figure 6: (a) The construction of G; (b) e-cut (bold edges) leaving from P ′i ; (c) e-cut (bold
edges) leaving from three T .
By Claim 7 we may suppose v2 is an internal vertex of P ′r−1 that has length at least
3. Then the subpath of P ′r−1 between both neighbors of v1 with two incident edges forms
326 Ars Math. Contemp. 23 (2023) #P2.09 / 315–334
a cycle, denoted by Cr−1. Thus G can be constructed from r − 1 disjoint even cycles
C1, C2, . . . , Cr−1 by using r− 1 disjoint paths to connect them in a cyclic way. More pre-
cisely, each P ′i connects vertex vi,1 of Ci−1 and vertex vi,2 of Ci, where i = 1, 2, . . . , r−1
and C0 = Cr−1 (see Figure 6(a)). Note that P ′2, . . . , P
′
r−2 remain unchanged, but P
′
1 is
lengthened by one edge and P ′r−1 is shorten. Then vi,1 and vi,2 have different colors in G.
Otherwise, there exists an e-cut decomposition from G to empty graph by taking ∂(V (P ′i ))
(see Figure 6(b)) as the first e-cut that has four edges. By Claim 1, cf(G) ≤ 2c(G) − 2,
a contradiction. Further, vi,2 and vi+1,1 have different colors in G, where vr,1 = v1,1.
Otherwise, two edges leaving from even cycle Ci are forbidden edges of G, which contra-
dicts that G is matching covered. Finally we claim that vi,2 and vi+1,1 are adjacent in Ci.
Otherwise, since vi,2 and vi+1,1 have different colors, two paths between vi,2 and vi+1,1
in Ci have length at least 3 (see Figure 6(c)). Let v5 and v6 be the two neighbors of vi,2
in Ci. Then v5, v6 and vi,1 are all of the same color in G. Let T be the tree induced by
{v5, v6} ∪ V (P ′i ). Then ∂(V (T )) is an e-cut of G of four edges. So there exists an e-cut
decomposition from G to empty graph by taking ∂(V (T )) as the first e-cut. By Claim 1,
we have cf(G) ≤ 2c(G)− 2, a contradiction.
Let x+P1 be an even cycle formed by the paths P ′i and edges vi,2vi+1,1, i = 1, 2, . . . ,
r − 1, and let Pi+1 be the path between vi,2 and vi+1,1 in Ci of length at least three.
Then the edges vi,2vi+1,1, i = 1, 2, . . . , r − 1, are contained in a frame of x + P1 and
G = x+ P1 + P2 + · · ·+ Pr is an ear decomposition of G described as in (ii).
4 Wheels and cylinders
In this section, we first present some lower bounds on the complete forcing numbers of
some special types of graphs. We then derive some closed formulas for the complete forc-
ing numbers of wheels and cylinders, respectively. Our main idea is to apply an e-cut de-
composition on a given graph to construct a complete forcing set whose cardinality attains
a lower bound on the complete forcing number.
Lemma 4.1. Let G be a graph that admits a perfect matching. If there is a set C of nice
cycles of G such that every edge of G lies in exactly two nice cycles of C, then cf(G) ≥ |C|.
Proof. For a nice cycle C of C, let T1(C) and T2(C) be the two frames of C. Let S be a
minimum complete forcing set of G. By Theorem 2.1, we have
|S ∩ Ti(C)| ≥ 1, i = 1, 2, for each nice cycle C of C.
Summing all the above inequalities together, we have
2|S| =
∑
C∈C
(|S ∩ T1(C)|+ |S ∩ T2(C)|) ≥ 2|C|,
because each edge of S belongs to exactly two nice cycles of C. Then we have
cf(G) = |S| ≥ |C|.
For a plane elementary bipartite graph G, all facial cycles (including the exterior facial
cycle) of G are nice cycles [28]. Since each edge of G lies in exactly two of these facial
cycles, by Lemma 4.1, we have
Corollary 4.2. Let G be a plane elementary bipartite graph with n faces. Then cf(G) ≥ n.
X. He and H. Zhang: Complete forcing numbers of graphs 327
This result is a generalization of a lower bound on the complete forcing numbers of
normal hexagonal systems (see [11]).
A wheel Wn (n ≥ 4) is a graph formed by connecting a single vertex (called the hub) to
all vertices of a cycle (called the rim) with n− 1 vertices. We can check that W2n (n ≥ 2)
is matching covered by the definition.
Theorem 4.3. For n ≥ 2, cf(W2n) = 2n− 1.
Proof. We denote by v0 the hub of W2n and by v1, v2, . . . , v2n−1 the vertices in the
rim of W2n along one of two directions of it. We can see that the set C of 4-cycles
{viv0vi+2vi+1vi|i = 1, 2, . . . , 2n − 1} consisting of 2n − 1 nice cycles of W2n, where
v2n = v1 and v2n+1 = v2. Moreover, each edge of W2n lies in exactly two nice cycles
of C. By Lemma 4.1, cf(W2n) ≥ |C| = 2n − 1. On the other hand, let D1 = ∂W2n(v0).
Then D1 is an e-cut of W2n with 2n − 1 edges. We use D1 to do an e-cut operation on
G1 = W2n and obtain G2. Since G2 is an odd cycle, by Lemma 2.3, D1 is a complete
forcing set. So cf(W2n) ≤ |D1| = 2n− 1. Consequently, cf(W2n) = 2n− 1.
The cartesian product G×H of two graphs G and H is a graph with vertex set V (G)×
V (H) specified by putting (u, v) adjacent to (u′, v′) if and only if (1) u = u′ and vv′ ∈
E(H), or (2) v = v′ and uu′ ∈ E(G). Let Pm = u1u2 · · ·um be a path with m vertices.
Recently, Chang et al. [5] obtained that cf(Pm × Pn) = ⌊n2 ⌋(m − 1) + ⌊
m
2 ⌋(n − 1).
It is natural to consider the complete forcing numbers of m × n cylinders. Let Cn =
v1v2 · · · vnv1 be a cycle with n vertices. An m × n cylinder Pm × Cn consists of m − 1
concentric layers of quadrangles (i.e. each layer is a cyclic chain of n quadrangles), capped
on each end by an n-polygon (see G1 of Figure 7 for an example). If both m and n are
odd, then Pm × Cn has an odd number of vertices and thus has no perfect matchings. So
we only consider the complete forcing number of Pm × Cn with even mn. The operation
of inserting a new vertex of degree two on an edge of a graph is called a subdivision of the
edge.
Lemma 4.4. If m is even, then
cf(Pm × Cn) ≥
{
mn− n2 , if n is even,
2mn+m−n−1
2 , if n is odd.
Proof. For 1 ≤ i ≤ m − 1, let Ri be the subgraph of Pm × Cn induced by {(ui, vj),
(ui+1, vj)| j = 1, 2, . . . , n} and E1i = {(ui, vj)(ui+1, vj)|j = 1, 2, . . . , n}. For 1 ≤ j ≤
n, let Lj be the subgraph of Pm × Cn induced by {(ui, vj), (ui, vj+1)|i = 1, 2, . . . ,m}
and E2j = {(ui, vj)(ui, vj+1)|i = 1, 2, . . . ,m}, where vn+1 = v1. Let S be a minimum
complete forcing set of Pm × Cn. Since Ri has n nice quadrangles of Pm × Cn and
each quadrangle of Ri has a frame completely contained in E1i, by Theorem 1.1, each
quadrangle of Ri has an edge in S ∩E1i. If n is even, then |S ∩E1i| ≥ n2 . And if n is odd,
then |S∩E1i| ≥ n+12 . Since Lj has m−1 nice quadrangles of Pm×Cn and each quadrangle
of Lj has a frame completely contained in E2j , by Theorem 1.1, each quadrangle of Lj
has an edge in S ∩ E2j . Since m is even, |S ∩ E2j | ≥ m2 . Thus we have if n is even, then
cf(Pm ×Cn) = |S| ≥
∑m−1
i=1 |S ∩E1i|+
∑n
j=1 |S ∩E2j | ≥
n(m−1)
2 +
mn
2 = mn−
n
2 .
And if n is odd, then cf(Pm × Cn) = |S| ≥
∑m−1
i=1 |S ∩ E1i| +
∑n
j=1 |S ∩ E2j | ≥
(m−1)(n+1)
2 +
mn
2 =
2mn+m−n−1
2 .
328 Ars Math. Contemp. 23 (2023) #P2.09 / 315–334
Lemma 4.5 (Pick’s theorem [8]). Let P be a simple polygon constructed on a polyomino
such that all the polygon’s vertices are polyomino’s vertices. Let the number of polyomino’s
vertices in the interior of P be i and the number of polyomino’s vertices on the boundary
of P be b. Then the area of P is given by A = b2 + i− 1.
