Graphs and Their Coverings

Home , Covering graph, Cycle rank, Graph (topology), Graph automorphism, Heawood graph, Pappus graph

GRAPHS AND THEIR COVERINGS

Jin Ho Kwak

Department of Mathematics, POSTECH, Pohang, 790–784 Korea [email protected]

Roman Nedela

Matej Bel University, Banska Bystrica, Slovakia Mathematical Institute, Slovak Academy of Sciences, Banska Bystrica, Slovakia [email protected]

This work is supported by Com2MaC-KOSEF, Korea.

Preface

Combinatorial treatment of graph coverings had its primary incentive in the solution of Heawood’s Map Colour Problem due to Ringel, Youngs and others [Ri74]. That coverings underlie the techniques that led to the eventual solution of the problem was recognized by Alpert and Gross [GA74]. These ideas further crystalized in 1974 in the work of Gross [Gr74] where voltage graphs were introduced as a means of a purely combinatorial description of regular graph coverings. In parallel, the very same idea appeared in Biggs’ monograph [Bi74]. Much of the theory of combinatorial graph coverings in its own right was subse- quently developed by Gross and Tucker in the seventies. We refer the reader to [GT87, Wh84] and the references therein. The theory was extended to combinatorial graph bundles introduced by Pisanski and Vrabec [PV82] in the eighties. After two decades, since the book of Gross and Tucker [GT87] was published, it remains the most popular textbook as well as the reference book covering the topic. The authors of the presented monograph have several ambitions. Firstly, following the ideas in [MNS00] to establish and extend the combinatorial theory of graph coverings and voltage assignments onto a more general class of graphs which include edges with free ends (called semiedges). The new concept of a graph proved to be useful in applications as well as in theoretical considerations. Some of the applications are shown in the book. From the theoretical point of view, graphs with semiedges arise as quotients of standard graphs by groups of automorphisms which are semiregular on vertices and darts (arcs) but may fix edges. They may be viewed as 1-dimensional analogues of 2- and 3- dimensional orbifolds. The fruitful concept of an orbifold comes from studies of geometry of n-manifolds. Orbifolds and related concepts are implicitly included in the work of pioneers such as Henri Poincare, or Paul Köbe. The first formal definition of an orbifold-like object was

i ii

given by Ichiro Satake in 1956 in the context of Riemannian geometry. William Thurston, later in the mid 1970s defined and named the more general notion of orbifold as part of his study of hyperbolic structures. It is not a surprise that the concept of graphs with semiedges was discovered in context of theoretical physics, as well (see [GK98] for instance). Secondly, following the ideas in [GS93] and in the subsequent papers [MNS00, Ma98, Sir01, Ma02, MMP04, MP06] we develop and organize a material on the combinatorial treatment of the lifting automorphism problem - a central problem considered in this book. The study of the automorphism lifting problem in the context of regular coverings of graphs had its main motivation in constructing infinite families of highly transitive graphs. The first notable contribution along these lines appeared, incidentally, in 1974 in Biggs’ monograph [Bi74] and in a paper of Djoković[Dj74]. Whereas Biggs gave a combinatorial sufficient condition for a lifted group to be a split extension, Djokovićfound a criterion, in terms of the fundamental group, for a group of automorphisms to lift at all. A decade later, several different sources added further motivations for studying the lifting problem. These include: counting isomorphism classes of coverings and, more generally, graph bundles, as considered by Hofmeister [Ho91] and Kwak and Lee [KL90, KL92]; constructions of regular maps on surfaces based on covering space techniques due to Archdeacon, Gvozdjak, Nedela, Richter, Siráˇn,ˇ Skovieraˇ and Surowski [ARSS94, AGJ97, GS93, NS96, NS97, Su88]; and construction of transitive graphs with prescribed degree of symmetry, for instance by Du, Malniˇc,Nedela, Maruˇsiˇc,Scapellato, Seifter, Trofi- mov and Waller [DMW> 06, MM93, MMS> 06, MS93, ST97]. Lifting and/or projecting techniques play a prominent role also in the study of imprimitive graphs, cf. Gardiner and Praeger [GP95] among others. The lifting problem in terms of voltages in the context of general topological spaces was considered by Malniˇc[Ma98]. Venkatesh [Ve> 06] obtained structural results which refine the work of Biggs [Bi74, Bi84] and Djoković[Dj74] on lifting groups. Recently, many original contribu- tions on the topic was published, both solving particular problems and developing pieces of theory. However, there is no monograph covering (more or less completely) fundamentals of the theory. The topic is omit- ted in the book by Gross and Tucker and in other related monographs iii

in topological graph theory. So far, the most complete contribution to the theory was done in the thesis of A. Malniˇc[Ma96], some parts are covered in the paper [MNS00]. The study of the lifting problem within the combinatorial framework was of course preceded by that in the topological setting. We refer the reader to the work of Armstrong, Birman, Hilden, MacBeath and Zieschang [Ar92, BH72, Mac61, Zi73] and many others. Finally, it is our ambition to shift by one dimension up, namely, to study 2-cell embeddings of graphs into surfaces. The corresponding 2- dimensional objects are called maps and they present a central object of topological graph theory. For the purpose of this future project, it was necessary first to rebuilt some classical parts of graph theory and the presented monograph can be used as preliminary material for investigation of maps and their symmetries. The book contains three chapters. First one introduce basic concepts such as graphs and groups acting on graphs. Some useful concepts and statements from permutation groups are briefly mentioned. The second chapter is central. There, a theory of graph coverings is developed. It includes action of the fundamental group, classical approach to the theory of graph coverings and the associated theory of voltage spaces with some applications. The lifting automorphism problem is studied in detail, theory of voltage spaces us unified and generalized to graphs with semiedges. This way ‘a gap’ in the familiar and popular Gross and Tucker’s book ‘Topological graph theory’ is filled in. In the final chapter the lifting and covering techniques are used to approach some problems of classical graph theory. In particular, a material on flows on graphs and enumeration of graph coverings is included. The book can be used as a material for a course on graph coverings as well as it could be of interest of specialist interesting in the topic. We would like to take this opportunity to thank all people who give us valuable comments and suggestions for revising this book, including many students. Finally we are grateful to Korea Science and Engineering Founda- tion (KOSEF) in Korea for supporting us during preparation of this book.

Jin Ho Kwak iv

Roman Nedela E-mail: [email protected] [email protected] July 2005, in Pohang, Korea Contents

Preface i

1 Graphs and Groups 1 1.1 Introduction ...... 1 1.2 Graphs and their fundamental groups ...... 2 1.3 Group actions ...... 7 1.4 Vertex-transitive graphs ...... 13

2 Coverings and liftings of automorphisms 23 2.1 Graph coverings and two group actions on the ﬁbre . . . 23 2.2 Voltage spaces ...... 30 2.3 Covering isomorphism problem ...... 37 2.4 Lifting automorphism problem - classical approach . . . 42 2.5 Lifting of graph automorphisms in terms of voltages . . . 50 2.6 Lifting problem, case of abelian CT(p)...... 59 2.7 Lifting problem, case of elementary abelian CT(p).... 69

3 Applications 81 3.1 Enumeration of Graph Coverings ...... 81 3.2 Some applications of graph coverings ...... 90 3.2.1 Existence of regular graphs with large girth . . . 91 3.2.2 Large graphs of given degree and diameter 2 . . . 91 3.2.3 Spectrum of a graph and its covering graphs . . . 93 3.3 Flows ...... 95

Index 109

Chapter 1

Graphs and Groups

We begin with some basic deﬁnitions and elementary properties of graphs and coverings.

