Nicolas Bougard Regulargraphsandconvexpolyhedrawithprescribednumbersoforbits

Universit´ e Libre de Bruxelles Facult´ e des Sciences

D´ epartement de Math´ ematiques

Regular graphs and convex polyhedra with prescribed numbers of orbits

Nicolas Bougard

Th` ese pr´ esent´ ee en vue de l’obtention du grade de docteur en sciences

Promoteur: Jean Doyen Juin 2007

Contents

Contents 3

Remerciements 5

Introduction 7

1 Regular graphs 15

1.1 Definitions . . . . 15

1.2 Connected graphs . . . . 15

1.2.1 Necessary condition . . . . 15

1.2.2 Preliminary results . . . . 15

1.2.3 1st step: ν − 1 ≤ ≤ 2ν + 1 . . . . 18

1.2.4 2nd step: k = 4 . . . . 20

1.2.5 3rd step: k = 5 . . . . 21

1.2.6 4th step: k ≥ 6 . . . . 22

1.2.7 Proof of Theorem 0.2 . . . . 24

1.3 Non connected graphs . . . . 24

1.3.1 Vertex stabilizer in a p-regular graph realizing (1, 1) . . . . 24

1.3.2 Covering graph . . . . 26

1.3.3 Countably many k-regular graphs realize (2, 1) . . . . 27

1.3.4 Countably many k-regular graphs realize (1, k) . . . . 35

1.3.5 Proof of Theorem 0.3 . . . . 36

2 Convex polyhedra 41 2.1 The sets FS (G) = F (G) . . . . 41

2.1.1 Preliminaries . . . . 41

2.1.2 Lemmas . . . . 43

2.1.3 Corollaries of Theorem 0.4 . . . . 46

2.1.4 The group C

C

. . . . 61

2.1.5 The groups D

C

, D

D

, D

and T . . . . 72

2.1.6 The isometry groups of prisms, antiprisms, platonic solids and the groups D

C

. . . 105

2.2 The sets F

(G) and FS

(G) . . . 122

2.2.1 The sets FS

(G) . . . 122

2.2.2 The sets F

(G) . . . 136

2.3 The sets F

and FS

. . . 141

2.4 Planar graphs . . . 143

Bibliography 145

3

Remerciements

Tout d’abord, je tiens ` a exprimer ma profonde gratitude envers mon promoteur Jean Doyen pour le temps qu’il m’a consacr´ e tout au long de cette th` ese : ses encouragements, sa patience et la pertinence de ses remarques ont ´ et´ e pour moi d’un grand secours. Merci de m’avoir remis les yeux en face des orbites quand il le fallait.

Richard Weiss (Tufts University, Boston) et Vladimir Trofimov (Institute of Mathematics and Mechanics, Ekaterinburg) m´ eritent ma reconnaissance : leur disponibilit´ e n’a eu d’´ egal que la valeur de leurs r´ esultats. Notre correspondance fut d’une aide cruciale.

Mes coll` egues et amis de l’ULB ont rendu la r´ ealisation de cette th` ese tr` es agr´ eable : un grand merci pour l’ambiance chaleureuse qui r` egne dans notre d´ epartement et pour votre joyeuse compagnie; Gwena¨ el, tout sp´ ecialement, pour ses divertissements math´ ematiques toujours int´ eressants.

Toute ma reconnaissance va ´ egalement ` a Philippe Lumen et G´ erald Troessaert, leur flex- ibilit´ e m’a permis de clˆ oturer cette th` ese dans un d´ elai raisonnable.

Je remercie le Fonds pour la formation ` a la Recherche dans l’Industrie et dans l’Agricul- ture (FRIA) pour la confiance et le financement qu’il m’a octroy´ es.

Ma th` ese n’aurait jamais ´ et´ e possible sans l’amour et le soutien de mes proches. Je saisis cette occasion pour manifester ma chaleureuse gratitude ` a mes parents, Mich` ele et Claudy,

`

a mes soeurs, B´ er´ enice et Charlotte, ` a Nicole, Robert, Florian sans oublier Lidwine pour sa patiente relecture. Enfin, je sors indemne de cette th` ese, en math´ ematiques de surcroˆıt, grˆ ace

`

a l’amour de mon ´ epouse, C´ elia et de mes enfants, Syrielle et Jolan qui m’ont aid´ e ` a ne pas devenir un ”chercheur fou”. Merci ´ egalement pour la douceur et le bonheur quotidien que vous faites vivre au sein de notre foyer.

5

A mes enfants,

avec tout mon amour.

Introduction

Automorphism groups are a powerful tool for investigating mathematical structures. In particular, the automorphism group Aut( S ) of any mathematical structure S partitions S into orbits consisting of all the elements of S having ”the same properties”. For example, if the structure S is a finite group G, we can compute the number ω(G) of orbits of Aut(G) on the elements of G. Conversely, given a positive integer ω, is there a finite group G such that ω(G) = ω? The answer is yes for every integer ω > 0. Indeed, as shown in [9], two elements of the cyclic group Z

(p an odd prime and e a positive integer) are in the same orbit of the automorphism group Aut( Z

) if and only if they have the same order. Hence ω( Z

) = ω.

The mathematical structures considered in this thesis will always be finite incidence structures. An incidence structure S of rank d consists of d pairwise disjoint non empty set X

, . . . , X

, together with incidence relations I

between X

and X

(0 ≤ i < j < d). The elements of X

are said to be of type i. We say that the structure S is finite if all the X

’s are finite. An easy example of an incidence structure of rank 2 is a finite graph (undirected, without loops and multiple edges), the two types of elements being vertices and edges. An example of an incidence structure of rank 3 is a convex polyhedron, where the 3 types of elements are vertices, edges and faces and where incidence is the inclusion. Given a class C of finite incidence structures, we will be interested in the following Problem (P): for which d-tuples (ω

, ω

, . . . , ω

) of integers does there exist S ∈ C such that Aut( S ) has exactly ω

orbits on the set X

for each i ∈ { 0, . . . , d − 1 } ?