Theorem 4.6.
cf(Pm × Cn) =

mn− n+ 2, if m is odd and n is even (m ≥ 1, n ≥ 4),
mn− n2 , if both m and n are even (m ≥ 2, n ≥ 4),
2mn+m−n−1
2 , if m is even and n is odd (m ≥ 2, n ≥ 3).
Proof. Since mn is even, we can see that each edge of Pm × Cn is allowed, so Pm × Cn
is matching covered. To construct a complete forcing set of Pm × Cn, by Lemma 2.3, we
can directly apply e-cut decomposition on Pm × Cn.
We divide our proof into the following three cases.
Case 1. m is odd and n is even (m ≥ 1, n ≥ 4).
If m = 1, then Pm × Cn is an even cycle and cf(Pm × Cn) = 2 by Theorem 3.1, and
the conclusion holds. In the following, we suppose that m ≥ 3.
By Corollary 4.2, cf(Pm × Cn) ≥ mn− n+ 2. So it suffices to construct a complete
forcing set of Pm × Cn of size mn− n+ 2.
1 1( , )u v
1( , )nu v
1 4( , )u v1 1( , )nu v -
2( , )mu v
1( , )mu v
( , )
m n
u v
( , )
m n
u v
1 2( , )u v
4( , )mu v
2 5( , )u v
1 5( , )mu v-
2 1( , )nu v -
1 1( , )m nu v- -
1G 2G
1( , )nu v
Figure 7: m is odd and n is even.
X. He and H. Zhang: Complete forcing numbers of graphs 329
Let
D1 ={(u2i+1, v1)(u2i+1, v2), (u2i+1, v2)(u2i+1, v3) | i = 0, 1, 2, . . . ,
m− 1
2
}∪
{(uj , v1)(uj+1, v1), (uj , v3)(uj+1, v3) | j = 1, 2, . . . ,m− 1}∪
{(u2k, vn)(u2k, v1), (u2k, v3)(u2k, v4) | k = 1, 2, . . . ,
m− 1
2
}
(see bold edges of G1 of Figure 7). Then D1 is an e-cut of G1 = Pm × Cn. We
use D1 to do an ED operation on G1 and obtain G2 = Pm × Pn−3 (see Figure 7).
Let D2, D3, . . . , D (m−1)(n−4)
4 +1
be ∂G2((u2i, v5)), ∂G2((u2i, v7)),. . . , ∂G2((u2i, vn−1))
(i = 1, 2, . . . , m−12 ), respectively. Then we continue to do ED operations from G2 by
D2, D3, . . . , D (m−1)(n−4)
4 +1
in turn and obtain G (m−1)(n−4)
4 +2
. Note that Di is an e-cut
of Gi for i = 1, 2, . . . ,
(m−1)(n−4)
4 + 1. We find that G (m−1)(n−4)
4 +2
can be obtained by
subdividing every edge of Pm−1
2 +1
× Pn−4
2 +1
as shown in the thin edges of G2 in Fig-
ure 7. Let C be a cycle of G (m−1)(n−4)
4 +2
. Suppose that C encloses some region R in
the plane, let A be the area of R, b be the number of vertices of G2 on C, and i be the
number of vertices of G2 in the interior of C. Then A is divisible by 4. We can see that
C is obtained by subdividing every edge of a cycle C ′ of Pm−1
2 +1
× Pn−4
2 +1
. Since C ′
is a cycle of even length and |V (C)| = 2|V (C ′)|, b is divisible by 4. By Lemma 4.5, i is
odd. Then G1 − V (C) has no perfect matchings. So each cycle of G (m−1)(n−4)
4 +2
is not
a nice cycle of G1. By Lemma 2.3, D1 ∪ D2 ∪ . . . D (m−1)(n−4)
4 +1
is a complete forcing
set of G1. Since |D1| = (m + 1) + 2(m − 1) + (m − 1) = 4m − 2 and |Di| = 4 for
i = 2, 3, . . . , (m−1)(n−4)4 + 1, cf(G1) ≤ |D1 ∪D2 ∪ . . . D (m−1)(n−4)
4 +1
| = mn− n+ 2.
Consequently, cf(Pm × Cn) = mn− n+ 2.
Case 2. Both m and n are even (m ≥ 2, n ≥ 4).
By Lemma 4.4, it suffices to construct a complete forcing set of Pm × Cn of size
mn− n2 . Let
D1 ={(u2i+1, v1)(u2i+1, v2), (u2i+1, v2)(u2i+1, v3) | i = 0, 1, 2, . . . ,
m− 2
2
}∪
{(uj , v1)(uj+1, v1), (uj , v3)(uj+1, v3) | j = 1, 2, . . . ,m− 1)}∪
{(u2k, vn)(u2k, v1), (u2k, v3)(u2k, v4) | k = 1, 2, . . . ,
m
2
}.
Then D1 is an e-cut of G1 = Pm × Cn. We use D1 to do an ED operation on G1 =
Pm × Cn and obtain G2 = Pm × Pn−3 (see Figure 8). Let D2, D3, . . . , Dm(n−4)
4 +1
be ∂G2((u2i, v5)), ∂G2((u2i, v7)),. . . , ∂G2((u2i, vn−1)) (i = 1, 2, . . . ,
m
2 ), respectively.
Continuously doing ED operations from G2 by D2, D3, . . . , Dm(n−4)
4 +1
in turn, we obtain
Gm(n−4)
4 +2
. Note that Gm(n−4)
4 +2
can be obtained by subdividing every edge of Pm−2
2 +1
×
Pn−4
2 +1
as shown in Figure 8. Let C be a cycle of Gm(n−4)
4 +2
. Suppose that C encloses
some region R in the plane, let A be the area of R, b the number of vertices of G2 on C, and
i be the number of vertices of G2 in the interior of C. Then A is divisible by 4. We can see
that C is obtained by subdividing every edge of a cycle C ′ of Pm−2
2 +1
×Pn−4
2 +1
. Since C ′
is a cycle of even length and |V (C)| = 2|V (C ′)|, b is divisible by 4. By Lemma 4.5, i is
330 Ars Math. Contemp. 23 (2023) #P2.09 / 315–334
1 1( , )u v
1( , )nu v
1 4( , )u v1 1( , )nu v -
2( , )mu v
1( , )mu v
( , )
m n
u v
( , )
m n
u v
1 2( , )u v
4( , )mu v
2 5( , )u v
5( , )mu v
2 1( , )nu v -
1( , )m nu v -
1G 2G ( 4) 2
4
m n
G
-
+
1( , )nu v
Figure 8: Both m and n are even.
odd. So G1 − V (C) has no perfect matchings. Thus each cycle of Gm(n−4)
4 +2
is not a nice
cycle of G1. By Lemma 2.3, D1 ∪ D2 ∪ . . . Dm(n−4)
4 +1
is a complete forcing set of G1.
Since |D1| = m + 2(m − 1) +m = 4m − 2, |Di| = 4 for i = 2, 3, . . . , (m−2)(n−4)4 + 1
and |Dj | = 3 for j = (m−2)(n−4)4 + 2,
(m−2)(n−4)
4 + 3, . . . ,
m(n−4)
4 + 1, cf(G1) ≤
|D1 ∪D2 ∪ . . . D (m−1)(n−4)
4 +1
| = mn− n2 . Consequently, cf(Pm × Cn) = mn−
n
2 .
Case 3. m is even and n is odd (m ≥ 2, n ≥ 3).
By Lemma 4.4, it suffices to prove cf(G1) ≤ 2mn+m−n−12 .
2( , )mu v 2( , )mu v
1 2( , )u v
3( , )mu v 3( , )mu v
1( , )mu v
1( , )mu v
1 1( , )u v
2 3( , )u v
2 2( , )u v 2 1( , )u v
1 3( , )u v
( )a ( )b
1D
m
D
1
2
m
D
+
2
m
D
Figure 9: m is even and n = 3.
X. He and H. Zhang: Complete forcing numbers of graphs 331
Subcase 3.1. n = 3.
Let G1 = Pm × C3 and D1, D2, . . . , Dm2 be ∂G1((u2i+1, v3)) (i = 0, 1, . . . ,
m−2
2 ),
respectively (see Figure 9(a)). Then we use D1, D2, . . . , Dm2 to do ED operations on
G1 in turn and obtain Gm2 +1. Let Dm2 +1, Dm2 +2, . . . , Dm be ∂Gm2 +1((u2i+1, v2)) (i =
0, 1, . . . , m−22 ), respectively. Then we use Dm2 +1, Dm2 +2, . . . , Dm to do ED operations in
turn and obtain Gm+1. We can see that Gm+1 consists of m2 disjoint cycles of length 3 and
c(Gm+1) =
m
2 as shown in Figure 9(b). Thus each cycle of Gm+1 is not a nice cycle of
G1. By Lemma 2.3, cf(G1) ≤ c(G1)+m−c(Gm+1) = 3(m−1)+1+m− m2 =
7m−4
2 .