1.1 Introduction

In considerations in graph theory and more generally in combinatorics one has to deal with a problem of representation and controlling properties of large structures, potentially based on unbounded number of elementary objects (vertices, points, edges). Several essentially different techniques to deal with large objects were developed. Most usual are inductive constructions and arguments employing inductive arguments, counting arguments which led to the birth and development of the prob- abilistic method in graph theory and finally, employing symmetries of structures one can use finite groups to construct highly symmetrical graphs, designs, maps, geometries, etc, satisfying various combinatorial properties. The latest approach is well-documented by a fruitful concept of a Cayley graph, introduced originally by group theorists in 19-th century to investigate groups. Topological graph theory deals with graphs as 1-dimensional topological objects which are usually embedded to surfaces or to a more complex CW-complexes. The aim of this introductory chapter is to develop a consistent combinatorial theory of such graphs. In particular, epimorphisms between graphs which are locally bijective, called graph coverings, will be investigated. We demonstrate the usefulness of

1 2 Chapter 1. Graphs and Groups

graph coverings to solve several classical problems. Between the applications the lifting automorphism problem is emphasized. There are two reasons: firstly, the lifting automorphism problem and its applications in constructions of highly symmetrical graphs and maps received last decade a lot of attention in literature. Secondly, a purely combinatorial treatment of this classical problem is not covered by the existing monographs in topological graph theory, including White [Wh84], Gross and Tucker [GT87]. Because of certain natural applications in the theory of Cayley graphs as well as in the theory of maps on surfaces, and for the sake of convenience, we extend our definition of a graph to allow multiple edges, loops and even semiedges. This slight modification has some far- reaching consequences. For instance, the fundamental group of such a graph may not be a free group(!). Therefore, it has been necessary to carefully reexamine the whole basic theory of coverings as it is known in topology and, so far, in topological graph theory. On the other hand, our revision has an additional pleasant consequence in that it enables a smooth and unified transition to the theory of voltage spaces. As a result, a genuine combinatorial theory of coverings with no direct topological counterpart is obtained. The terminology not explicitly defined in this book (but freely used in the sequel) is tacitly adopted from graph theory as well as group theory and topology [Ma91, Ro82, GT87].

1.2 Graphs and their fundamental groups

A graph is an ordered quadruple X = (D,V ; I, λ) where D is a set of darts, V is a nonempty set of vertices, which is required to be disjoint from D, I is a mapping of D onto V , called the incidence function, and λ is an involutory permutation of D, called the dart-reversing involution. For convenience or if λ is not explicitly speciﬁed we sometimes write x−1 instead of λx. Intuitively, the mapping I assigns to each dart its initial vertex, and the permutation λ interchanges a dart and its reverse. The terminal vertex of a dart x is the initial vertex of λx. The set of all darts initiated at a given vertex u is denoted by Du, called the neighborhood of u. The cardinality |Du| of Du is the valency of the vertex u. The orbits of λ are called edges; thus each dart determines 1.2. Graphs and fundamental groups 3

uniquely its underlying edge. An edge is called a semiedge if λx = x, a loop if λx 6= x and Iλx = Ix, and it is called a link otherwise. We represent a graph, as defined above, by a topological space in the usual way as a 1-dimensional CW-complex. Note that from a topological point of view a semiedge is identical with a pendant edge except that its free endpoint is not listed as a vertex. There is an alternative definition of our graph discovered in context of theoretical physics (see [GK98]). They introduce a graph as a triple (D; ∼, λ), where D is the set of darts, ∼ is an equivalence relation on D and λ is an involutory permutation acting on D. The equivalence relation ∼ gives rise to a decomposition of D into equivalence classes [x], x ∈ D. We may set V = {[x] | x ∈ D} as the quotient set and I : x → [x]. On the other hand, for a given graph X = (D,V ; I, λ) the corresponding equivalence relation is defined by the decomposition −1 D = ∪v∈V I (v). Thus the two definitions are equivalent. Our definition of a graph is more convenient for purposes of this book than the classical definition of a graph consisting of vertices V and edges E ⊆ 2V of cardinality two, since in topological graph theory multiple edges, loops or semiedges naturally arise. In what follows our investigation is restricted to unoriented graphs. However, in some cases an orientation of edges is used as a technical tool. An orientation of a graph X = (D,V ; I, λ) is a transversal D+ of the set of edges E(X) = {{x, x−1} | x ∈ D}. It follows that D+ intersects each edge in exactly one dart. With each orientation D+ a unique (reverse) orientation D− such that D+ ∪ D− = D is associated. The intersection D+ ∩ D− is formed by the darts underlying the semiedges of X and it is empty for graphs without semiedges. An oriented or directed graph is a graph with a specified orientation. A morphism of graphs f :(D,V ; I, λ) → (D0,V 0; I0, λ0) is a function f : D ∪ V → D0 ∪ V 0 such that fD ⊆ D0, fV ⊆ V 0, fI = I0f and fλ = λ0f. Thus, a morphism is an incidence-preserving mapping which takes vertices to vertices and edges to edges. Note that the image of a link can be a link, a loop or a semiedge, the image of a loop can be a loop or a semiedge, and the image of a semiedge can be just a semiedge. Composition of morphisms is defined as composition of functions from right to left. This defines the category Grph. Note that the condition 4 Chapter 1. Graphs and Groups

0 fI = I f implies that f is completely determined by its restriction f|D on the set of darts. Moreover, if X is a simple graph (one without loops, semiedges or multiple adjacencies) then f|V coincides with the usual notion of a graph morphism used commonly by graph theorists. The isomorphism between two graphs and the automorphism (group) of a graph are deﬁned in a natural way. The automorphism group of a graph X will be denoted by Aut(X).

A walk of length n ≥ 1 is a sequence of n darts W = x1x2 . . . xn such that, for each index 1 ≤ k ≤ n − 1, the terminal vertex of xk coincides with the initial vertex of xk+1. Moreover, we define each vertex to be a trivial walk of length 0. The initial vertex of W is the initial vertex of x1, and the terminal vertex of W is the terminal vertex of xn. The walk is closed if the initial and the terminal vertex coincide. In this case we say that the walk is based at that vertex. If W has initial vertex u and terminal vertex v, then we usually write W : u → v. The walk −1 −1 −1 −1 W = xn xn−1 . . . x1 is the inverse of W . Let W1 and W2 be two walks such that the terminal vertex of W1 coincides with the initial vertex of W2. We define the product W1W2 as the juxtaposition of the two sequences. The product of a walk W and a trivial walk is W itself. A graph is connected if for every pair of vertices u and v there exists a walk u → v. A walk W is reduced if it contains no subsequence of the form xx−1. Clearly, each walk gives rise to a unique reduced walk by a repeated cancellation of all occurrences of the form xx−1. For instance, if x is a dart underlying a semiedge, then xx is a closed walk which reduces to the trivial walk at the initial vertex of x. By declaring two walks to be (combinatorially) homotopic provided that they give rise to the same reduced walk, we obtain an equivalence relation similar to the usual homotopy relation known in topology. The essential difference from the homotopy on the associated 1-CW complex is that a walk of length 1 traversing a semiedge is not homotopically trivial. By π(X) we denote the fundamental groupoid of a graph X, that is, the set of all reduced walks equipped with the product W1 · W2 (often denoted simply by W1W2) being the reduction of W1W2, whenever defined. Note that, for every vertex u, the trivial walk 1u behaves as the local identity in the groupoid.

Lemma 1.1. The subset π(X, u) ⊆ π(X) of all reduced closed walks 1.2. Graphs and fundamental groups 5

based at a vertex u ∈ X forms a group. If X is connected, then the groups π(X, u), π(X, v) at any two vertices u, v are isomorphic.

The group πu(X) := π(X, u) is called the fundamental group of X at u. Sometimes it is denoted simply by πu.