A linear space is an incidence structure of rank 2, consisting of points and lines in such a way that each pair of points is incident with exactly one line and each line with at least two points. Problem (P ) for the class of finite linear spaces was solved in 1987 by Blokhuis, Brouwer, Delandtsheer and Doyen [1]: there is a finite linear space L such that Aut( L ) has exactly π orbits on the set of points and λ orbits on the set of lines if and only if 1 ≤ π ≤ λ.

They proved a similar result for quasilinear spaces (each pair of points is incident to at most one line): there is a finite quasilinear space L such that Aut( L ) has exactly π orbits on the set of points and λ orbits on the set of lines for every pair (π, λ) of positive integers. Given an integer k ≥ 3, one would like to solve Problem (P ) for the class of finite linear spaces where all lines have exactly k points, that is for the class of 2 − (v, k, 1) designs (where k is fixed but the number v of points is not fixed). This is still an open problem. Partial results were obtained for k = 3 (i.e. Steiner triple systems) by Saxl [22]: he proved that if the automorphism group of a Steiner triple system has π orbits on the set of points and π orbits on the set of lines, then π ≤ 3.

A hypergraph H is an incidence structure of rank 2, consisting of points and blocks, the blocks being non empty sets of points and incidence being the inclusion. An edge of H is any set of two points contained in a block of H . The underlying graph of H is the graph G ( H ) whose vertices are the points of H and whose edges are those of H . H is connected iff G ( H ) is connected; H is k-uniform iff all the blocks of H have the same size k ≥ 2.

Let H

(resp. H

) be the set of all pairs (π, β) of integers for which there exists a finite

7

8 INTRODUCTION k-uniform hypergraph (resp. a finite connected k-uniform hypergraph) whose automorphism group has exactly π orbits on the set of points and β orbits on the set of blocks.

Theorem 0.1 (Delandtsheer [8]).

(π, β ) ∈ H

⇐⇒ 1 ≤ π ≤ kβ + 1, (π, β ) ∈ H

⇐⇒ 1 ≤ π ≤ (k − 1)β + 1.

The particular case k = 2 (i.e. the case of graphs) had been considered earlier by Buset [6].

Chapter 1 of this thesis is concerned with Problem (P ) in certain classes of graphs (con- sidered as incidence structures of rank 2, the two types of elements being vertices and edges).

In other words, they are finite undirected, without loops and multiple edges.

Let G be a graph. If Aut( G ) acts transitively on the set of vertices of G (resp. on the set of edges of G ), we say that G is vertex-transitive (resp. edge-transitive). We will say that a graph G realizes a given pair (ν, ) of integers if Aut( G ) has exactly ν orbits on the set of vertices of G and orbits on the set of edges of G .

A graph is k-regular if all its vertices have the same degree k. Let G

(resp. G

) be the set of all pairs (ν, ) of integers for which there exists a finite k-regular graph (resp. a finite connected k-regular graph) which realizes (ν, ). We will prove the two following theorems.

Theorem 0.2 (Bougard [3]).

1. (ν, ) ∈ G

⇐⇒ (ν, ) = (1, 0), 2. (ν, ) ∈ G

⇐⇒ (ν, ) = (1, 1), 3. (ν, ) ∈ G

⇐⇒ (ν, ) = (1, 1),

4. (ν, ) ∈ G

(k ≥ 3) ⇐⇒ ν − 1 ≤ ≤ (k − 1)ν + 1, ν ≥ 1 and ≥ 1.

Theorem 0.3 (Bougard [2]).

1. (ν, ) ∈ G

⇐⇒ (ν, ) = (1, 0), 2. (ν, ) ∈ G

⇐⇒ (ν, ) = (1, 1), 3. (ν, ) ∈ G

⇐⇒ ν = ≥ 1,

4. (ν, ) ∈ G

(k ≥ 3) ⇐⇒ 1 ≤ ν ≤ 2 ≤ 2kν.

These results are completely trivial when k ≤ 2 because a connected 0-regular graph is just an isolated vertex, a connected 1-regular graph is an isolated edge and a finite connected 2-regular graph is a cycle. In the case of k-regular graphs, it is not so difficult to prove that the conditions given in Theorem 0.2 and 0.3 are necessary. Indeed the inequalities 0 ≤ ν − 1 ≤ (for G

) and 1 ≤ ν ≤ 2 (for G

) are a consequence of Theorem 0.1. Let G be a finite k-regular graph G realizing (ν, ). Since each vertex of G belongs to exactly k edges, each vertex-orbit of Aut( G ) gives rise to at most k edge-orbits, and so ≤ kν . Moreover, the hypergraph H whose points are the edges of G and whose blocks are the sets of k edges containing a given vertex is k-uniform and Aut( H ) has orbits on points and ν orbits on blocks. By Theorem 0.1, we get 1 ≤ ≤ (k − 1)ν + 1 if G (and H ) is connected. The real difficulty is to prove that these conditions are sufficient when k ≥ 3. We will successively prove:

(a) Theorem 0.2 for k = 3 and 4,

INTRODUCTION 9 (b) (ν, ) ∈ G

when ν − 1 ≤ ≤ 2ν + 1 and k ≥ 5,

(c) (ν, ) ∈ G

when (k − 2)ν + 2 ≤ ≤ (k − 1)ν + 1 and k ≥ 5,

(d) Theorem 0.2 by induction on k ≥ 3 using (a), (b), (c) and the fact that (ν, ) ∈ G

implies (ν, ν + ) ∈ G

.