Consequently, cf(Pm × C2n) = 7m−42 .
1 4( , )u v1( , )nu v
4( , )mu v( , )m nu v
1G 2G
1( , )nu v
( , )
m n
u v
1( , )mu v
2( , )mu v
2D
1
2
m
D
+
3
2 2
m n
D
-
+
Figure 10: m is even and n is odd (n ≥ 5).
Subcase 3.2. n ≥ 5.
Let
D1 ={(u2i+1, v1)(u2i+1, v2), (u2i+1, v2)(u2i+1, v3) | i = 0, 1, 2, . . . ,
m− 2
2
)}∪
{(uj , v1)(uj+1, v1), (uj , v3)(uj+1, v3)|j = 1, 2, . . . ,m− 1}∪
{(u2k, vn)(u2k, v1), (u2k, v3)(u2k, v4)|k = 1, 2, . . . ,
m
2
}.
Then we use D1 to do an ED operation on G1 = Pm × C2n and obtain G2 =
Pm × Pn−3 (see Figure 10). Let Dt (t = 2, 3, . . . , m2 ) be
{(u2t−2, v2i+2)(u2t−1, v2i+2) | i = 1, 2, . . . ,
n− 3
2
}∪
{(u2t−2, vj+3)(u2t−2, vj+4) | j = 1, 2, . . . , n− 4}∪
{(u2t−3, v2k+3)(u2t−2, v2k+3) | k = 1, 2, . . . ,
n− 3
2
}.
332 Ars Math. Contemp. 23 (2023) #P2.09 / 315–334
Then we use D2, D3, . . . , Dm2 to do ED operations on G1 in turn and obtain Gm2 +1 which
is P2 × Pn−3. Let Dm2 +1, Dm2 +2, . . . , Dm2 +n−32 be ∂Gm2 +1((um, v2s+3))
(s = 1, 2, . . . , n−32 ), respectively. Then we use Dm2 +1, Dm2 +2, . . . , Dm2 +n−32 to do ED op-
erations on Gm
2 +1
in turn and obtain Gm
2 +
n−3
2 +1
which is the empty graph. By Lemma 2.3,
D1 ∪D2 ∪ · · · ∪Dm
2 +
n−3
2
is a complete forcing set of G1 and cf(G1) ≤ c(G1) + m2 +
n−3
2 − 0 =
2mn+m−n−1
2 . Consequently, cf(Pm × Cn) =
2mn+m−n−1
2 .
At the end of this paper, by some simple calculations, we present the relationship
between the cyclomatic number and complete forcing number for wheels and cylinders.
For a wheel W2n, c(W2n) = |E(W2n)| − |V (W2n)| + 1 = 2(2n − 1) − 2n + 1 =
2n − 1. By Theorem 4.3, cf(W2n) = c(W2n). For a cylinder Pm × Cn, c(Pm × Cn) =
|E(Pm × Cn)| − |V (Pm × Cn)| + 1 = n(m − 1) +mn −mn + 1 = mn − n + 1. By
Theorem 4.6, we can see that cf(Pm ×Cn) = c(Pm ×Cn) + 1 if m is odd and n is even,
cf(Pm × Cn) = c(Pm × Cn) + n2 − 1 if both m and n are even, and cf(Pm × Cn) =
c(Pm × Cn) + m+n−32 if m is even and n is odd.
ORCID iDs
Xin He https://orcid.org/0000-0002-5853-9653
Heping Zhang https://orcid.org/0000-0001-5385-6687
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ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.10 / 335–348
https://doi.org/10.26493/1855-3974.2857.07b
(Also available at http://amc-journal.eu)
Locally s-arc-transitive graphs arising from
product action*
Michael Giudici
Department of Mathematics and Statistics, The University of Western Australia,
Perth WA 6009, Australia
Eric Swartz †
Department of Mathematics, William & Mary,
P.O. Box 8795, Williamsburg, VA 23187-8795, USA
Received 30 March 2022, accepted 14 September 2022, published online 13 December 2022
Abstract
We study locally s-arc-transitive graphs arising from the quasiprimitive product action
(PA). We prove that, for any locally (G, 2)-arc-transitive graph with G acting quasiprimi-
tively with type PA on both G-orbits of vertices, the group G does not act primitively on
either orbit. Moreover, we construct the first examples of locally s-arc-transitive graphs of
PA type that are not standard double covers of s-arc-transitive graphs of PA type, answering
the existence question for these graphs.
Keywords: Locally s-arc-transitive graph, quasiprimitive group, product action.
Math. Subj. Class. (2020): 20B25, 05C25, 05E18
1 Introduction
For an integer s ⩾ 1, an s-arc in a graph Γ is an (s+ 1)-tuple (α0, α1, . . . , αs) of vertices
such that αi ∼ αi+1 and αi ̸= αi+2 for each i. We say that Γ is s-arc-transitive if Γ
contains an s-arc and the automorphism group of Γ acts transitively on the set of all s-arcs.
*This paper formed part of the Australian Research Council’s Discovery Project DP120100446 of the first au-
thor. The authors would also like to thank Ákos Seress, who made this collaboration possible in 2012 by allowing
the second author to visit Australia, and Luke Morgan, for providing a proof of Lemma 3.2 and encouraging the
completion of the project. Finally, the authors wish to thank the anonymous referees for their detailed comments
and suggestions that greatly improved the final version of this paper.
†Corresponding author.
E-mail addresses: michael.giudici@uwa.edu.au (Michael Giudici), easwartz@wm.edu (Eric Swartz)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
336 Ars Math. Contemp. 23 (2023) #P2.10 / 335–348
If Γ is s-arc-transitive and each (s − 1)-arc can be extended to an s-arc then any s-arc-
transitive graph is also (s − 1)-arc-transitive. The study of s-arc-transitive graphs goes
back to the pioneering work of Tutte [32, 33], who showed that if Γ has valency three then
s ⩽ 5. Weiss [35] later showed that if the valency restriction is relaxed to allow valency at
least three then s ⩽ 7, with equality holding for the generalised hexagons arising from the
groups G2(q) for q = 3f .
Praeger [25] initiated a programme for the study of finite connected s-arc-transitive
graphs by first showing that if G ⩽ Aut(Γ) acts transitively on the set of all s-arcs of Γ
and N ◁ G has at least three orbits on the set of vertices, then the quotient graph ΓN
whose vertices are the orbits of N is also s-arc-transitive. Moreover, Γ is a cover of ΓN .
This reduces the study of finite connected (G, s)-arc-transitive graphs to two basic types:
• those where G is quasiprimitive on the set of vertices, that is, where all nontrivial
normal subgroups of G are transitive on vertices;
• those where G is biquasiprimitive on the set of vertices, that is, where all nontrivial
normal subgroups of G have at most two orbits on vertices and there is a normal
subgroup with two orbits.
Praeger showed that of the eight types of finite quasiprimitive groups, only four — HA
(affine), TW (twisted wreath), AS (almost simple) and PA (product action) — can act 2-
arc-transitively on a graph [25]. We use the types of quasiprimitive groups as given in [27]
and define type PA, the main focus of this paper, in Section 2. These are slight variations
on the types of primitive permutation groups given by the O’Nan–Scott Theorem. All
graphs of type HA were classified by Praeger and Ivanov [18] while those of type TW were
studied by Baddeley [1]. The 2-arc-transitive graphs for some families of almost simple
groups have all been classified, for example the Suzuki groups [9], Ree groups [8] and
PSL(2, q) [16]. The first examples of 2-arc-transitive graphs of PA type were given by
Li and Seress [22] and studied further by Li, Seress, and Song [23]. Another family of
quasiprimitive 2-arc-transitive graphs of PA type were constructed by Li, Ling, and Wu in
[21].
In the biquasiprimitive case the graph is bipartite and such graphs were investigated in
[26, 28]. An alternative way to study such graphs is via the notion of local s-arc-transitivity.
We say that a graph Γ is locally (G, s)-arc-transitive for a group G ⩽ Aut(Γ) if for each
vertex α, the vertex stabiliser Gα acts transitively on the set of all s-arcs starting at α.