Proof. We prove the second part. Since X is connected, any two vertices u, v are connected by a walk W : u → v. Then the mapping S 7→ W −1SW takes a u-based closed walk S ∈ π(X, u) onto a closed v-based walk in π(X, v). It is easily seen that it preserves the homotopy classes, composition of classes and the identity is mapped to u −1 −1 the identity. Moreover, if S1,S2 ∈ π then W S1W = W S2W im- −1 u v plies S1 = S2. Hence S 7→ W SW gives a monomorphism π → π . Clearly, its inverse is S 7→ WSW −1. Thus πu ∼= πv. 2

In contrast to the classical case of graphs without semiedges, the fundamental group need not be a free group in our case. Nevertheless, it is a free product of cyclic groups, each of which is isomorphic to Z or Z2.

x - x1 - xk - + W I P ) Q ? N Xk Yk k

I IQk

u u

(a) A fundamental cycle (b) Factoring into fundamental cycles

Figure 1.1: A generator of πu 6 Chapter 1. Graphs and Groups

Lemma 1.2. Let X be a finite connected graph with e edges, s ≤ e semiedges and v vertices. Then the fundamental group π(X, u) is a free product of e − v + 1 cyclic groups. More precisely, it is a free product of e − s − v + 1 copies of Z and s copies Z2. Proof. A minimal generating set of π(X, u) can be constructed by the standard algorithm employing a spanning tree T of X (an inclusion- minimal connected spanning subgraph) as in the classical case. Let {x, λx} be a cotree edge and P , Q be the unique u → Ix, u → Ix−1 paths in T . We form a generating set from the homotopy classes [P xQ−1], where x ranges through the cotree darts choosing exactly one of {x, λx} for each cotree edge. See Figure 1.1(a). Such u-based closed walk P xQ−1 is called the fundamental cycle associated with the cotree dart x. The proof is finished by showing that each homotopy class of u-based closed walks can be expressed in terms of generators. We first reduce a u-based closed walk to a minimal representative W u of the homotopy class in π . Let W = ··· x1 ··· x2 ··· xk ··· , where x1, x2, . . . , xk are the cotree darts, or their inverses, listed in the order as they appear in W . Let Wi be the subwalk joining xi to xi+1. By the −1 minimality, Wi is a path for all i = 1, 2, . . . , k−1 and Wk = YkQk QkXk is a concatenation of four paths as in Figure 1.1(b), where Qk belongs to T and joins u to the path Wk = YkXk in T , the terminal vertex of Yk and Qk coincides. Let Qi be the unique path in T joining u to Wi in the shortest length. The terminal vertex of Qi is met by Wi exactly once. Thus Wi = YiXi where Yi and Xi are, respectively, the subpaths terminating and initiating at the terminal vertex of Qi. Then replacing each −1 Wi by YiQi QiXi we do not change the homotopy class of W . This −1 Qk−1 −1 shows that W ∼ QkXkx1Y1Q1 i=1 QiXixi+1Yi+1Qi+1 expressing W as a product of the generating walks. Clearly, no generating homotopy class can be expressed as a product of the others. Hence the generating set is a minimal one. Moreover, since the order of the class [P xQ−1] is two if and only if x is a semiedge (in that case P = Q), the above algorithm yields a one-to-one correspondence between the set of semiedges of X and the set of free factors in π(X, u) isomorphic to Z2. 2

The number β(X) = e − v + 1 is called the betti number of a graph X. Group actions 7

Remark: By the abelianization of a group G we mean a factor group H = G/[G, G], where [G, G] is the derived group. Abelianization of the fundamental group π(X, u) is called the (ﬁrst) homology group of a graph X and denoted by H1(X). By Lemma 1.1 H1(X) does not depend on the choice of the base vertex. By Lemma 1.2 the (ﬁrst) homology group H1(X) of a graph X is isomorphic to the free product of the free abelian group of rank e − s − v + 1 and s copies of Z2.

Exercises 1.2.1. Describe connected graphs with abelian fundamental groups. 1.2.2. Prove that each class of homotopic equivalent walks contains a unique reduced walk. 1.2.3. Let X be a ﬁnite connected graph and T ≤ X be a spanning tree. Prove that for any two distinct cotree darts xi+1, xj+1, the u-based −1 −1 closed walks QiXixi+1Yi+1Qi+1 and QjXjxj+1Yj+1Qj+1, deﬁned in the proof of Lemma 1.2, are not homotopic equivalent.

1.3 Group actions

Graph coverings and group actions are closely related. It is therefore impossible to develop a consistent theory of graph coverings and not to refer to some basic statements from the theory of group actions. In what follows we introduce basic notions and statements without proofs. For a more detailed explanation the reader is referred to [DM96, Wie]. Let Ω be a set. By SymL Ω and SymR Ω we denote the left and the right symmetric group on a (nonempty) set Ω, respectively. The right symmetric group on the set {0, 1, . . . , n − 1} is commonly denoted by Sn. A (right) action of a group G on a set Ω is defined by a function σ :Ω × G → Ω, with the common notation σ(z, g) = z · g, such that z · 1G = z and z · (gh) = (z · g) · h for any z ∈ Ω and g, h ∈ G. Note that the function z 7→ z · g is a bijection on Ω for each g in G. Therefore, one may equivalently define a group action of G on a set Ω as a group homomorphism G → SymR Ω. When a group G acts on a set Ω, we call Ω a G-space. The left action is defined in a similar manner, as a function G×Ω → Ω, or a group homomorphism G → SymL Ω. 8 Chapter 1. Graphs and Groups

The subgroup Gz = {g ∈ G | z · g = z} is called the stabilizer of z ∈ Ω under the action of G. The set {z · g ∈ Ω | g ∈ G} is called the orbit of z. An action of a group is called semi-regular if all the stabilizers are trivial. It is transitive if it has only one orbit, that means for any two z2, z1 ∈ Ω there is g ∈ G such that z2 = z1 · g. The stabilizers in a transitive action are all conjugate, thus the structure of Gz does not depend on the choice of the point z. A transitive and semi-regular action is called regular.

Let G act on Ω. The core CoreΩ(G) = ∩z∈ΩGz ≤ G is formed by the elements of G letting all the elements ﬁxed.

If a G-action on Ω is deﬁned by a homomorphism α : G → SΩ, then the core CoreΩ(G) is nothing but the kernel of α. Hence, the core is a normal subgroup of G. If CoreΩ(G) = 1 then the action of G on Ω is faithful. If G acts faithfully on Ω then the assignment g 7→ αg, where αg(x) = x · g, is a monomorphism G → SΩ, called a permutation representation of G on Ω and one can identify G with a subgroup of SΩ. In a general case, G/CoreΩ(G) is isomorphic to a subgroup of SΩ. The size |Ω| is called a degree of the permutation representation. A group of permutations G is of degree n if G ≤ SΩ and |Ω| = n. We shall later need the concept of a group action extended to an action of a fundamental groupoid which can be introduced in the obvious way, see Proposition 2.2.

Let Ωi be a Gi-space for i = 1, 2. We call a group isomorphism f : G1 → G2 admissible or consistent with the respective actions if there exists a bijection φ :Ω1 → Ω2 such that φ(z · g) = φ(z) · f(g) for all z ∈ Ω and g ∈ G, that is, the diagram

- Ω1 × G1 Ω1

φ × f φ ? ? - Ω2 × G2 Ω2

commutes. For a Gi-space Ωi (i = 1, 2), if there is an admissible isomorphism f : G1 → G2, then two actions are called isomorphic and the pair (φ, f) is called an isomorphism of the actions. 1.3. Group actions 9

As a special case, let G1 = G2 := G, and f = idG be the identity. We define a morphism from Ω1 to Ω2 to be a function φ :Ω1 → Ω2 such that φ(z · g) = φ(z) · g for all g ∈ G and all z ∈ Ω1. Morphisms of G-spaces are also called equivariant. If an equivariant morphism φ :Ω1 → Ω2 is bijective then its inverse is also equivariant, and φ is an equivalence or G-isomorphism. The two G-spaces Ω1 and Ω2 are isomorphic. Transitive permutation representations are of crucial importance for investigation of highly transitive combinatorial structures. The following two statements are well-known, see [DiMo, Robinson p.35]. Let H ≤ G be a subgroup. For each g ∈ G define τg by τg : Hx 7→ Hxg. Then τ defines a transitive action of G on the right cosets T −1 (by right translation) with kernel HG = g Hg. It follows that x∈A p : g 7→ τg is a transitive permutation representation of the quotient group G/HG. In particular, if H = 1 then the action is regular, and the statement is known as a Cayley Theorem. The next theorem shows that all transitive actions can be obtained as above up to equivalence. Theorem 1.3. Any transitive action of a finite group G on a set Ω is equivalent to the action of G on the right cosets of some subgroup H of G by right translation. Since the kernel of the action of G on the right cosets of H is the T −1 core CoreG(H) = g∈G g Hg for some H ≤ G, it follows that for an abelian group G, the right translation of G is the only (faithful) transitive representation of G up to equivalence.

g Deﬁnition 1.1. Let G ≤ SΩ, ∆ ⊆ Ω. If for any g ∈ G, either ∆ = ∆ or ∆g ∩ ∆ = ∅, then we call ∆ a block of G.