We would like to point out that all our proofs will be constructive: we will explicitly exhibit a connected k-regular graph G for which Aut( G ) has exactly ν vertex-orbits and edge-orbits for every pair (ν, ) such that ν − 1 ≤ ≤ (k − 1)ν + 1, ν ≥ 1 and ≥ 1.

In order to prove that the conditions given in Theorem 0.3 are sufficient, we first note that Theorem 0.2 gives the answer when ν − 1 ≤ ≤ (k − 1)ν + 1. A k-regular graph having ν pairwise non isomorphic components, each of them realizing (1, k), realizes (ν, kν).

If (k − 1)ν + 2 ≤ < kν, then (ν, ) is realized by a k-regular graph having − (k − 1)ν + 1 pairwise non isomorphic components, one of them realizing (kν − , (kν − )(k − 1)) (such a graph exists by Theorem 0.2) and the other ones realizing (1, k). Moreover, (2, ) is realized by a k-regular graph having pairwise non isomorphic components, each of them realizing (2, 1). When + 2 ≤ ν < 2, it is sufficient to take a k-regular graph having ν − + 1 pairwise non isomorphic components, one of them realizing (2 − ν, 2 − ν) (such a graph exists by Theorem 0.2) and the other ones realizing (2, 1). These constructions are sufficient to prove Theorem 0.3 but they work only if there are countably many non isomorphic k-regular graphs realizing (1, k) and countably many non isomorphic k-regular graphs realizing (2, 1) (that is edge- but not vertex-transitive k-regular graphs). Section 1.3 is essentially devoted to the most difficult part of the proof, namely the construction of edge- but not vertex-transitive regular graphs (such graphs are usually called semisymmetric graphs).

This problem was first studied by Folkman [11] in 1967: he was able to construct sev- eral infinite families of semisymmetric graphs. However the degree of the vertices of all semisymmetric graphs constructed by Folkman is never a prime. He concluded his paper by the following open problem: ”Is there a semisymmetric graph whose vertices have a prime degree?”. The answer was given by Bouwer who proves that the Gray graph (discovered by Marion C. Gray in 1932) is 3-regular and semisymmetric [4] and that there is a k-regular semisymmetric graph for every k ≥ 3 [5]. But as far as we know, the question whether there are countably many k-regular semisymmetric graphs, where k ≥ 5 is a fixed prime, was still open. In the case k = 3, many examples are known (see [15], [16], [17], [20] or [21]). In this thesis, we will construct countably many non isomorphic k-regular graphs realizing (2, 1) for every odd prime k. Our proof is based on the following fact (proved in Section 1.3.1):

Let G be a finite connected p-regular graph for some odd prime p and let x be a vertex of

G . We denote by Aut( G )

the stabilizer of x in Aut( G ). If G is vertex- and edge-transitive,

then | Aut( G )

| ≤ p!((p − 1)!)

. This is not the best possible upper bound for | A

|

(for example, Tutte [29] found a better one in the case of 3-regular graphs). However, the

important fact is that this bound on | Aut( G )

| does not depend on the number of vertices of

G but only on p. We use this result to prove that our k-regular graphs cannot be vertex- and

edge-transitive if the number of vertices is large enough. Note that deep group-theoretical

theorems, proved by Weiss [32] in 1981 and Trofimov [28] in 2003, are crucial ingredients in

our proof of this result.

10 INTRODUCTION Chapter 2 of this thesis is concerned with Problem (P ) in the class of polyhedra (con- sidered as incidence structures of rank 3, the 3 types of elements being vertices, edges and faces of convex polyhedra in Euclidean 3-space R

). More generally, we prove a result which is true for all convex polyhedra viewed as sets of points in R

. In what follows, the word polyhedron will always refer to a convex polyhedron in R

.

In Chapter 2, it will be convenient to say that two groups G and H are isomorphic (written G ∼ = H) if they are conjugate subgroups of the group Isom( R

) of all isometries of R

. Moreover, every group will be supposed to be a finite group of isometries of R

.

Let P be a polyhedron. We denote by Isom( P ) the group consisting of all isometries of R

leaving P invariant (in this thesis, such isometries will be called isometries of P and Isom( P ) the isometry group of P ). Let G be a subgroup of Isom( P ). The f

-vector of P is the vector f

( P ) = (ν, , ϕ) where ν, and ϕ denote the number of vertex-, edge- and face- orbits of G respectively. The f

-vector f

(P) and the f

-vector f

(P) are simply called the f -vector and the f

-vector of P and are denoted by f( P ) and f

( P ) respectively (1 denotes the identity as well as the identity group). For any group G, we define F (G) as the set of all triples (ν, , ϕ) of integers for which there exists a polyhedron P and a subgroup H of Isom( P ) such that G ∼ = H and f

( P ) = (ν, , ϕ). When G = 1, we simply write F for F (1), namely the set of all triples (ν, , ϕ) for which there is a polyhedron having exactly ν vertices, edges and ϕ faces.

Theorem 0.4 (Steinitz [24], see also [13], p.190).

(ν, , ϕ) ∈ F ⇐⇒

⎧ ⎨

⎩

(i) 4 ≤ ν ≤ 2ϕ − 4, (ii) 4 ≤ ϕ ≤ 2ν − 4,

(iii) ν − + ϕ = 2 (Euler’s formula).

The 1-skeleton G ( P ) of a polyhedron P is the graph whose vertices are those of P , two vertices being adjacent if and only if they are joined by an edge of P . It is well- known that a finite graph is the 1-skeleton of a polyhedron if and only if it is planar and 3-connected (Steinitz [25] and Steinitz and Rademacher [26]). Clearly, each isometry of P induces an automorphism of G(P ). The converse is not always true. For example, consider a tetrahedron T having as 1-skeleton the complete graph on 4 vertices. If T is regular, then every automorphism of G ( T ) induces an isometry of T . But if T is rigid (i.e. Isom( T ) is reduced to the identity), then the identity automorphism of G(T ) is the only automorphism of G ( T ) which induces an isometry of T . However, Mani proved in 1971 the following remarkable theorem:

Theorem 0.5 (Mani [18]). Given a finite 3-connected planar graph G , there exists a poly- hedron P with 1-skeleton G and such that every automorphism of G induces an isometry of P .