If G also acts transitively on the set of vertices then Γ is s-arc-transitive. If Γ is locally
(G, s)-arc-transitive but G is intransitive on the set of vertices, then G has two orbits on
vertices and Γ is bipartite. One way to construct locally s-arc-transitive graphs is to start
with an s-arc-transitive graph Γ and take its standard double cover Σ, which has vertex set
V Γ × {1, 2} and (α, i) ∼ (β, j) precisely when i ̸= j and α ∼ β in Γ. Then Aut(Γ)
acts as automorphisms on Σ with two orbits on vertices and Σ is locally (Aut(Γ), s)-arc-
transitive [11].
If Γ is a bipartite graph and G ⩽ Aut(Γ) acts transitively on the set of vertices, then
Γ is locally (G+, s)-arc-transitive where G+ is the index two subgroup that stabilises each
part of the bipartition. Hence the study of locally s-arc-transitive graphs encompasses the
study of all bipartite s-arc-transitive graphs and hence the biquasiprimitive case in Praeger’s
programme. It is also a wider class of graphs as the known generalised octagons are locally
9-arc-transitive but not vertex-transitive, and it has been shown by van Bon and Stellmacher
[34] that this is best possible.
M. Giudici and E. Swartz: Locally s-arc-transitive graphs arising from product action 337
A programme for the study of finite connected locally s-arc-transitive graphs was mapped
out by Giudici, Li and Praeger [11]. If Γ is locally (G, s)-arc-transitive with G having two
orbits on vertices and N ◁ G is intransitive on both G-orbits, then the quotient graph ΓN
is also locally s-arc-transitive. Moreover, Γ is a cover of ΓN . This reduces the study of
finite connected locally (G, s)-arc-transitive graphs for which G is vertex-intransitive into
two basic types:
• those where G is quasiprimitive on each of its two orbits on vertices;
• those where G is quasiprimitive on only one of its two orbits on vertices.
In the second case, it was shown [11] that the quasiprimitive action must be of type HA, HS,
AS, PA or TW. These were further studied in [12] where all examples where the quasiprim-
itive action has type PA preserving a product structure or type HS were classified. An in-
finite family of examples where the quasiprimitive action has type TW was given by Kaja
and Morgan [19]. In the first case, either the two quasiprimitive actions have the same
quasiprimitive type and are one of HA, AS, TW or PA, or they are different with one of
type SD and one of type PA [11]. All 2-arc-transitive graphs of the latter type were classi-
fied in [13] and there are locally 5-arc-transitive examples in this case [14]. It was shown in
[17, Lemma 3.2] that all locally 2-arc-transitive graphs where the quasiprimitive action is
of type HA on both orbits are actually vertex-transitive but a complete classification has not
been obtained – see [18, Section 2] for further discussion. All locally (G, 2)-arc-transitive
graphs have been classified in the cases where G is an almost simple group whose socle is
a Ree group [7], Suzuki group [31], or PSL(2, q) [3], while the sporadic group case was
studied in [20]. Examples also exist in the PA and TW cases as we can take standard double
covers of s-arc-transitive graphs of type PA and TW respectively.
The aim of this paper is to study locally s-arc-transitive graphs of PA type. We prove
that, for any locally (G, 2)-arc-transitive graph with G acting quasiprimitively with type PA
on both G-orbits of vertices, the group G does not act primitively on either orbit. Moreover,
in the spirit of [22], we solve the existence problem for locally 2-arc-transitive graphs of
PA type. In particular, we construct the first examples of locally s-arc-transitive graphs of
PA type that are not standard double covers of s-arc-transitive graphs of PA type.
2 PA type
Let G act quasiprimitively on a set Ω. We say that G has type PA if there exists a G-invariant
partition B of Ω such that G acts faithfully on B and we can identify B with ∆k for some
set ∆ and k ⩾ 2 such that G ⩽ H wrSk acts in the usual product action of a wreath
product on ∆k, where H ⩽ Sym(∆) is an almost simple group acting quasiprimitively on
∆. Moreover, if T = soc(H) then G has a unique minimal normal subgroup N = T k.
Note that since G is quasiprimitive, N acts transitively on Ω and hence on B. Thus G =
NGα = NGB , where B ∈ B is a block containing α ∈ Ω. As N is minimal normal
in G we have that G transitively permutes the simple direct factors of N and hence so do
both Gα and GB . Thus given B = (δ, . . . , δ) ∈ B we may assume that NB = T kδ and for
α ∈ B we have that Nα is a subdirect subgroup of NB , that is, the projection of Nα onto
each direct factor is isomorphic to Tδ .
Let R = Tδ . Following the terminology of [22], if Nα ∼= R then we call Nα a diagonal
subgroup of NB = Rk. Then there exists automorphisms φ2, φ3, . . . , φk of R such that
Nα = {(t, tφ2 , . . . , tφk) | t ∈ R}.
338 Ars Math. Contemp. 23 (2023) #P2.10 / 335–348
If each of the φi is the trivial automorphism then we call Nα a straight diagonal subgroup
while if some φi is nontrivial then we call Nα a twisted diagonal subgroup. Furthermore, if
Nα ̸∼= R then we refer to Nα as being a nondiagonal subgroup. We refer to the quasiprim-
itive permutation group G of type PA as being of straight diagonal, twisted diagonal, or
nondiagonal type according to the type of Nα.
Note that unlike for primitive groups of type PA, G does not necessarily preserve a
product structure on Ω, only on some G-invariant partition B. Indeed the following result
shows that for locally 2-arc-transitive graphs this partition must be nontrivial on each of the
bipartite halves.
Theorem 2.1. Let Γ be a locally (G, 2)-arc-transitive connected graph with G quasiprim-
itive of type PA on both orbits Ω1 and Ω2. Let N = T k = soc(G) and for i = 1, 2, let Bi
be a G-invariant partition of Ωi such that G preserves a product structure ∆ki on each Bi.
Then Bi ̸= Ωi for each i.
Proof. Suppose that Bi = Ωi for some i. Without loss of generality suppose that i = 1.
Also note that there is an almost simple group H with socle T such that G ⩽ H wrSk.
Let α = (ω, . . . , ω) ∈ Ω1. Then Nα = T kω with Tω ̸= 1 and Gα = G ∩ (Hω wrSk).
By [11, Lemma 3.2], GΓ(α)α is 2-transitive so either all neighbours of α lie in the same
block of B2 or in distinct blocks. If they all lie in the same block then for each β ∈ Ω1
we have that the neighbours of β lie in the same block. However, this contradicts Γ being
connected. Hence for each α ∈ Ω1, the neighbours of α lie in distinct blocks. Hence
Gα acts 2-transitively on the set X of blocks of B2 that contain neighbours of α. By [11,
Lemma 6.2], NΓ(α)α is a transitive subgroup of the 2-transitive group G
Γ(α)
α and so Nα also
acts transitively on X . Let B = (δ1, δ2, . . . , δk) ∈ B2 be a block containing a neighbour γ
of α. Then X = (δ1, δ2, . . . , δk)Nα = δTω1 × δ
Tω
2 × · · · × δ
Tω
k . By [29, Theorem 1.1(b)],
the stabiliser G1 in G of the first simple direct factor of N projects onto H in the first
coordinate and so (G1)α projects onto Hω in the first coordinate. Hence δTω1 = δ
Hω
1 . Since
Gα ⩽ Hω wrSk and transitively permutes the k simple direct factors of N , it follows that
δTωi = δ
Tω
1 for each i. In particular, X = A
k for some set A and we could have chosen
B = (δ, . . . , δ) for some δ ∈ ∆2. Thus Gαγ ⩽ Gα,B ⩽ Hωδ wrSk. However, for
δ′ ∈ A\{δ} there is no element of Hωδ wrSk mapping (δ′, δ, . . . , δ) to (δ′, δ′, δ, . . . , δ),
contradicting Gα acting 2-transitively on X . Thus B1 ̸= Ω1.
Corollary 2.2. Let Γ be a locally (G, 2)-arc-transitive connected graph with G quasiprim-
itive of type PA on both orbits. Then G is not primitive on either orbit.
3 Constructions
Let G be a finite group with subgroups L and R. Let ∆1 be the set [G : L] of right cosets
of L in G and ∆2 be the set [G : R] of right cosets of R in G. We define the coset graph
Γ = Cos(G,L,R) to be the bipartite graph with vertex set the disjoint union ∆1 ∪ ∆2
such that {Lx,Ry} is an edge if and only if Lx ∩ Ry ̸= ∅, or equivalently xy−1 ∈ LR.