Obviously, Ω, ∅ and one-element subset {α} are blocks of G; they are called the trivial blocks.

Deﬁnition 1.2. Let G ≤ SΩ. If G has a nontrivial block ∆, then G is called an imprimitive group, and ∆ is called an imprimitivity set. Otherwise, we say G a primitive group. When |Ω| = 2, we make the following convention: if G is identity, we also call G imprimitive. 10 Chapter 1. Graphs and Groups

In other words, a permutation group G on a set Ω is imprimitive if Ω may be split into nontrivial blocks (of the same size if the action is transitive) such that each g ∈ G maps a block into a block. For example, all intransitive groups are imprimitive. Obviously, every primitive group is transitive. Until the end of the section, we assume that G is a transitive permutation group on Ω.

Proposition 1.4. Let G ≤ SΩ. Assume that G is transitive but not primitive. Let ∆ be an imprimitivity set of G. Write H = G{∆} = {g ∈ G | ∆g = ∆}. Then

(1) the subgroup H is transitive on ∆. S (2) Assume that G = r∈R Hr is a decomposition of G into a union S r 0 of right cosets of H. Then Ω = r∈R ∆ , and for any r 6= r in R, we have ∆r ∩ ∆r0 = ∅. (3) |∆| divides |Ω|.

The set {∆r | r ∈ R} is called a complete system of imprimitivity.

Proof. (1) For any α, β ∈ ∆, since G is transitive, there is a g ∈ G such that αg = β. Thus β ∈ ∆ ∩ ∆g, and ∆ ∩ ∆g 6= ∅. Since ∆ is a block, g we get ∆ = ∆ . Hence g S∈ H, and H is transitive on ∆. (2) Assume that G = r∈R Hr is a decomposition of G into a union of right cosets of H. Let α ∈ ∆, β ∈ Ω. Since G is transitive, there is a g ∈ G such that β = αg ∈ ∆g. Let g = hr, where h ∈ H, r ∈ R. Then hr r S r we have β ∈ ∆ = ∆ and Ω = r∈R ∆ . Moreover, if ∆r ∩ ∆r0 6= ∅ then ∆rr0−1 ∩ ∆ 6= ∅. Since ∆ is a imprimitivity set, we have ∆ = ∆rr0−1 . Hence rr0−1 ∈ H, Hr = Hr0. Since R is a complete set of representatives of H, we have r = r0. (3) Since |∆| = |∆r|, the conclusion follows from (2). 2

Corollary 1.5. A transitive group of prime degree is primitive.

The kernel K of an action of G on Ω is a subgroup formed by elements of G ﬁxing each block of imprimitivity in ∆.

Proposition 1.6. The kernel K of an action of G with respect to imprimitivity system ∆ is a normal subgroup of G 1.3. Group actions 11

The following theorem is well-known in a permutation group theory and will be used later. Theorem 1.7. (see [DM96, Theorem 4.2A]) Let G be a transitive subgroup of Sym(Ω), and let C be the centralizer of G in Sym(Ω). Then ∼ (1) C is semiregular and C = NG(Gx)/Gx for any x ∈ Ω. (2) C is transitive if and only if G is regular, (3) if C is transitive then it is conjugate to G in Sym(Ω) and hence C is regular. So far we have considered action of a group on its subgroups by left and right translations. There is another important action of a group on its subgroups, namely the action by conjugation. If H < G is a g −1 subgroup then the stabilizer NG(H) = {g ∈ G|H = H = g Hg} is called the normalizer of H in G. Clearly, the normalizer is a normal subgroup of G containing H. Moreover, it is the least subgroup of H with this property. Orbits of the action of G are called conjugacy classes. Two groups H and Hg in the same conjugacy class are called conjugate subgroups. Counting conjugacy classes of subgroups of finite index in a finitely generated group is closely related to enumeration of isomorphism classes of combinatorial objects. To this end we present the following counting lemma recently proved by A. Mednykh [Me06]. Theorem 1.8. Let G be a finitely generated group. Then the number of conjugacy classes of subgroups of index n in the group G is given by the formula 1 X X N (n) = Epi(K, Z ), G n ` `| n Kcyclic group Z` of order `. We show how to compute the number of homomorphisms and epimorphisms of an arbitrary finitely generated group into cyclic Z`. De- note by Hom(G, Z`) the set of homomorphisms ofP a group G into the cyclic group Z` of order `. Since |Hom(G, Z`)| = |Epi(G, Zd)|, by d| ` the Möbiusinversion formula we have the following result 12 Chapter 1. Graphs and Groups

Lemma 1.9. (G. Jones [Jo95])

X ` |Epi(G, Z )| = µ( )|Hom(G, Z )|, ` d d d| ` where µ(n) is the M¨obiusfunction.

This lemma essentially simplifies the calculation of |Epi(G, Z`)| for a finitely generated group G. Indeed, let H1(G) = G/[G, G] be the first homology group of G. Suppose that H1(G) = r Zm1 ⊕ Zm2 ⊕ ... ⊕ Zms ⊕ Z . Then we have

Lemma 1.10. X ` |Epi(G, Z )| = µ( )(m , d)(m , d) ... (m , d) dr, ` d 1 2 s d| ` where (m, d) is the greatest common divisor of m and d.

Note that |Hom(Zm, Zd)| = (m, d) and |Hom(Z, Zd)| = d. Since the group Zd is Abelian, we obtain

|Hom(G, Zd)| = |Hom(H1(G), Zd)| r = (m1, d)(m2, d) ... (ms, d) d .

Then the result follows from Lemma 1.9.

Corollary 1.11. Let Fr be a free group of rank r. Then H1(Fr) = Zr P ` r and Epi(Fr, Z`) = µ( d )d . d| `

Exercises

1.3.1. Let P1 : Hg 7→ Hgx and P2 : Kg 7→ Kgx be two actions of a group G on right cosets of subgroups H ≤ G and K ≤ G, respectively. Show that P1 and P2 are equivalent if and only if H and K are conjugate in G. 1.3.2. (Orbit-Stabilizer Theorem) Prove: Let G act on a set Ω. Then for each z ∈ Ω, |G/Gz| equals to the size of the orbit of z. Vertex-transitive graphs 13

1.3.3. (Burnside Lemma) Prove: Let G act on a set Ω. For g ∈ G, the set of ﬁxed points of g is ¯ ﬁxΩ(g) = {x ∈ Ω ¯ x · g = x}.

Then the numberP of orbits is the average of the number of fixed points, i.e., |Ω/G| = 1 |fix (g)|. In particular, if G is transitive on Ω, P|G| g∈G Ω then |G| = g∈G |fixΩ(g)|; and when |Ω| > 1, G has a regular element, that is an element without fixed points.