A polyhedron P such that each automorphism of the 1-skeleton G ( P ) induces an isometry of P will be called full symmetric. For any group G, we define FS (G) as the set of all triples (ν, , ϕ) of integers for which there exists a full symmetric polyhedron P and a subgroup H of Isom( P ) such that G ∼ = H and f

( P ) = (ν, , ϕ).

Let G be a group. A direct consequence of Theorem 0.5 is that (ν, , ϕ) ∈ F (G) if and only

if there is a polyhedron P and a subgroup H of Isom( P ) such that G and H are isomorphic,

f

( P ) = (ν, , ϕ) and every automorphism of G ( P ) induces an isometry of P . Therefore,

F (G) = FS (G) for every group G. In order to determine the sets F (G) = FS (G), we will

use the classification of all finite groups G of isometries of R

.

INTRODUCTION 11 Let C

be the cyclic rotation group of order n (generated by a rotation of 360

/n) and D

the dihedral rotation group of order 2n (generated by a rotation of 360

/n and an involutory rotation with perpendicular axis). We denote by T , O and I the rotation group of a regular tetrahedron, a regular octahedron and a regular icosahedron respectively. It is well-known that any finite group of isometries of R

has a fixed point c. We denote by σ

the inversion about c defined by σ

(p) = q if and only if c is the midpoint of p and q. If H is a rotation group with elements α

, α

, . . . , α

fixing a point c, then H is the group whose elements are α

, α

, . . . , α

, σ

α

, σ

α

, . . . , σ

α

. Note that O (resp. I ) is the isometry group of a regular octahedron (resp. icosahedron). Suppose that there is a rotation group K having H as a subgroup and containing the additional rotations γ

, γ

, . . . , γ

not in H. Let c be a fixed point of K. Then KH denotes the group of order 2h whose elements are α

, α

, . . . , α

, σ

γ

, σ

γ

, . . . , σ

γ

. The group OT is the isometry group of a regular tetrahedron.

Theorem 0.6 (Hessel [14], see also [19], Theorem 17.11). Any finite group of isometries of R

is isomorphic to one of the following:

C

, C

, C

C

, (n = 1, 2, . . . ) D

, D

, D

D

, D

C

, (n = 2, 3, . . . ) T, O, I, T , O, I, OT.

Chapter 2 is essentially devoted to the characterization of the sets F (G) = FS (G) for all groups G.

Solving Problem (P) requires the characterization of other sets. For any group G, let F

(G) (resp. FS

(G)) be the set of all triples (ν, , ϕ) of integers for which there exists a polyhedron (resp. a full symmetric polyhedron) P such that G ∼ = Isom( P ) and f

( P ) = (ν, , ϕ). In section 2.2, we investigate these sets.

In section 2.3, we completely describe the sets F

and FS

, where F

(resp. FS

) is the set of all triples (ν, , ϕ) of integers for which there exists a polyhedron (resp. a full symmetric polyhedron) P such that f

(P) = (ν, , ϕ). Of course,

F

=

G

F

(G) ⊆

G

F (G) and FS

=

G

FS

(G) ⊆

G

FS (G).

These properties are used to find conditions satisfied by the triples (ν, , ϕ) ∈ F

or FS

, without knowing all the sets F

(G) and FS

(G) but only the sets F (G) = FS (G). Our main results are as follows:

Theorem 0.7.

(ν, , ϕ) ∈ FS

⇐⇒

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

(i) 1 ≤ ν ≤ 2ϕ + 1, (ii) 1 ≤ ϕ ≤ 2ν + 1,

(iii)(a) ν − 1 ≤ ≤ ν + ϕ − 1, if ν = 2ϕ + 1, (b) ν − 1 ≤ ≤ ν + ϕ, if ν = 2ϕ, (c) ν − 1 ≤ ≤ ν + ϕ + 1, if ϕ < ν < 2ϕ, (d) ν = ϕ ≤ ≤ ν + ϕ + 1, if ν = ϕ, (e) ϕ − 1 ≤ ≤ ν + ϕ + 1, if ν < ϕ < 2ν, (f ) ϕ − 1 ≤ ≤ ν + ϕ, if ϕ = 2ν, (g) ϕ − 1 ≤ ≤ ν + ϕ − 1, if ϕ = 2ν + 1,

(iv) (ν, , ϕ) = (1, 2, 1), (1, 3, 1),(1, 3, 2), (2, 3, 1), (2, 5, 2),

(2, 6, 3) and (3, 6, 2).

12 INTRODUCTION Theorem 0.8.

(ν, , ϕ) ∈ F

⇐⇒

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

(i) 1 ≤ ν ≤ 2ϕ + 1, (ii) 1 ≤ ϕ ≤ 2ν + 1,

(iii)(a) ν − 1 ≤ ≤ ν + ϕ − 1, if ν = 2ϕ + 1, (b) ν − 1 ≤ ≤ ν + ϕ, if ν = 2ϕ, (c) ν − 1 ≤ ≤ ν + ϕ + 1, if ϕ < ν < 2ϕ, (d) ν = ϕ ≤ ≤ ν + ϕ + 1, if ν = ϕ, (e) ϕ − 1 ≤ ≤ ν + ϕ + 1, if ν < ϕ < 2ν, (f ) ϕ − 1 ≤ ≤ ν + ϕ, if ϕ = 2ν, (g) ϕ − 1 ≤ ≤ ν + ϕ − 1, if ϕ = 2ν + 1.