Then G acts by right multiplication on both ∆1 and ∆2, and induces automorphisms of
Γ. Note that the vertices in ∆1 have valency |L : L ∩ R| while the vertices in ∆2 have
valency |R : L ∩ R|. We say that Γ has valency {|L : L ∩ R|, |R : L ∩ R|}. Conversely,
if Γ is a graph and G ⩽ Aut(Γ) acts transitively on the set of edges of Γ but not on the set
of vertices then Γ can be constructed in this way [11, Lemma 3.7]. We refer to the triple
(L,R,L ∩R) as the associated amalgam.
M. Giudici and E. Swartz: Locally s-arc-transitive graphs arising from product action 339
We collect the following properties of coset graphs. We say that a subgroup H of a
group G is core-free if ∩g∈GHg = 1.
Lemma 3.1 ([11, Lemma 3.7]). Let G be a group with proper subgroups L and R, and let
Γ = Cos(G,L,R).
(1) Γ is connected if and only if G = ⟨L,R⟩.
(2) G acts faithfully on both [G : L] and [G : R] if and only if both L and R are core
free in G.
(3) G acts transitively on the set of edges of Γ.
(4) Γ is locally (G, 2)-arc-transitive if and only if L acts 2-transitively on [L : L ∩ R]
and R acts 2-transitively on [R : L ∩R].
We also need the following result, which essentially follows from the definition of a
completion (and the universal completion) of an amalgam (see [15]) and results on covers
of graphs (see, e.g., [2, Chapter 19]). The result is truly “folklore”: while it seems to be
taken for granted in the field, we also cannot find an explicit proof in the literature. We
have included a proof here provided by Luke Morgan [24].
Lemma 3.2. If Γ is a locally s-arc-transitive graph with amalgam (L,R,L ∩ R) and
s ⩾ 2, then any other graph with amalgam (L,R,L ∩R) is locally s-arc-transitive.
Proof. Let G := L ∗L∩RR be the universal completion of (L,R,L∩R) and let Γ∗ denote
the universal tree on which G acts edge-transitively. We identify L and R with their images
in G, and label an edge {α, β} so that Gα = L, Gβ = R, and Gαβ = L ∩ R. Since Γ
is locally s-arc-transitive for s ⩾ 2, it is locally 2-arc-transitive and so the actions of L on
the set of right cosets of L ∩ R in L, and of R on the set of right cosets of L ∩ R in R are
2-transitive [11, Lemma 3.2]. In particular, Γ∗ is locally (G, 2)-arc-transitive.
Now let Σ be a graph with edge-transitive group of automorphisms H such that the
amalgam (Hγ , Hδ, Hγδ) is isomorphic to (L,R,L∩R), where {γ, δ} is an edge of Σ. By
the universal property of G and of Γ∗, there is a map ϕ : G → H such that the following
diagrams commute:
L G R G L ∩R G
Hγ H Hδ H Hγδ H
.
Let N be the kernel of ϕ. Then, Σ = Γ∗N , the quotient graph, and the kernel of the
action of G on Σ is exactly N .
In particular, ϕ(Gα) = Hγ and ϕ(Gβ) = Hδ . Further, since ϕ(Gαβ) = Hγδ, we have
commutative diagrams of the following groups:
Gα G
Γ∗(α)
α Gβ G
Γ∗(β)
β
Hγ H
Σ(γ)
γ Hδ H
Σ(δ)
δ
,
where GΓ
∗(α)
α denotes the induced action of Gα on Γ∗(α), etc.
340 Ars Math. Contemp. 23 (2023) #P2.10 / 335–348
We now claim that for ε = γ, δ and ζ ∈ Γ∗(ε), we have ζN ∩ Γ∗(ε) = {ζ}. Indeed,
this follows since |Gα : Gαβ | = |Hγ : Hγδ| and |Gβ : Gαβ | = |Hδ : Hγδ|.
Now suppose Γ∗ is locally (G, r)-arc-transitive and Σ is locally (H, t)-arc-transitive.
By [11, Lemma 5.1(3)], we have t ⩾ r.
Assume that r < t. We will show that Γ∗ would be locally (G, r + 1)-arc-transitive in
this case, contradicting the maximality of r.
Suppose P and P ′ are (r + 1)-paths in Γ∗ with initial vertex α or β. Since r ⩾ 1,
without loss of generality we may assume P = (α, β1, . . . , βr, βr+1) and P ′ = (α, β1, . . . ,
βr, β
′
r+1), where β1 = β.
Consider the images of PN and (P ′)N in Σ. Note that the images are two (r+1)-paths,
since the equality βNi−1 = β
N
i+i would contradict our claim above. Hence, there is h ∈ Hγ
such that (PN )h = (P ′)N . Since ϕ(Gα) = Hγ , we can take h = ϕ(g) for g ∈ Gα, so g
fixes α. Now, (PN )h = (P ′)N implies (βN )g = βN . Thus, g fixes βN , and, since g fixes
α, g fixes the unique vertex in Γ∗(α) ∩ βN , which is β; so, g ∈ Gαβ . Continuing in this
way, we see that g ∈ Gαβ1...βr . Now, (βNr+1)h = (β′r+1)N , and so β
g
r+1 lies in the N -orbit
of β′r+1, and at the same time must be adjacent to βr, since g ∈ Gβr . Once more, the claim
implies βgr+1 = β
′
r+1.
We have thus shown that Gα is transitive on (r + 1)-arcs with initial vertex u. A
similar argument establishes that same result for Gβ , and hence Γ∗ is locally (G, r + 1)-
arc-transitive. This contradicts the maximality of r, and, therefore, r = t, as desired. In
particular, taking Σ = Γ we see that r = s. Hence Γ∗, and so any graph with amalgam
(L,R,L ∩R), is locally s-arc-transitive.
Lemma 3.1 enables us to construct locally (G, 2)-arc-transitive graphs where G has
two orbits ∆1 and ∆2 on vertices and acts quasiprimitively of type PA on each. Recall the
three types straight diagonal, twisted diagonal and nondiagonal of quasiprimitive groups of
type PA. Analogously to [22], we refer to a locally (G, 2)-arc-transitive graph Γ where G
is quasiprimitive of type PA on each orbit by the type of the two PA actions. For example,
if G is of straight diagonal type on ∆1 and twisted diagonal type on ∆2 then we refer to Γ
as being of straight-twisted type.
3.1 Straight-twisted type
Construction 3.3. We begin with the following: let (L,R,L ∩ R) be an amalgam for a
locally s-arc-transitive graph, and suppose further that L = L1⋊K and R = R1⋊K such
that K acts trivially on R1. Note that this implies L ∩R = (L1 ∩R1)K.
Let H be an almost simple group with socle T , and subgroups H1 and H2 such that
• H1 ∼= L1, H2 ∼= R1, H1 ∩ H2 ∼= L1 ∩ R1, i.e., ϕ : H1 → L1, τ : H2 → R1 are
isomorphisms with restrictions each sending H1 ∩H2 → L1 ∩R1,
• H = ⟨H1, H2⟩, and
• not all automorphisms of L1 in K extend to automorphisms of T .
We will abuse notation slightly and assume L1, R1 ⩽ H. Let k = |K| and let
F = {f : K → H} ∼= Hk.
M. Giudici and E. Swartz: Locally s-arc-transitive graphs arising from product action 341
For each ℓ ∈ L1 and r ∈ R1, define fℓ, fr ∈ F such that
fℓ(κ) = ℓ
κ,
fr(κ) = r
for all κ ∈ K. Furthermore, we let
Nα := {fℓ | ℓ ∈ L1} ∼= L1,
Nβ := {fr | r ∈ R1} ∼= R1.
Since K acts trivially on R1, we have that
Nα ∩Nβ = {fr | r ∈ R1 ∩ L1} ∼= L1 ∩R1.
Let N := ⟨Nα, Nβ⟩.
Now K acts on F via fσ(κ) = f(σκ) for each σ, κ ∈ K. Then for ℓ ∈ L1 we have
that
(fℓ)
σ(κ) = fℓ(σκ) = ℓ
σκ = fℓσ (κ).
Hence (fℓ)σ = fℓσ and so K normalises Nα. Similarly, (fr)σ = fr for all r ∈ R1 so K
normalises Nβ and hence also N . Define
Gα := Nα ⋊K,
Gβ := Nβ ⋊K,
G := ⟨Gα, Gβ⟩.
Finally, we define Γ := Cos(G,Gα, Gβ).
Lemma 3.4. Let Γ be a graph yielded by Construction 3.3. Then Γ is a connected locally
(G, s)-arc-transitive graph such that G acts quasiprimitively with type PA on each orbit of
vertices. Moreover, the action of G on [G : Gβ ] is straight diagonal, and the action of G
on [G : Gα] is twisted diagonal, that is, Γ is of straight-twisted type.