1.4 Vertex-transitive graphs

Group actions can be used to construct highly symmetrical graphs. One can use them to illustrate the notions and statements defined above. As we shall see later, the construction of a Cayley graph naturally generalize to Cayley voltage graphs and graphs presented here will provide a wide class of useful examples. Cayley graphs. Given a group G and a set of generators S = S−1, 1 ∈/ S, we define a Cayley graph Cay(G, S) = (D,V ; I, λ) by setting D = G×S, V = G, I(g, x) = g and λ(g, x) = (gx, x−1). Clearly, G acts on V by left multiplication g · v = gv. Let τg denotes the permutation given by the left multiplication by g. It is easy to check τg is a graph automorphism, and the set A = {τg | g ∈ G} with the composition of permutations form a subgroup of Aut(Cay(G, S)) isomorphic to G. Since A acts transitively (even regularly) on V , Cayley graphs form a family of highly symmetrical examples of graphs. Note that G also acts transitively (even regularly) on V by right multiplication, v · g = vg, but it may not be a graph automorphism. Although A ∼= G the left and right actions are not isomorphic, in general. There is a natural coloring f of edges of a Cayley graph Cay(G, S) by the pairs of generators of the form {x, x−1} defined by f(gh) = {x, x−1} if g−1h ∈ {x, x−1}. The monochromatic factor determined by {x, x−1} is either a 2-factor formed by |G|/|x| cycles of length |x| > 2, or |x| = 2 and it is a perfect matching. It is easy to show that the left action of G on vertices preserves the coloring of edges.

The two actions GL ≤ SG, GR ≤ SG of G by left and right translation on itself can be viewed as a particular instance of Theorem 1.7 14 Chapter 1. Graphs and Groups

with GRx = 1 and C = GL ≤ Aut(Cay(G, S)). The following characterization of Cayley graphs is well-known.

Proposition 1.12. (Sabidussi) A simple graph X is a Cayley graph if and only if its automorphism contains a subgroup H ≤ Aut(X) acting regularly on vertices.

Proof. Let X = Cay(G, S) be a Cayley graph. It was already observed that H = {τg | g ∈ G} ≤ Aut(X) is a transitive group of automorphisms. Indeed, if g and h are two vertices of Cay(G, S) then there is an x ∈ G such that h = xg = τx(g) proving the transitivity. Assuming τgx = x for some vertex x ∈ G, we get gx = x implying g = 1. Hence, Gx = {τ1} is trivial and the action of H is regular. For the suﬃciency, let H ≤ Aut(X) be a subgroup acting regularly on the vertices of X. Fix a vertex w. Since the action is regular, for each vertex u there is a unique element g ∈ H taking w onto u. It gives a bijective labelling ` : V → H such that u = `(u)w for each vertex u and we may identify the vertices of X with elements of H. Let uv be an arbitrary edge, and let x = `(u) takes an edge wa onto uv. Then u = xw and v = xa. Let g = `(a). By taking labellings in H, we can say that x = `(u) takes an arc (1, g) to an arc (x, xg) which is the label of (u, v). Hence the adjacency of vertices is deﬁned by right multiplication by the labels of the neighbours of w with `(w) = 1. Set S to be the set of labels of the neighbours of w. If g = `(a) belongs to S then g−1 takes a 7→ w and w 7→ b, where g−1 = `(b) ∈ S. Hence S = S−1 is closed under taking inverses and we get X = Cay(G, S). 2

Note that a representation of a graph X as a Cayley graph may not be unique. For instance, the complete graph Kn can be deﬁned as a Cayley graph Cay(G; G \{1}) for any group of order n.

Example 1.1. (Octahedral graphs) The octahedral graph is deﬁned by

On = K2n − {{k, n + k} | k = 1, . . . , n} = K2n − {n disjoint edges}.

For example, O2 is the cycle C4 and O3 is the underlying graph of the regular octahedron. The octahedral graph can be deﬁned as a Cayley 1.4. Vertex-transitive graphs 15

graph On = Cay(Z2n; {±1, ±2,..., ±(n − 1)}) as well. However, the 3-dimensional octahedral graph O3 can be deﬁned alternatively as

O3 = Cay(S3; {(1, 2, 3), (1, 3, 2), (1, 2), (2, 3)}).

Both representations of O3 are depicted on Fig. 1.3 with r = (1 2 3) and s = (1 2) in S3.

6 7 8 3 1 5 1 6 1

4 2 4 2 5 2 3 4 3

O2 O3 O4

Figure 1.2: Octahedral graphs On

2 r ...... −1 1...... 3 s...... sr...... 5 ...... sr ...... 0 4 1 r2

(a) Cay(Z6; ±1, ±2) (b) Cay(S3; ±r, ±s)

Figure 1.3: Two presentations of the octahedral graph O3 16 Chapter 1. Graphs and Groups

Example 1.2. (Generalized Petersen graphs) All Cayley graphs are vertex-transitive but the converse is not true. A simple family of vertex-transitive graphs that are not Cayley graphs can be constructed as follows. A generalized Petersen graph GP (n, k), k ≤ n/2 is a simple graph whose vertices are elements of Zn × Z2 and each vertex (i, j) has exactly three neighbours (i + kj, j), (i − kj, j) and (i, j + 1). It follows that a mapping ρ taking (i, j) 7→ (i + 1, j) is a graph automorphism. Hence, either Aut(GP (n, k)) acts on the vertex set with two orbits 2 Zn ×{0}, Zn ×{1}; or it is vertex-transitive. If k ≡ ±1(mod n) there is an additional automorphism α of GP (n, k) taking α :(i, j) 7→ (ki, j+1). This can be easily checked by computing images of the three kinds of pairs of adjacent vertices. Hence the graph is vertex-transitive in this case. The goal is that with a single exception the opposite implication holds as well.

The following statements classify those GP (n, k) that are vertex- transitive or Cayley graphs. Theorem 1.13. (Frucht, Graver, Watkins [FGW71]) A generalized Pe- tersen graph GP (n, k) is vertex-transitive if and only if either (n, k) = (10, 2), or k2 ≡ ±1(mod n). Theorem 1.14. [NS95] A generalized Petersen graph GP (n, k) is a Cayley graph if and only if k2 ≡ 1(mod n). Proof. We prove only the suﬃciency. Assume k2 ≡ 1(mod n). Take the subgroup H = hρ, αi ≤ Aut(GP (n, k)). It follows that

αρα(i, j) = αρ(ki, j + 1) = α(ki + 1, j + 1) = (k(ki + 1), j) = (i + k, j) = ρk(i, j).

Hence, we have αρα = ρk, hρi is a normal subgroup of H of index 2 and so |H| = 2n. Therefore, the action of H is regular on vertices. 2

In view of Theorem 1.3 we generalize the construction of a Cayley graph as follows. Coset graphs. Let G be a ﬁnite group and H a proper subgroup of g G with ∩g∈GH = 1. By a double coset we mean a set of elements 1.4. Vertex-transitive graphs 17

of the form HxH, where x ∈ G. Let B be a union of some dou- −1 ble cosets of H in G such that B = B and G = hH,B¯i. Denote ¯ by X = C(G; H,B) the¯ coset graph with V (X) = {gH g ∈ G}, D(X) = {(gH, gbH) ¯ b ∈ B, g ∈ G}, I(gH, gbH) = gH and the dart-reversing involution L(gH, gbH) = (gbH, gbb−1H) = (gbH, gH). Clearly, G acts faithfully and vertex-transitively on the coset graph by left multiplication as group of graph automorphisms with a vertex stablizer isomorphic to H. Conversely, let X = (D,V ; I, λ) be a simple vertex-transitive graph and let A ≤ Aut(X) acts vertex-transitively. Given a fixed vertex v in X, let H := Av be the stabilizer. Then the left cosets A/H can be identified with the vertex set V by correspondence gH 7→ vg. Let {d1, d2, . . . , d`} be the neighbors of v with terminal vertices vg1, vg2, . . . , vg`. Each vertex vgj gives rise to the double coset HgjH. ` ˜ Set B = ∪i=1HgiH, D = {(gH, gbH) | b ∈ B, g ∈ A} and let the dart-reversing involution be L(gH, gbH) = (gbH, Hg). Then the graph X is isomorphic to the coset graph C(A; H,B). The above discussion generalizes the Sabidussi Theorem as follows. Theorem 1.15. Every coset graph X = C(A; H,B) is vertex-transitive. Conversely, every connected simple vertex-transitive graph X is isomorphic to a coset graph C(A; H,B), where A ≤ Aut(X) acts vertex- transitively on X, H is a stabilizer of a vertex, B = B−1 and A = hH,Bi. Note that for H = 1 the definitions of coset graph and Cayley graph coincide.