Here is another consequence of Theorem 0.5: a full symmetric polyhedron P is such that f

( P ) = (ν, , ϕ) if and only if P (viewed as a rank 3 incidence structure S ) is such that Aut( S ) has ν orbits on the set of vertices X

, orbits on the set of edges X

and ϕ orbits on the set of faces X

. Therefore, the set FS

solves Problem (P ) for the class of polyhedra viewed as incidence structures.

We will end Chapter 2 by applying our results to finite planar graphs:

Theorem 0.9. (ν, ) is realized by a finite 3-connected planar graph if and only if ν − 1 ≤ ≤ 3ν and ≥ 1.

We could also try to solve Problem (P ) for the class of d-polytopes (i.e. of convex polytopes in the Euclidean space R

). A d-polytope P is an incidence structure of rank d where X

is the set of i-facets (the facets of dimension i) of P and where incidence is the inclusion between the facets. This problem is much more difficult when d ≥ 4. To see this, consider the following easier problem: given d, determine all the d-tuples (ω

, ω

, . . . , ω

) for which there exists a d-polytope whose number of i-facets is ω

. We have seen that the case d = 3 has been solved by Steinitz with Theorem 0.4, but the cases d ≥ 4 are still unsolved!

(see [13] or [33] for a survey of partial known results).

Chapter 1

Regular graphs

In this thesis, all graphs are supposed to be finite, undirected, without loops and multiple edges.

1.1 Definitions

We begin by clarifying some definitions.

We denote by G (x) the neighborhood of x in G . Given an integer s ≥ 0, a walk W of length s from x

to x

is an (s + 1)-tuple (x

, x

, . . . , x

) of vertices of G with x

∈ G (x

) if 1 ≤ i ≤ s. Apath P of length s from x

to x

is a walk (x

, x

, . . . , x

) for which x

= x

for every 0 ≤ i < j ≤ s. If x

= x

, we say that P is a circuit at base point x

= x

.

Given two vertices v, w of G, we denote by d(v, w) the length of the shortest path from v to w. In a graph, a circuit of length n will be called an n-circuit. A connected 2-regular graph is called a cycle that is a graph having one component which is a circuit.

If x and y are two vertices or two edges of a graph G and if x and y are in the same orbit of Aut( G ), we will say that x and y are G -equivalent and we will write x ≡

y.

1.2 Connected graphs

1.2.1 Necessary condition

Proposition 1.1.

(ν, ) ∈ G

(k ≥ 3) = ⇒ ν − 1 ≤ ≤ (k − 1)ν + 1, ν ≥ 1 and ≥ 1.

Proof. If (ν, ) ∈ G

, then there exists a connected k-regular graph G which realizes (ν, ).

Since G is connected, Theorem 0.1 (for k = 2) implies that 1 ≤ ν ≤ + 1.

Let H be the hypergraph whose points are the edges of G and whose blocks are the sets of k edges containing a given vertex. H is k-uniform and Aut( H ) has orbits on points and ν orbits on blocks. By Theorem 0.1, 1 ≤ ≤ (k − 1)ν + 1.

1.2.2 Preliminary results

Proposition 1.2. For any integer k ≥ 2, (i) (ν, ) ∈ G

= ⇒

ν

, ν +

ν(ν + 1)

∈ G

;

15

16 CHAPTER 1. REGULAR GRAPHS (ii) if there exists a connected k-regular graph on n vertices realizing (ν, ), in which each edge belongs to at most n circuits of length m (m < 8), then there exists a connected (k + 1)-regular graph on 2n

vertices realizing

ν

, ν +

ν(ν + 1)

, in which each edge belongs to at most n circuits of length m.

Proof. (i) Let G be a connected k-regular graph realizing (ν, ) and let n be the number of vertices of G . We may identify the vertices of G with the elements of Z

. We will construct a connected (k + 1)-regular graph G

realizing

ν

, ν +

ν(ν + 1)

. The set of vertices of G

will be V = Z

× Z

, and the set of edges will be E = E

∪ E

∪ E

, where

1. E

= {{ (i, j, 0), (i, l, 0) }|{ j, l } is an edge of G , i, j, l ∈ Z

} ; 2. E

= {{ (i, j, 1), (l, j, 1) }|{ i, l } is an edge of G , i, j, l ∈ Z

} ; 3. E

= {{ (i, j, 0), (i, j, 1) }| (i, j) ∈ Z

} .

Clearly, since G is connected and k-regular, G

is also connected and all its vertices have degree k + 1.

We prove first that no edge e

= { (a, b, 0), (a, c, 0) } ∈ E

is in the same orbit of Aut( G

) as an edge e

= { (d, e, 0), (d, e, 1) } ∈ E

. It is easy to check that e

is not included in a circuit of length less than 8. Note also that all the 8-circuits containing e

have the following form:

(d, e, 0), (d, e, 1), (d

, e, 1), (d

, e, 0), (d

, e

, 0), (d

, e

, 1), (d, e

, 1), (d, e

, 0), (d, e, 0) where { d, d

} and { e, e

} are edges of G . There are k

such 8-circuits. Moreover, for any two edges e

, e

such that |e

∩ e

| = |e

∩ e

| = 1 and e

∩ e

= ∅, there is only one 8-circuit containing e

, e

and e

. This property is not true for the edge e

. Indeed consider the two edges e

= { (a, b, 0), (a, b, 1) } and e

= { (a, c, 0), (a, c, 1) } of G

: we have e

∩ e

= ∅ , but there are k circuits of length 8 containing e

, e

et e

, namely

(a, b, 0), (a, c, 0), (a, c, 1), (a

, c, 1), (a

, c, 0), (a

, b, 0), (a

, b, 1), (a, b, 1), (a, b, 0)

where {a, a

} is an edge of G. Since k ≥ 2, our assertion is proved. A similar argument shows that no edge e

∈ E

is in the same orbit of Aut( G

) as an edge e

∈ E

.