Proof. Let FT = {f ∈ F | f(κ) ∈ T for all κ ∈ K} ∼= T k. For each κ ∈ K, let
πκ : F → H
f 7→ f(κ).
Since ⟨R1, L1⟩ = H , we have that πκ(N) = H for all κ ∈ K and so by
[30, page 328, Lemma], N ∩ FT is a direct product of diagonal subgroups, each isomor-
phic to T . Since there are elements κ ∈ K that do not extend to an automorphism of T , it
follows that N ∩FT is not itself a diagonal subgroup and so N ∩FT ∼= T j for some integer
2 ⩽ j ⩽ k.
Since the action of K on Nα is isomorphic to the action of K on L1 we see that
Gα ∼= L and similarly, Gβ ∼= R. Moreover, Gα∩Gβ ∼= ⟨L1∩R1,K⟩ = L∩R. Therefore
Γ := Cos(G,Gα, Gβ) is a connected graph with amalgam (L,R,L ∩ R) and is thus a
locally s-arc-transitive graph.
Finally, since K transitively permutes the simple direct factors of FT it also transitively
permutes the simple direct factors of N ∩ FT . Thus soc(G) ∼= T j and G ≲ H wrSj
for some integer j ⩾ 2. Since πκ(Nα) = L1 for all κ ∈ K it follows that Nα is a
342 Ars Math. Contemp. 23 (2023) #P2.10 / 335–348
subdirect subgroup of Lj1 and similarly, Nβ is a subdirect subgroup of R
j
1. Therefore, G
acts quasiprimitively with type PA on both [G : Gα] and [G : Gβ ], and, by construction,
the action of G on [G : Gβ ] is straight diagonal, and the action of G on [G : Gα] is twisted
diagonal.
Example 3.5. This example is based on [22, Example 4.1]. First, (AGL(1, 5)× C2, S3 ×
C4, C4×C2) is an amalgam admitting a locally 2-arc-transitive connected graph of valency
{3, 5}: indeed, a GAP computation shows that in the group S7 we can take
L = ⟨(4, 5, 6, 7), (3, 4, 5, 7, 6), (1, 2)⟩ ∼= AGL(1, 5) × C2 and R = ⟨(1, 2), (1, 2, 3),
(4, 5, 6, 7)⟩ ∼= S3 × C4 such that ⟨L,R⟩ = S7, and L ∩R ∼= C4 × C2 [10].
Let T = PSL(2, p), where p is a prime and p ≡ ±1 (mod 60). Thus we may select
D < T such that D ∼= D60, with D = ⟨h, d | h30 = d2 = 1, hd = h−1⟩. First, define
L1 := ⟨h3⟩ ∼= C10 ∼= C5 × C2. Noting that D has a subgroup B := ⟨h15, d⟩ ∼= C22 , there
exists an element x of T such that Bx = B and dx = h15 [6]. Define R1 := ⟨(h10)x, dx⟩
to be a subgroup of Hx isomorphic to S3. Hence ⟨L1, R1⟩ = T and L1 ∩ R1 = C2.
Finally, the order four elements of AGL(1, 5) cannot be extended to automorphisms of T
since Aut(T ) = PGL(2, p) has no elements of order four normalising but not centralising
a subgroup of order five. Thus we let K = ⟨k⟩ ∼= C4 and L = L1 ⋊ K. Note, as
in [22, Example 4.1], that the action of k2 on elements of T is the same as conjugation
by d. Therefore, by Lemma 3.4, there is a locally 2-arc-transitive graph with amalgam
(AGL(1, 5)× C2, S3 × C4, C4 × C2) of straight-twisted type.
Theorem 3.6. There is an infinite family of locally 5-arc-transitive graphs with valencies
{4, 5} of straight-twisted type.
Proof. By [20], there is an amalgam admitting a locally 5-arc-transitive connected graph
of valency {4, 5} from the Mathieu group M24, with L = C42 ⋊ (A4 ×C3), R = A5 ×A4,
and L ∩ R = A4 × A4. Note that L = L1 ⋊K and R = R1 ×K where L1 = C42 ⋊ C3,
R1 = A5 and K = A4.
Let n ⩾ 2 be an integer and T = PSL(2, 22n). Then T contains a subgroup R1 ∼=
A5 ∼= PSL(2, 4) (see [6], for instance). Furthermore, T contains a subgroup Y isomorphic
to C2n2 ⋊ C22n−1, and 22n − 1 ≡ 0 (mod 3). Let Y = Y2 ⋊ Y1, where Y2 ∼= C2n2 and
Y1 = ⟨y1⟩ ∼= C22n−1. Thus Y1 has a cyclic group of order three, which we will denote
by Y3 = ⟨y(2
2n−1)/3
1 ⟩, acting semiregularly on the nonidentity elements of Y2. Moreover,
we may choose R1 such that Y0 := R1 ∩ Y ∼= A4 and Y3 ⩽ Y0. By [6, Theorem 260],
we see that NT (Y0) ⩽ Y , and, noting that Y1 acts regularly on the nonidentity elements of
Y2, we see that NT (Y0) = Y0. By [6, Theorem 255], for each divisor m of 2n, all subfield
subgroups of T isomorphic to PSL(2, 2m) are conjugate. This implies that Y0 is contained
in a unique subfield subgroup Tm isomorphic to PSL(2, 2m) for each divisor m of 2n, m
even (if m is odd, then 22 − 1 = 3 does not divide 2m − 1). Note also that this implies
that the maximal subgroup of Tm isomorphic to Cm2 ⋊ C2m−1 is actually Tm ∩ Y . We
claim that no subfield subgroup Tm containing Y0, for m a proper even divisor of 2n, also
contains Y y10 . If some Tm contains Y
y1
0 , then, since the elements of order two in Y0 and
Y y10 commute and Y0 ∩ Y
y1
0 = Y3, we have that ⟨Y0, Y
y1
0 ⟩ ⩽ Tm ∩ Y ∼= Cm2 ⋊ C2m−1,
where Tm ∩ Y1 acts regularly on the nonidentity elements of Tm ∩ Y2. However, Y1 acts
regularly on the nonidentity elements of Y2, so y1 is the unique element of Y1 mapping, say,
y2 ∈ Y0 ∩ Y2 to yy12 ∈ Y
y1
0 ∩ Y2. On the other hand, y1 ̸∈ Tm ∩ Y1 = ⟨y
(22n−1)/(2m−1)
1 ⟩,
so we have a contradiction.
M. Giudici and E. Swartz: Locally s-arc-transitive graphs arising from product action 343
Let L1 := ⟨Y0, Y ′y10 ⟩. Then L1 ∼= 24:3 (SmallGroup(48,50) in the GAP [10] small
groups library) which is isomorphic to the subgroup L1 in L, hence the abuse of notation.
Moreover, L1 ∩ R1 ∼= A4 and, since L1 is not contained in any subfield subgroup, we
have that T = ⟨L1, R1⟩. Since PΓL(2, 22n) does not contain a subgroup isomorphic to L
([6, Theorem 260] and noting that the outer automorphism group of PSL(2, 22n) is cyclic),
it follows that not all automorphisms of L1 in L extend to automorphisms of T . Hence
by Lemma 3.4, Construction 3.3 yields a locally 5-arc-transitive graph of straight-twisted
type.
3.2 Twisted-twisted type
If G acts quasiprimitively with straight PA type on a set Ω, then there exists α ∈ Ω such that
Nα = {(r, r, . . . , r) | r ∈ R}, where N = T k is the unique minimal normal subgroup of
G. If g = (t1, t2, . . . , tk) ∈ Rk ⩽ N then Nαg = (Nα)g = {(rt1 , rt2 , . . . , rtk) | r ∈ R},
which is a twisted diagonal subgroup if ti /∈ CT (R) for some i. Thus the examples given in
the previous section can also be viewed as being of twisted-twisted type. However, if G acts
quasiprimitively of type twisted PA on a set Ω then Nα is a twisted diagonal subgroup of
Rk for some R but there may not be a β ∈ Ω such that Nβ is a straight diagonal subgroup.
Thus not all twisted-twisted type examples arise in this way. In this section we give an
alternative construction.
Construction 3.7. Let (L,R,L ∩ R) be an amalgam for a locally s-arc-transitive graph,
and suppose further that L = L1 ⋊K and R = R1 ⋊K such that K = KL ×KR where
KL ⩽ Aut(L1) such that KL ∩ Inn(L1) = {1}, KL acts trivially on R1, KR ⩽ Out(R1)
and KR acts trivially on L1. Let H be an almost simple group with socle T , and subgroups
H1 and H2 such that
• H1 ∼= L1, H2 ∼= R1, H1 ∩H2 ∼= L1 ∩R1,
• H = ⟨H1, H2⟩, and
• not all elements of K extend to automorphisms of T .