Example 1.3. For example, let A = S5 and let H = {1, (123), (132), (12), (13), (23), (45), (123)(45), (132)(45), (12)(45), (13)(45), (23)(45)} of order 12 which is isomorphic S{1,2,3} × S{4,5}. Set a = (14)(25) and B = HaH. Then hH, ai generates S5, and C(A; H,B) is the Petersen graph GP (5, 2).

A graph X is called arc-transitive or dart-transitive if its automorphism group acts transitively on darts. Clearly, all arc-transitive graphs 18 Chapter 1. Graphs and Groups

are vertex-transitive. But the converse is not true. A coset graph C(G; H,B) is arc-transitive if and only if B forms a single double coset. The following ‘orbital graph’ construction gives another point of view on vertex-transitive graphs. Orbital graphs. If G ≤ Aut(X) is transitive on darts of a graph Γ = (D,V ; I, λ), then G acts on V transitively and induces a natural action on V × V by (x, y)g = (xg, yg) for g ∈ G, in which D is an orbit. Conversely, a transitive G-action on a set V induces an action on V × V by (x, y)g = (xg, yg) for g ∈ G. Its orbits are called orbitals. The diagonal ∆ is a trivial orbital; all others in ∆c = (V × V ) \ ∆ are nontrivial. An orbital O is called self-paired if O = O−1, where (x, y) ∈ O−1 if and only if (y, x) ∈ O . For any self-paired orbital O, (O,V ; I, λ), where λ(x, y) = (y, x) and I(x, y) = x, is an arc-transitive graph, called an orbital graph. An orbital graph X = (O,V ; I, λ) determined by a transitive action of G on V is isomorphic to a coset graph C(G; H,B), where A ≤ Aut(X) acts vertex-transitively on X, H is a stabilizer of a vertex and B is generated by a single element a. In this case we write O(G; H, a) for the orbital graph. This is in fact isomorphic to C(A; H, HaH). Clearly, the graph is connected if and only if G = hH, ai. For instance O(S5; H, a) is the Petersen graph given in the previous example. All connected arc-transitive graphs are orbital graphs up to isomorphism.

Theorem 1.16. Every orbital graph X = O(G; H, a) is arc-transitive. Conversely, every connected simple arc-transitive graph X is isomorphic to an orbital graph O(G; H, a), where A ≤ Aut(X) acts vertex- transitively on X, H is a stabilizer of a vertex and a ∈ A \ H such that A = hH, ai.

Assume the orbital generated by a is non-self-paired. Then G acts vertex and edge transitively but not arc-transitively on the orbital graph X = O(G; H, a). Generally, an action of a group of automorphisms of graph is called half-arc-transitive if it is edge and vertex transitive, but not arc-transitive. A half-arc transitive action of G on darts of X has two orbits D = D+ ∪D−. We call D+ a G-orientation of 1.4. Vertex-transitive graphs 19

X. Since a G-orientation is balanced, each vertex has an even valency 2k for some k ≥ 1. The arc-transitivity of a graph X can be generalized obviously to a transitivity on the walks of a given length. We say that a group G ≤ Aut(X) is t-transitive, t ≥ 1, if it acts transitively on the set of irreducible walks of length t. The automorphism group of a cycle is t- transitive for any t. Surprisingly, with the exception of cycles, maximal t for which Aut(X) is t-transitive is bounded by 7 in general [We81], and by 5 in case of cubic graphs (see Tutte [Tu47]). More precisely Weiss proved the following. Theorem 1.17. (Weiss [We81]) Let X be a simple t-transitive connected graph of valency > 2. Then t ≤ 7. Moreover, a 6-transitive graph is 7-transitive as well. Weiss’s proof depends on the classification of finite simple non- abelian groups which is a major achievement of group theory of 20th century. No independent proof is known yet. Highly symmetric graphs can be defined as incidence graphs of finite geometries or block designs. Few examples follow.

Example 1.4. The Pappus graph 93 is 3-arc-transitive cubic graph on 18 vertices, being the incidence graph (D,V ; I,L) of the Pappus conﬁguration

B = {123, 456, 789, 147, 258, 369, 158, 348, 267}, which is a union of three parallel classes of lines in the aﬃne geometry AG(2, 3), with exactly one set of three parallel lines missing. In other words, V = P ∪ B, where P = {1, 2,..., 9},(x, y) is a dart if x ∈ P and x ∈ y ∈ B, or y ∈ P and y ∈ x ∈ B, I is the projection on the ﬁrst coordinate and L(x, y) = (y, x). The automorphism group of 93 has order 216, and is a semi-direct product of a non-abelian group of order 27 and exponent 3 by a dihedral group of order 8. Another remarkable property of 93 is that it has a highly symmetrical hexagonal embedding in the torus, (the map {6, 3}3,0 in the notation of Coxeter and Moser, see Figure 1.4). 20 Chapter 1. Graphs and Groups

Figure 1.4: Hexagonal embedding of the graph of Pappus conﬁguration in the torus

Example 1.5. The Heawood graph F014 is the incidence graph of the Fano plane P = {123, 345, 156, 147, 257, 367, 246}, ∼ with automorphism group PSL(3, 2) : Z2 = P GL(2, 7). The graph is 4-arc regular, and also admits a one-arc regular action of a subgroup of order 42. There is a well-known hexagonal embedding of the Heawood graph, (the map {6, 3}2,1 in the notation of Coxeter and Moser; see Figure 1.5).

Example 1.6. Vertices of the Coxeter graph F028 may be taken as antiﬂags of the Fano plane P (that is, ordered pairs (p, `) consisting of a line ` and a point p not incident to `), and two vertices γ = (p, `) and δ = (q, m) are adjacent if P = ` ∪ m ∪ {p, q}. The automorphism group 1.4. Vertex-transitive graphs 21

Figure 1.5: Hexagonal embedding of the Heawood graph in the torus of the Coxeter graph has 336 elements and is isomorphic to P GL(3, 2) : ∼ Z2 = P GL(2, 7). This graph is 3-regular.

Example 1.7. Constellations. Motivated by some investigations in low-dimensional topology and theory of Riemann surfaces, the following combinatorial objects called constellations are frequently investigated. A constellation is a sequence of permutations [g1, g2, . . . , gk], where gi ∈ Sn, G = hg1, g2, . . . , gki is transitive on the set of n points and the product is g1g2 . . . gk = id. It follows that G has a right action on 0 the set of points. Two constellations C = [g1, g2, . . . , gk] and C = 0 0 0 0 [g1, g2, . . . , gk] based on sets E and E , respectively, are isomorphic 22 Chapter 1. Graphs and Groups

0 0 −1 if there is a bijection h : E → E such that gi = h gih, for i = 1, 2, . . . , k. The automorphism group of a constellation is formed by the permutations centralizing all gi, i = 1, 2, . . . , k. It follows that the automorphism group of C is the centralizer of G in the symmetric group Sn. If gi act from right (left) then automorphisms act on E from left (right). Generally, G is much larger than Aut(C) and , by Theorem 1.7 G ∼= Aut(C) if and only if |G| = |Aut(C)|.

Exercises

1.4.1. Find the least Cayley graph such that the left and right actions of G on the vertex set are non-isomorphic. 1.4.2. Find all Cayley graphs with at most 8 vertices and distinguish them up to equivalence of group actions and up to graph isomorphisms. 1.4.3. Prove that the Petersen graph GP (5, 2) and the dodecahedral graph GP (10, 2) are not Cayley graphs. 1.4.4. Deﬁne the 1-skeletons of the ﬁve Platonic solids as orbital graphs. 1.4.5. Prove that each octahedral graph On can be represented as a Cayley graph based on a non-abelian group. 1.4.6. For a coset graph G = G(A; H,B), show that (1) A acts arc-transitively if and only if B = HbH, ∀b ∈ B. (2) If B = H`H for some ` ∈ A then G is arc-transitive of valency |H|/|H ∩ `H`−1|. 1.4.7. Prove Theorem 1.15. 1.4.8. Find all non-isomorphic constellations on 6 points. 1.4.9. Prove that the action of the automorphism group on the set of points of a constellation is semi-regular. Chapter 2

Coverings and liftings of automorphisms

2.1 Graph coverings and two group actions on the ﬁbre

The aim of this section is to introduce a graph covering and two natural group actions on the vertex set of the covering graph: One is by the covering transformation group (from the left) and the other is by the fundamental groupoid (from the right). A graph covering is roughly speaking, an epimorphism between graphs which is locally bijective. A natural way to construct graph covering is to consider a projection of a graph to its quotient graph deﬁned by a semiregular group of automorphisms.