If ρ is an automorphism of G , the permutations α(ρ) and β(ρ) defined by α(ρ)(i, j, l) = (ρ(i), j, l)

β(ρ)(i, j, l) = (i, ρ(j), l)

are clearly automorphisms of G

. Note also that γ : G

→ G

: (i, j, l) → (j, i, l + 1) is an automorphism of G

.

We will say that the G -edge of { (a, b, c), (d, e, c) } ∈ E

∪ E

is { a, d } (if b = e) or { b, e } (if a = d) and that its G -vertex is b (if b = e) or a (if a = d). Using the automorphisms defined above, it is easy to prove that two edges {(a, b, c), (d, e, c)} and {(a

, b

, c

), (d

, e

, c

)}

of E

∪ E

are G

-equivalent iff their G -edges and their G -vertices are both G -equivalent. For

example, if there is an automorphism σ of G

mapping the edge { (a, b, 1), (d, b, 1) } on the edge

{(a

, b

, 1), (d

, b

, 1)}, then σ maps the set V

= {(i, b, 1)|i ∈ Z

} on the set {(i, b

, 1)|i ∈ Z

}

and the set V

= { (a, i, 0) | i ∈ Z

} on either { (a

, i, 0) | i ∈ Z

} or { (d

, i, 0) | i ∈ Z

} because

the edges of E

∪ E

and those of E

are not in the same edge-orbit of Aut( G

). Thus the

restrictions of σ to V

and to V

induce two automorphisms of G mapping respectively { a, d }

on { a

, d

} and b on b

. On the other hand, if there are two automorphisms ρ

, ρ

of G such

1.2. CONNECTED GRAPHS 17

Figure 1.1: The 1-skeleton of the great rhombicuboctahedron.

that ρ

( { a, d } ) = { a

, d

} and ρ

(b) = b

, then the automorphism α(ρ

) ◦ β(ρ

) maps the edge { (a, b, 1), (d, b, 1) } on the edge { (a

, b

, 1), (d

, b

, 1) } . Other cases are done in a similar way.

Therefore, to each ordered pair consisting of an edge-orbit and a vertex-orbit of Aut( G ), there corresponds an edge-orbit of Aut(G

); this gives rise to ν edge-orbits of Aut(G

).

It remains to check that two edges { (a, b, 0), (a, b, 1) } and { (a

, b

, 0), (a

, b

, 1) } of E

are G

-equivalent iff either a ≡

a

and b ≡

b

, or b ≡

a

and a ≡

b

; this gives rise to

ν(ν + 1) edge-orbits of Aut( G

).

We conclude that the number of edge-orbits of Aut( G

) is ν +

ν(ν + 1).

It is easy to check that (a, b, c)

≡

(d, e, f ) iff a ≡

d and b ≡

e when c = f (resp. a ≡

e and b ≡

d when c = f ). It follows that the number of vertex-orbits of Aut( G

) is ν

.

(ii) describe straightforward properties of G

.

Proposition 1.3. For any integer k ≥ 3, (1, k) ∈ G

. Moreover, there exists a connected k-regular graph realizing (1, k) in which each edge belongs to at most one 4-circuit.

Proof. We proceed by induction on k. If k = 3, the 1-skeleton of the great rhombicuboc- tahedron (see figure 1.1) has the desired properties. If k ≥ 3, suppose that there exists a connected k-regular graph realizing (1, k) and in which each edge belongs to at most one 4-circuit. By Proposition 1.2, (1, k + 1) ∈ G

and there exists a connected (k + 1)-regular graph realizing (1, k + 1), in which each edge belongs to at most one 4-circuit.

Proposition 1.4. (ν, ) ∈ G

= ⇒ (ν, ν + ) ∈ G

.

Proof. Let G be a connected k-regular graph realizing (ν, ) and let n be the number of

vertices of G (considered as elements of Z

). Let m be an odd integer > n. We will construct

a connected (k + 2)-regular graph G

realizing (ν, + ν). The set of vertices of G

will be

V = Z

× Z

, the edges of G

being defined as follows:

18 CHAPTER 1. REGULAR GRAPHS 1. { (i, j), (l, j) } is an edge of G

iff { i, l } is an edge of G (where i, l ∈ Z

, j ∈ Z

).

2. { (i, j), (i, j + 1) } is an edge of G

for every (i, j) ∈ V .

G

is clearly connected and (k + 2)-regular. Note that the shortest path from a vertex (a, b) to a vertex (c, b) is included in the set { (i, b) | i ∈ Z

} . Indeed if we replace by b the second component of every vertex of a path from (a, b) to (c, b), we get a shortest path linking these two vertices.

No edge of G

of the form { (a, b), (c, b) } is in the same orbit of Aut( G

) as an edge of the form { (d, e), (d, e + 1) } , otherwise { (a, b), (c, b) } would be included in an m-circuit C.

Moreover, the shortest path between two vertices of C is included in C (because { (d, e), (d, e+

1) } has this property), and this path is unique because m is odd. Since G has n vertices and C has m > n edges, C contains two edges { (f, g), (f, g + 1) } and { (h, g), (h, g + 1) } . Let P be the shortest path between (f, g) and (h, g), which is included in C and in { (i, g) | i ∈ Z

} . If we add 1 (modulo m) to the second component of all vertices of P , we get the shortest path from (f, g + 1) to (h, g + 1). This path is also included in C, and so the length of C is even, a contradiction.

Clearly g : (i, j) → (i, j + 1) is an automorphism of G

. It is not difficult to check that (i, j)

≡

(i

, j

) iff i ≡

i

. Therefore Aut( G

) has ν vertex-orbits and it is easy to prove that Aut( G

) has ν + edge-orbits.