We will abuse notation slightly and assume L1, R1 ⩽ H. Let k = |K| and let F =
{f : K → H} ∼= Hk. For each ℓ ∈ L1 ∪ R1, define fℓ ∈ F such that fℓ(κ) = ℓκ for all
κ ∈ K. Furthermore, we let Nα := {fℓ | ℓ ∈ L1} ∼= L1 and Nβ = {fr | r ∈ R1} ∼= R1.
Moreover, Nα ∩Nβ = {fr | r ∈ R1 ∩ L1} ∼= L1 ∩R1. Let N := ⟨Nα, Nβ⟩.
Now K acts on F via fσ(κ) = f(σκ) for each σ, κ ∈ K. As in Construction 3.3, K
normalises both Nα and Nβ , and hence also N . Define Gα := Nα ⋊K, Gβ := Nβ ⋊K
and G := ⟨Gα, Gβ⟩. Let Γ = Cos(G,Gα, Gβ).
Lemma 3.8. Let Γ be a graph yielded by Construction 3.7. Then Γ is a connected locally
(G, s)-arc-transitive graph such that G acts quasiprimitively with type PA on each orbit
on vertices. Moreover, the action of G on both [G : Gα] and [G : Gβ ] is twisted diagonal,
that is, Γ is of twisted-twisted type.
Proof. The proof is analogous to that of Lemma 3.4.
Example 3.9. First, (C71:C70 × C9, C19:C18 × C35, C630) is an amalgam that admits a
344 Ars Math. Contemp. 23 (2023) #P2.10 / 335–348
locally 2-arc-transitive graph; indeed, if G = A89,
L := ⟨(1, 2, 8, 28, 14, 30, 34, 3, 20, 54, 36, 33, 40, 41, 9, 56, 26, 51, 60, 18, 42, 29, 39, 17, 46, 58,
47, 10, 15, 70, 62, 13, 32, 59, 57, 31, 66, 22, 24, 67, 48, 27, 35, 50, 45, 12, 23, 11, 52, 4, 64,
7, 53, 25, 16, 61, 21, 44, 6, 5, 68, 71, 19, 55, 38, 69, 65, 49, 63, 43, 37),
(2, 3, 4, . . . , 71)(72, 73, . . . , 89)⟩,
and
R := ⟨(1, 72, 73, 85, 74, 88, 86, 78, 75, 80, 89, 84, 87, 77, 79, 83, 76, 82, 81),
(2, 3, 4, . . . , 71)(72, 73, . . . , 89)⟩,
then, using GAP, we see that L ∼= C71:C70 × C9, R ∼= C19:C18 × C35, L ∩ R ∼= C630,
⟨L,R⟩ = G, and by Lemma 3.1, the coset graph Cos(G,L,R) is a connected locally
(G, 2)-arc-transitive graph.
Let T = M, the Monster Group. By [4], T contains subgroups L1 ∼= D142 and R1 ∼=
D38, and L1 and R1 may be selected such that L1 ∩ R1 ∼= C2 (here, the element of order
two is of type 2B). By [36] we see that M does not have a maximal subgroup of order
divisible by 71 and 19. Thus ⟨L1, R1⟩ = T . Let K = C315 = C35 × C9, and since T
does not contain an element of order 315 [5], not all elements of K lift to an automorphism
of T . Therefore, by Lemma 3.8, Construction 3.7 yields a locally 2-arc-transitive graph Γ
with amalgam (C71:C70×C9, C19:C18×C35, C630) of twisted-twisted type with valencies
{71, 19}.
3.3 Straight-nondiagonal type
We first include an example of an equidistant linear code from [22], which proves useful
in later constructions. A linear (n,k)-code C over GF(q) is a k-dimensional subspace of
GF(q)n, a codeword has weight w if it has exactly w nonzero coordinates, and a code C is
equidistant if all nonzero codewords have the same weight.
Example 3.10 ([22, Example 5.1]). Let V = GF(3)4, and let
C = ⟨(1, 1, 1, 0), (1, 2, 0, 1)⟩ < V.
Then, C is a linear (4, 2)-code, and it contains eight nonzero code words:
(1, 1, 1, 0), (1, 2, 0, 1), (2, 0, 1, 1), (0, 2, 1, 2), (2, 2, 2, 0), (2, 1, 0, 2), (1, 0, 2, 2), (0, 1, 2, 1),
and hence C is equidistant of weight 3.
Let τ = (σ, 1, σ, σ)(1, 2, 3, 4) ∈ GL(1, 3)wrS4 < GL(V ). Then, τ4 = (σ, σ, σ, σ),
|τ | = 8, and τ permutes the eight nonzero words of C in the order given above.
Our next result constructs examples of straight-nondiagonal type.
Theorem 3.11. For each integer n ⩾ 3, there exists a locally 2-arc-transitive graph of
straight-nondiagonal type with valencies {n, 9}.
Proof. We adapt the construction of [22, Lemma 5.2]. Let H = Sn+2. Then H contains
subgroups L ∼= S2×Sn and R ∼= S3×Sn−1 such that ⟨L,R⟩ = H and L∩R ∼= S2×Sn−1
M. Giudici and E. Swartz: Locally s-arc-transitive graphs arising from product action 345
(this is realized by letting L be the stabilizer of {1, 2} and letting R be the stabilizer of
{1, 2, 3}).
Based on the equidistant linear code defined in Example 3.10, we define Nα :=
⟨(ℓ, ℓ, ℓ, ℓ) | ℓ ∈ L⟩. Moreover, if R = R1 × R2, where R1 ∼= S3, R2 ∼= Sn−1, and R1 =
⟨h, σ|h3 = σ2 = hhσ = 1⟩, we define Nβ := ⟨(h, h, h, 1), (h, h−1, 1, h), (x, x, x, x)|x ∈
⟨σ⟩×R2⟩. By choosing σ ∈ L we have Nα∩Nβ ∼= S2×Sn−1, and, as in [22, Lemma 5.2],
Nβ ∼= (C23 :C2) × Sn−1 ̸∼= R. Let N := ⟨Nα, Nβ⟩. Since ⟨L,R⟩ ∼= Sn+2 it follows that
N projects onto Sn+2 in each of its four coordinates. Moreover, given any two of the four
coordinates, Nβ contains an element that is the identity in one coordinate and a nonidentity
element of An+2 in another. Thus A4n+2 ◁ N . Note that N is not necessarily all of S
4
n+2;
indeed, the elements of Nβ that do not have all entries equal have even permutations as
their entries.
Define τ := (σ, 1, σ, σ)(1, 2, 3, 4). Then τ4 = (σ, σ, σ, σ) and so τ8 = 1. Furthermore,
τ centralizes Nα and normalises Nβ . Let Gα := ⟨Nα, τ⟩, Gβ := ⟨Nβ , τ⟩, and G :=
⟨Gα, Gβ⟩. By similar reasoning as in [22, Lemma 5.2], A4n+2 ≲ G and G induces C4
on the 4 simple direct factors. Moreover, Gβ ∼= AGL(1, 32) × Sn−1. We also see that
Gα ∼= C8 × Sn, and Gα ∩Gβ ∼= C8 × Sn−1.
Let Γ := Cos(G,Gα, Gβ). Since Gα acts on [Gα:Gα ∩ Gβ ] as Sn does on n points
and Gβ acts on [Gβ :Gα∩Gβ ] as AGL(1, 32) does on GF(32), we see that Γ is a connected
locally 2-arc-transitive graph with valencies {n, 9}. Clearly, the action of G on [G:Gα] is
straight diagonal, and the action of G on [G:Gβ ] is nondiagonal (as in [22, Lemma 5.2]).
Therefore, Γ is a locally 2-arc-transitive graph of straight-nondiagonal type with vertex
valencies {n, 9}.
3.4 Twisted-nondiagonal type
As discussed at the start of Section 3.2, the straight-nondiagonal examples given by Theo-
rem 3.11 can also be viewed as twisted-nondiagonal examples. We also have the following
construction of a graph of twisted-nondiagonal type.
Example 3.12. Let T = PSL(2, 61). By [6], T contains a maximal subgroup M ∼= D60.
Now, M contains a subgroup X isomorphic to C22 , and NT (X) ∼= A4. Now, NT (X)
contains an element g of order three that is not in M . Thus we may select subgroups
L ⩽ M and R ⩽ Mg such that L ∼= C10 ∼= C5 × C2, R ∼= C3:C2, ⟨L,R⟩ = T and
L ∩ R = X ∼= C2. Note that we may select presentations L = ⟨ℓ, x|ℓ5 = x2 = 1⟩ and
R = ⟨r, x|r3 = x2 = rrx = 1⟩.