Definition 2.1. Let X = (D,V ; I, λ) and X˜ = (D,˜ V,˜ I,˜ λ˜) be graphs. A graph epimorphism p : X˜ → X is called a covering projection if, for ˜ ˜ every vertex u˜ ∈ X, p maps the neighborhood Du˜ ofu ˜ bijectively onto the neighborhood Dpu˜ of pu˜. The graph X is usually referred to as the base graph or a quotient graph and X˜ is called the covering graph. By −1 −1 fibu = p u and fibx = p x we denote the fibre over u ∈ V and x ∈ D, respectively.

23 24 Chapter 2. Coverings and liftings of automorphisms

Example 2.1. (Two quotients of the n-cube Qn) The n-cube or the n- dimensional Hypercube Qn = (D,V ; I, λ) is defined as a Cayley graph n n Cay(Z2 ,B), where V = Z2 , the n-dimensional vector space over the field (Z2, +, ·) and B = {εi | i = 1, . . . , n} is the standard basis for V = n Z2 . That is, εi ∈ V is a vector of length n whose entries are all 0 except for the i-th entry which is 1. All translations by elements in V form a n group of automorphisms t(V ) of Qn = Cay(Z2 ,B), which acts regularly on the vertex-set and semiregularly on the dart-set of Qn. One can easily show that the quotient projection p : Qn → Qn/t(V ) is actually a covering projection. Figure 2.1 shows a covering p : Q3 → Q3/t(V ), where the quotient graph Q3/t(V ) has one vertex and 3 semiedges. In general, the quotient Qn/t(V ) has one vertex and n semiedges, and it ¯ ¯ ¯ ¯ ¯ ¯ can be described as Qn/t(V ) = (D, V ; I, λ), where V = {0¯}, D = Zn, ¯ ¯ I : Zn → {0¯} and λ = id. This graph is called the n-semistar and denoted by stn. Another instance of this situation one can get a quotient graph by the antipodal mapping x 7→ −x on V . The quotient graph is called the halved cube. In particular if n = 3 the antipodal action on Q3 defines the quotient graph isomorphic to the complete graph K4 on 4 vertices.

7 c7 a7 b7

c3 a6 b5 3 5 6 a3 a5 c6 b b3 c5 b6

b1 b4 a c c1 a2 2 c2 a4 1 4 1 b 4 a1 2 c4

b0 a0 c0 0

Figure 2.1: A covering projection Q3 → st3, given by xi → x 2.1. Graph coverings and Two actions 25

Example 2.2. (The n-cube covers the n-dipole) For another covering projection from the n-cube, we use the same notation as in Ex- n ample 2.1. Let T : Z2 → Z2 be a linear transformation defined by Pn−1 (x0, x1, x2, . . . , xn−1) 7→ i=0 xi. Then the kernel K := Ker(T ) consists of the vectors expressible as a sum of even number of vectors in the n standard basis B for V = Z2 . All translations by elements in K form an automorphism group t(K) of Qn, which acts semiregularly on the vertex and dart sets of cube Qn. Note that the translation group t(K) is n−1 isomorphic to Z2 , generated by t(ε0 +ε1), t(ε1 +ε2), . . . , t(εn−2 +εn−1), where εi ∈ B. We may form a quotient graph described by Qn/t(K) = ¯ ¯ ¯ ¯ ¯ ¯ ¯ (D, V ; I, λ), where V = Z2, D = Z2 × Zn, the incidence function I is the projection on the first coordinate and the dart-reversing involution λ¯ takes (i, j) 7→ (i + 1, j). This graph is called the n-dipole and denoted by D(n). It has two vertices and joined by n parallel edges Define p : Qn → D(n) by p(a, εi) = (0, i) if a ∈ K, and p(a, εi) = (1, i), otherwise. Clearly p : Qn → Qn/t(K) is a covering projection. We n have fib0 = K, fib1 = Z2 \ K, fib(0,εj ) = {(a, εj) | a ∈ K} and n fib(1,εj ) = {(a, εj) | a ∈ Z2 \ K} for each j ∈ Zn. See Figure 2.3.

Consider an arbitrary dart x ∈ X with its initial vertex u. By deﬁnition, for each vertex u˜ ∈ ﬁbu there exists a unique dart x˜ initiating atu ˜ which is the lift of x, that is, px˜ = x. This means that every walk of length 1 starting at u lifts uniquely to a walk starting at u˜. Consequently, by induction we obtain the following “Unique walk lifting theorem”. (For the classical topological variant, see [Ma91, pp. 151].)

Proposition 2.1. [Walk lifting theorem] Let p : X˜ → X be a covering projection of graphs and let W : u → v be an arbitrary walk in X. Then: ˜ (1) For every vertex u˜ ∈ ﬁbu there is a unique walk W which projects to W and has u˜ as the initial vertex.

(2) Homotopic walks lift to homotopic walks.

(3) If X is connected, then the cardinality of ﬁbu does not depend on u ∈ X. 26 Chapter 2. Coverings and liftings of automorphisms

Proof. (1) Use induction on the length of W . If |W | = 1 then the statement holds by the definition of p. Let W = W1x where x is a ˜ dart. By the induction hypothesis W1 lifts to a unique walk W1 with an initial vertex u˜ and a terminal vertexv ˜. Then W lifts to the unique ˜ walk W1x˜, wherex ˜ is the unique lift of x based at v. (2) Let P and Q be homotopic walks. It is enough to prove the statement for the case when Q arises from P by cancelling or adding a subsequence of the form xx−1. Let Ix = v and letv ˜ be the lift of v traversed by the unique lifts P˜ and Q˜. By part (1) xx−1 lifts to a unique v˜-based closed walk x˜x˜−1. It follows that P˜ and Q˜ are homotopic. (3) Since X is connected, for any two vertices u, v there exists a walk W joining u to v. Assume W = x is of length 1. Then there are |fibu| lifts of x. The terminal points of these lifts are all distinct, otherwise we get a pair of lifts of x˜−1 based at the same vertex contradicting part −1 (1) . Hence |fibu| ≤ |fibv|. Replacing W by W we get |fibv| ≤ |fibu|, hence |fibu| = |fibv|. Since X is connected, by induction the statement extends to a pair of vertices at any distance. 2

Note that Proposition 2.1 may fail when considering a path starting at the free endpoint of a semiedge in the 1-CW complex associated with a graph. This difficulty does not occur with combinatorial graphs since we only consider walks which start and end at vertices. It follows from Proposition 2.1 that the cardinality |fibu| of a covering projection p : X˜ → X, where X is connected, does not depend on the vertex u. If |fibu| = n, we say that the covering projection p is n-fold. Let p : X˜ → X be a covering projection. To simplify the notation we write π = π(X) and πu = π(X, u). As an immediate consequence of the unique walk lifting we have the following simple but useful observation.

Proposition 2.2. Let p : X˜ → X be a covering projection of graphs. Then there exists a right action of the groupoid π on the vertex set of X˜ defined by u˜ · W =v, ˜ where W : pu˜ → pv˜ is a walk in π and v˜ is the endvertex of the unique (lifted) walk W˜ over W starting at u˜. Moreover, u˜ 7→ u˜·W is a bijection fibpu˜ → fibpv˜. 2.1. Graph coverings and Two actions 27

This right action of the fundamental groupoid π (or the fundamental group) on the vertex set of X˜ is known as the monodromy action. In view of Proposition 2.1, the action of the fundamental groupoid can, in fact, be extended to an action of all walks. In particular, the u ˜ fundamental group π acts on ﬁbu. If X (and hence X as well) is connected, then this action is transitive. To simplify the notation we u denote by πu˜ = (π )u˜ the stabilizer ofu ˜ ∈ ﬁbu under the action of u u˜ π . In this case, the stabilizer πu˜ is the image of π by the group u˜ u homomorphism p∗ : π → π . In fact, one can prove the following.