1.2.3 1st step: ν − 1 ≤ ≤ 2 ν + 1

The results of this section were essentially proved by Schmitz [23].

Lemma 1.5 (Bouwer [5]). For any integer k ≥ 3, (2, 1) ∈ G

.

Proposition 1.6. If ≥ 1, then ( + 1, ) ∈ G

for every integer k ≥ 3.

Proof. By Lemma 1.5, we may assume that ≥ 2. Let K = { k, k + 1, . . . , 2k − 2 } . We will construct a connected k-regular graph G realizing ( + 1, ). The set of vertices of G will be V

∪ V

∪ · · · ∪ V

, where V

= Z

, V

= Z

× Z

(2 ≤ i ≤ ) and V

= Z

× Z

× K , and the set of edges of G will be E

∪ E

∪ · · · ∪ E

, where

E

= {{ (i

, . . . , i

), (i

, . . . , i

, i

) }| (i

, . . . , i

, i

) ∈ Z

× Z

} (1 ≤ j ≤ − 1) and E

= {{ (i

, i

, . . . , i

), (i

, . . . , i

, j) }| (i

, i

, . . . , i

) ∈ Z

× Z

, j ∈ K} .

It is easily checked that each V

(resp. each E

) is a vertex-orbit (resp. an edge-orbit) of Aut( G ) (note for example that E

is an edge-orbit because the vertices of V

∪ V

are the only ones belonging to a 4-circuit). Therefore, G realizes ( + 1, ).

Lemma 1.7. For any integer k ≥ 3, (2, 2) ∈ G

.

Proof. The following graph realizes (2, 2): its set of vertices is V

∪ V

, where V

= { 1, 3 }×Z

and V

= {2, 4} × Z

, and its set of edges is E

∪ E

, where E

= {{(i, j), (i + 1, m)}|i ∈ { 1, 3 } , j ∈ Z

, m ∈ Z

} and E

= {{ (1, j), (3, j) }| j ∈ Z

} . It is easy to check that V

and V

(resp. E

and E

) are vertex-orbits (resp. edge-orbits).

Let k ≥ 3 be an integer and K = {k, k + 1, . . . , 2k − 2}. Given any integer n ≥ 2, we

will denote by G

the graph defined as follows: the set of vertices of G

will be

V

,

where V

= Z

, V

= Z

× Z

× Z

(2 ≤ i ≤ n) and V

= Z

× Z

× K (the

two vertices of V

will be denoted by v

( G

) and v

( G

)). The set of edges of G

will

be

E

, where E

= {{ (i

, . . . , i

), (i

, . . . , i

, i

) }| (i

, . . . , i

, i

) ∈ Z

× Z

× Z

}

1.2. CONNECTED GRAPHS 19 (1 ≤ j ≤ n − 1), E

= {{ (i

, i

, . . . , i

), (i

, . . . , i

, j) }| (i

, i

, . . . , i

) ∈ Z

×Z

× Z

, j ∈ K}

and E

= {{ (i

, . . . , i

, j), (i

, . . . , i

, l) }| (i

, . . . , i

) ∈ Z

× Z

, j, l ∈ K} .

It is easily seen that each vertex of G

has degree k, except the two vertices v

( G

) and v

( G

) which have degree 1. Moreover, the vertex-orbits of Aut( G

) are the V

’s and its edge-orbits are the E

’s.

Proposition 1.8. If ν ≥ 1, then (ν, ν) ∈ G

for every integer k ≥ 3.

Proof. The complete graph K

realizes (1, 1). If we replace each edge { x, y } of K

by the graph G

(where x and y are identified respectively with the vertices v

( G

) and v

( G

)), we get a graph realizing (n + 1, n + 1) (for any n ≥ 2). The graphs constructed in Lemma 1.7 realize (2, 2) for every k ≥ 3. Therefore, Proposition 1.8 is proved.

Lemma 1.9. For any integer k ≥ 3, (2, 3) ∈ G

.

Proof. The following graph realizes (2, 3): its set of vertices is V

∪ V

, where V

= Z

× Z

and V

= Z

, and its set of edges is E

∪ E

∪ E

, where E

= {{ (i, j), (i, m) }| i ∈ Z

, j, m ∈ Z

} , E

= {{ 0, (0, i) } , { 2, (0, i) } , { 1, (1, i) } , { 3, (1, i) }| i ∈ Z

} and E

= {{ 0, 1 } , { 2, 3 }} . It is easy to check that V

and V

(resp. E

, E

and E

) are vertex-orbits (resp. edge- orbits).

Proposition 1.10. If ν ≥ 1, then (ν, ν + 1) ∈ G

for every integer k ≥ 3.

Proof. The following graph realizes (1, 2): its set of vertices is V = Z

× Z

and its set of edges is E

∪ E

, where E

= {{ (i, j), (i, m) }| i ∈ Z

, j, m ∈ Z

} and E

= {{ (i, j), (i + 1, j) }| (i, j) ∈ Z

× Z

} . It is easy to check that V is a vertex-orbit and that E

and E

are edge-orbits.

If we replace each edge of E

by the graph G

(as explained in Proposition 1.8), we get a graph realizing (n + 1, n + 2) for any n ≥ 2. Lemma 1.9 covers the missing case, namely (2, 3).

An edge { a, b } of a graph G will be called l-symmetric (l ≥ 3) if the number of l-circuits of G containing a is equal to the number of l-circuits of G containing b. Two edges of G will be called adjacent if they have exactly one vertex in common.

Proposition 1.11.

(i) If ν ≥ 1 and n ≥ 0, then (ν + n, 2ν + n) ∈ G

for every integer k ≥ 3.

(ii) There exists a connected 3-regular graph realizing (ν+n, 2ν +n) and having the following property: for each edge e which is both 3-symmetric and 4-symmetric,

– if e does not belong to a 4-circuit nor a 3-circuit, then no edge adjacent to e is included in a 4-circuit;

– if e belongs to exactly one 4-circuit and if exactly two edges adjacent to e belong to a 4-circuit, then the other two edges adjacent to e are not 4-symmetric.