Note that L has an isomorphism ϕ defined by ϕ : ℓ 7→ ℓ2, x 7→ x. We define ℓ :=
(ℓ, ℓϕ, ℓϕ
2
, ℓϕ
3
) = (ℓ, ℓ2, ℓ4, ℓ3) and x := (x, x, x, x). Furthermore, we define Nα :=
⟨ℓ, x⟩, Nβ := ⟨(r, r, r, 1), (r, r−1, 1, r), x⟩, and N := ⟨Nα, Nβ⟩. As in [22, Lemma 5.2],
none of the coordinates of Nβ can be linked, so N ∼= T 4. Moreover, Nα ∼= L ∼= C5 × C2,
Nβ ∼= C23 :C2 and Nα ∩Nβ ∼= C2.
Define τ := (x, 1, x, x)(1, 2, 3, 4). Then τ4 = (x, x, x, x) and so τ8 = 1. Let Gα :=
⟨Nα, τ⟩, Gβ := ⟨Nβ , τ⟩, and G := ⟨Gα, Gβ⟩. We note that τ centralizes x, whereas
ℓ
τ
= (ℓ3, ℓ, ℓ2, ℓ4) = ℓ3, and so Gα ∼= C2.AGL(1, 5). By similar reasoning as in [22,
Lemma 5.2], we deduce that G ∼= PSL(2, 61)wrC4 and Gβ ∼= AGL(1, 32). We also see
that Gα ∩Gβ ∼= C8.
Let Γ := Cos(G,Gα, Gβ). Since Gα acts on [Gα:Gα ∩ Gβ ] as AGL(1, 5) does on
GF(5) and Gβ acts on [Gβ :Gα ∩ Gβ ] as AGL(1, 32) does on GF(32), we see that Γ is a
346 Ars Math. Contemp. 23 (2023) #P2.10 / 335–348
connected locally 2-arc-transitive graph with vertex valencies {5, 9}. Clearly, the action of
G on [G:Gα] is twisted diagonal, and the action of G on [G:Gβ ] is nondiagonal (as in [22,
Lemma 5.2]). Therefore, Γ is a locally 2-arc-transitive graph of twisted-nondiagonal type
with valencies {5, 9}.
3.5 Nondiagonal-nondiagonal type
Finally, in this subsection, we include a construction of a graph of nondiagonal-nondiagonal
type.
Example 3.13. Let T = J2, the second Janko group. By [5], T has two conjugacy classes
of elements of order three, labelled 3A and 3B, and two conjugacy classes of involutions,
labelled 2A and 2B. Moreover, the elements of type 3A are contained in a maximal sub-
group isomorphic to A5×D10 which contains involutions from class 2B, and the elements
of type 3B are contained in a maximal subgroup isomorphic to A5 which also contains
involutions of type 2B. Furthermore, within each of these maximal subgroups the elements
of order three are normalized by an involution of type 2B. Using GAP, there are subgroups
L,R < T , each isomorphic to S3, such that L ∩ R ∼= C2, L contains an element of or-
der three of type 3A, R contains an element of order three of type 3B, and ⟨L,R⟩ = T.
Furthermore, by [5], the two conjugacy classes of order three are not fused by any outer au-
tomorphism of T . Let L = ⟨ℓ, x|ℓ3 = x2 = ℓℓx = 1⟩ and R = ⟨r, x|r3 = x2 = rrx = 1⟩.
We again use the equidistant linear code as defined in Example 3.10. Define Nα :=
⟨(ℓ, ℓ, ℓ, 1), (ℓ, ℓ−1, 1, ℓ), (x, x, x, x)⟩ and Nβ := ⟨(r, r, r, 1), (r, r−1, 1, r), (x, x, x, x)⟩.
Note that L ∩ R ∼= C2, and, reasoning as in [22, Lemma 5.2], we deduce that Nα ∼=
Nβ ∼= C23 :C2 ̸∼= L,R. Also, given any two of the four coordinates, both Nα and Nβ con-
tain an element that is the identity in one coordinate and a nonidentity element in another,
so N := ⟨Nα, Nβ⟩ ∼= J42 .
Define τ := (x, 1, x, x)(1, 2, 3, 4). Then τ4 = (x, x, x, x) and so τ8 = 1. Let Gα :=
⟨Nα, τ⟩, Gβ := ⟨Nβ , τ⟩, and G := ⟨Gα, Gβ⟩. By similar reasoning as in [22, Lemma 5.2],
G ∼= J2 wrC4 and Gα ∼= Gβ ∼= AGL(1, 32). We also see that Gα ∩Gβ ∼= C8.
Let Γ := Cos(G,Gα, Gβ). Since Gα (respectively Gβ) acts on [Gα:Gα ∩ Gβ ] (re-
spectively [Gβ :Gα ∩ Gβ ]) as AGL(1, 32) does on GF(32), we see that Γ is a connected
locally (G, 2)-arc-transitive graph with valencies {9, 9}. Moreover, Γ cannot be a standard
double cover of a (G, 2)-arc-transitive graph since L and R are not conjugate subgroups
in Aut(J2). Clearly, the action of G on both [G:Gα] and [G:Gβ ] is nondiagonal (as in
[22, Lemma 5.2]). Therefore, Γ is a locally (G, 2)-arc-transitive graph of nondiagonal-
nondiagonal type that is regular of valency 9.
ORCID iDs
Michael Giudici https://orcid.org/0000-0001-5412-4656
Eric Swartz https://orcid.org/0000-0002-1590-1595
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xvi
MARKO PETKOVŠEK (1955 – 2023)
Marko Petkovšek was born in 1955 and
died in 2023. He completed his PhD in
1991 at the School of Computer Science,
Carnegie Mellon University, Pittsburgh, af-
ter which he worked at the University of
Ljubljana until retirement in 2021 as a pro-
fessor, researcher, the head of the mathe-
matics department and vice-dean.
Marko Petkovšek has an outstanding
worldwide reputation in the fields of dis-
crete mathematics and theoretical comput-
er science, which he earned through his re-
search and work in the field of symbolic
computation. He is best known as a coau-
thor of the well-known book A=B and the
author of the “Hyper” algorithm for solv-
ing linear recursive difference equations
with polynomial coefficients in terms of
hypergeometric forms, nowadays called
the Petkovšek’s algorithm.
In addition to fundamental results and
publications in symbolic computation,
Marko’s work in graph theory, where he intermittently collaborated with one of us over
a period of several decades, also contributes to his visibility. Let us say a little more about
his work in this area. He explored various classes of perfect graphs, graphs with non-
empty intersections of longest paths, hereditary graph classes, Fibonacci and Lucas cubes,
and attacked several related problems. One of the first challenges suggested by Marko was
the problem of the intersection of longest paths in graphs. We wrote a joint paper that
went mostly unnoticed for a quarter of a century, only to receive wide attention in the past
decade. Marko’s mathematical breadth was extremely helpful in the treatment of various
problems, as it often led to unexpected insights. Of this kind were his contributions to the
enumeration of the vertex and edge orbits of Fibonacci cubes and Lucas cubes. Marko’s
work also established new directions of development. In his paper [Marko Petkovšek, Let-
ter graphs and well-quasi-order by induced subgraphs, Discrete Mathematics 244 (2002)
375–388] he introduced the notion of letter graphs and proved that the class of k-letter
graphs is well-quasi-ordered by the induced subgraph relation, and that it has only finitely
many minimal forbidden induced subgraphs. This visionary paper preceded developments
in the field by a decade, and is today recognized as a fundamental reference on the topic.
Let us finish with a few personal thoughts about Marko. Our deep and unbroken friend-
ship began more than 30 years ago. One of us was lucky to share an office with Marko as
a freshly minted assistant, and the other as his student. He introduced us both to the world
of research and transferred his enthusiasm for it to us. He was the best possible friend.
Despite his broad mathematical knowledge and depth of thought, he was extremely mod-
est and downplayed his strengths and contributions. His was always pleasant and soothing
xvii
company, be it on mountain trails, Saturday evening bridge sessions, or just conversations
at and around work. In addition to mathematics, he had a broad general outlook. He held
himself to the highest ethical and moral standards, inspiring others to do the same. And on
mountain hikes he could always name every flower we saw in his mother tongue Slovene,
German, and Latin.
Unfortunately, Marko left us too soon. We shall always remember the beautiful mo-
ments we spent with him and keep him in our memories as a truly exceptional man.
Andrej Bauer and Sandi Klavžar
xviii

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