Moreover, each vertex of this graph belongs to at most two 4-circuits.

Proof. By Proposition 1.10, we may assume ν ≥ 2. Let N = {2ν − 1 − 1, 2(2ν − 1) − 1, 3(2ν − 1) − 1, 4(2ν − 1) − 1 } . We will construct a k-regular graph G realizing (ν, 2ν). The set of vertices of G will be V = Z

× Z

and its set of edges will be E

∪ E

∪ E

, where E

= {{ (i, j), (i, j + 1) }| (i, j) ∈ V } , E

= {{ (i, j), (l, j) }| i, l ∈ Z

, j ∈

Z

\ N

} and

E

= {{ (i, j), (l, j + 2(2ν − 1)) }| i ∈ Z

, l ∈ ( Z

\ { i } ) , j ∈ { 2ν − 1 − 1, 2(2ν − 1) − 1 }} .

20 CHAPTER 1. REGULAR GRAPHS Note that the E

’s are not edge-orbits of Aut( G ). Note also that the permutations g

, g

and g

, defined by g

((i, j)) = (i, j + 2ν − 1), g

((i, j)) = (i, − 2 − j) and g

((i, j)) = (i + 1, j) are automorphisms of G .

Let V

= Z

× N . If k = 3, V

is the set of vertices of G which do not belong to a 4-circuit. If k ≥ 4, V

is the set of vertices of G which do not belong to a 3-circuit. The automorphisms g

and g

show that V

is a vertex-orbit and that E

is an edge-orbit of Aut( G ).

As usual, we define the distance from a vertex x to V

as d(x, V

) = min

d(x, y) and the distance from an edge e to E

as d(e, E

) = min

d(e, e

).

Let V

= { x ∈ V | d(x, V

) = i } , i ∈ { 1, 2, . . . , ν − 1 } , F

= { e ∈ E

| d(e, E

) = i } , i ∈ { 1, 2, . . . , ν } and D

= { e ∈ E

| d(e, E

) = i } , i ∈ { 2, 3, . . . , ν } .

The automorphisms g

, g

and g

show that the V

’s are the other vertex-orbits of Aut( G ), and so the sets F

and D

are edge-orbits of Aut( G ) if j = ν. It remains to prove that the edges of F

are not in the same orbit of Aut( G ) as the edges of D

. If k ≥ 4, the edges of D

are contained in at least one 3-circuit; if k = 3 and ν ≥ 3, D

contains edges belonging to at least two 4-circuits, if k = 3 and ν = 2, each edge e ∈ D

belongs to a 6-circuit containing also the two edges of F

adjacent to e. These properties are not true for the edges of F

. Therefore, G realizes (ν, 2ν).

Given any integer n ≥ 2, if we replace each edge of F

by G

, we get a graph realizing (ν + n, 2ν + n). In order to get a graph realizing (ν + 1, 2ν + 1), add to G the vertices v

for every i ∈ Z

, j ∈ { 0, 1, 2, 3 } and replace each edge { (i, ν − 2 + j(2ν − 1)), (i, ν − 1 + j(2ν − 1)) } of F

(i ∈ Z

, j ∈ { 0, 1, 2, 3 } ) by the two edges { (i, ν − 2 + j(2ν − 1)), v

} and { v

, (i, ν − 1 + j(2ν − 1)) } . The last step to do is to add the edges { v

, v

} for every i, l ∈ Z

.

(ii) is a property of these graphs which is easy to check.

Lemma 1.12. For any integer k ≥ 3, (1, 3) ∈ G

.

Proof. By Proposition 1.3, we may assume k ≥ 4. By Proposition 1.10, (1, 2) ∈ G

for every k ≥ 3. Then, by Proposition 1.2, (1, 3) ∈ G

for every k ≥ 4.

Proposition 1.13. If ν ≥ 1, then (ν, 2ν + 1) ∈ G

for every integer k ≥ 3.

Proof. By Lemma 1.12, we may assume ν ≥ 2. Let N = { 2ν − 2, 2ν − 1, 4ν − 2, 4ν − 1, 6ν − 2, 6ν − 1, 8ν − 2, 8ν − 1 } . We construct a graph G whose set of vertices is V = Z

× Z

and whose set of edges is E

∪ E

∪ E

, where E

= {{ (i, j), (i, j + 1) }| (i, j) ∈ V } , E

= {{ (i, j), (l, j) }| i, l ∈ Z

, j ∈ ( Z

\ N ) } and E

= {{ (i, j), (l, j + 4ν) }| i ∈ Z

, l ∈ ( Z

\ { i } ) , j ∈ { 2ν − 2, 2ν − 1, 4ν − 2, 4ν − 1 }} . An argument similar to the proof of Proposition 1.11 shows that Aut( G ) has ν vertex-orbits and 2ν + 1 edge-orbits.

Proposition 1.14. If ν − 1 ≤ ≤ 2ν + 1, ν ≥ 1 and ≥ 1, then (ν, ) ∈ G

for every integer k ≥ 3.

Proof. Write = ν + r (where − 1 ≤ r ≤ ν + 1). Thanks to Propositions 1.6, 1.8, 1.10, 1.11 et 1.13, we may assume 2 ≤ r ≤ ν − 1 with ν ≥ 3. By Proposition 1.11, (r, 2r) ∈ G

because r ≥ 2. Since ν − r ≥ 1, we get (r + (ν − r), 2r + (ν − r)) = (ν, ν + r) ∈ G

.

1.2.4 2nd step: k = 4

Proposition 1.15. If ν ≥ 2 and n ≥ 0, then (ν + n, 3ν + n) ∈ G

.