Proposition 2.3. Let p : X˜ → X be a covering of connected graphs. Then the mapping p∗ : π(X,˜ u˜) → π(X, u) deﬁned by W˜ 7→ pW˜ = W (W˜ a closed u˜-based walk) is a monomorphism of groups. Conversely, let X be a connected graph and G ≤ π(X, u) be a subgroup of ﬁnite index. Then there is a connected graph X˜ and a covering p : X˜ → X such that G = p∗(π(˜u, X˜)).

As an application we can prove the following theorem by Schreier.

Theorem 2.4. Let Fβ be a free group of rank β. Let H < Fβ be a subgroup of index m. Then H is a free group of rank m(β − 1) + 1

Proof. By Lemma 1.2 Fβ can be identiﬁed with a fundamental group of a one-vertex graph X with β loops. By Proposition 2.3 H is a fundamental group of a graph p : X˜ → X covering X. Hence H = π(X˜) is a free group of rank V (X˜)−E(X˜)+1 = n−βn+1 = n(β−1)+1, where n is the number of folds. Since p∗ : π(X˜) → π(X) is a monomorphism, n = m and we are done. 2

Clearly, the actions of the fundamental groups at two distinct vertices of a connected graph are isomorphic. To show this, let W ∈ π be a walk from u to v. By Proposition 2.2 Wˆ :u ˜ 7→ u˜ · W is a bijection u v −1 and W∗ : π → π defined by S 7→ W SW is an isomorphism. Then ˆ u v (W,W∗) : (fibu, π ) → (fibv, π ) is an isomorphism of the actions. Another important group action on the fibre of a covering projection p : X˜ → X is defined with the ‘covering transformation group’. 28 Chapter 2. Coverings and liftings of automorphisms

An automorphism ϕ of X˜ satisfying p◦ϕ = p is called a covering transformation. The set of all covering transformations forms a group under the composition, denoted by CT(p) and called the covering transformation group. This group acts on Xe from the left. In fact, it acts on each fibre fibu. By definition, the (right) π-action on the vertex set of X˜ defined in Proposition 2.2 commutes with the (left) action of CT(p). Each fibre is fixed by CT(p) set-wise. Thus a group action of the covering transformation group on is defined on each fiber. As an immediate consequence of the unique walk lifting, we have the following observation.

Proposition 2.5. Let p : X˜ → X be a covering projection of connected graphs. Then CT(p) acts semiregularly on X˜; that is, CT(p) acts without ﬁxed points both on vertices and on darts of X˜.

Proof. It is enough to show that f˜ ∈ CT(p) has no ﬁxed vertices except the identity. Suppose that f˜(˜x) =x ˜. We need to show f˜(˜y) =y ˜ for any vertexy ˜ in X˜. To do this, take a walk W˜ fromx ˜ toy ˜. The uniqueness of walk lifting implies f˜(W˜ ) = W˜ , and hence f˜(˜y) =y ˜. 2

By Proposition 2.5, every covering transformation f˜ ∈ CT(p) is uniquely determined by the image of a single vertex of X˜. The fact that CT(p) acts semiregularly for a connected covering p is particularly important. Therefore, for the rest of this section, as well as for most of the book, the graphs X and X˜ will be assumed to be connected.

By the semiregularity of CT(p) we have |CT(p)| ≤ |fibu|, with equality occuring if and only if CT(p) acts regularly on each fibre. The covering projection is called regular if CT(p) acts regularly on each fibre. In Figure 2.2, the projection (b) is regular, but (a) is not.

There is an alternative approach to regular coverings of graphs. Given a connected graph X = (D,V ; I,L) let G ≤ Aut(X) be a subgroup acting semiregularly on the both sets D, V of darts and vertices of X. The regular quotient X/G = (D,¯ V¯ ; I,¯ L¯) is deﬁned by setting ¯ ¯ ¯ ¯ D = {[x]G|x ∈ D}, V = {[v]G|v ∈ V }, I[x]G = [Ix] and L[x]G = [Lx]G. 2.1. Graph coverings and Two actions 29

w3 w2

w1 u3 w2 v3 u2 v2 w1 u u2 v2 1 w0 v1

u1 v1 u0 v0

? ? w w

u v u v (a) (b)

Figure 2.2: Regular and irregular graph coverings

Proposition 2.6. (1) Let X˜ = (D,˜ V˜ ; I,˜ λ˜) be a connected graph and let A ≤ Aut(X˜) act semiregularly on the vertex set V˜ and also on ˜ ˜ ˜ the dart set D. Then the mapping q : X → X/A taking x 7→ [x]A is a regular covering projection with CT(q) = A. (2) Any regular covering projection p : X˜ → X between connected graphs is equivalent to a covering q : X˜ → X/CT˜ (p) to the regular quotient, that is, there is a graph isomorphism α : X/CT˜ (p) → X such that α ◦ q = p. Proof. (1) The semiregularity of A-action on the vertices and darts of X˜ implies that q is a covering X˜ → X. The elements of A permute the ˜ darts in fib[x] = [x] for each x ∈ D. By definition, A acts regularly on [x]A, implying CT(q) = A. (2) If p : X˜ → X is regular then A = CT (p) acts regularly on the fibres over each vertex and each dart. It follows that the fibres are formed by the orbits of A acting semiregularly on vertices and darts. Hence there is a one-to-one correspondence between the orbits of A on V˜ and D˜ and elements of V and D, respectively, determining the required graph isomorphism α. 2 30 Chapter 2. Coverings and liftings of automorphisms

Hence, the coverings in Examples 2.1, 2.2 are all regular with the n n covering transformation groups Z2 and K C Z2 , respectively.

Exercises

n 2.1.1. Prove that |Aut(Qn)| = 2 n! and it is transitive on vertices, arcs and ∼ n 2-arcs but not on 3-arcs. Hint: Aut(Qn) = Z2 o Sn consisting of the n (left) translations of V = Z2 and the permutations of n coordinates. 2.1.2. Let p : X˜ → X be a covering of graphs. Prove that the mapping p∗ : π(X,˜ u˜) → π(X, u) defined by W˜ 7→ pW˜ = W (W˜ a closedu ˜- based walk) is a monomorphism of groups. 2.1.3. * Let X be a connected graph and G ≤ π(X, u) be a subgroup. Prove that there is a graph X˜ and a covering p : X˜ → X such that G = p∗(π(˜u, X˜)). 2.1.4. * An inclusion of the trivial group into the fundamental group of a connected graph X induces a so-called universal cover over X. The universal cover X˜ over X is characterized by the following property: if Y → X is a covering of connected graphs then X˜ covers Y . Describe a procedure constructing the respective covering graph. De- scribe the universal cover over a two vertex graph, where the two vertices are joined by three internally disjoint paths of lengths 1, 2 and 3. 2.1.5. For which graphs the universal cover is a finite graph? (countable graph ?) 2.1.6. Prove Theorem 2.14. 2.1.7. Given covering p : X˜ → X the (right) monodromy action of the fundamental group on a fibre fibu determines a permutation representation of π(u, X)/Core(p∗π(v, X˜)) called the monodromy group. Prove that in a regular covering the monodromy group and group of covering transformations are isomorphic.

2.2 Voltage spaces

The goal of this section is to explain, unify and generalize the classical concepts of ordinary, permutation and relative voltage assignments introduced by Alpert, Biggs, Gross, Tucker and others [GT87] in the 1970’s. In fact, this uniﬁed approach works in the general setting 2.2. Voltage spaces 31

of fairly arbitrary topological spaces, see [Ma98]. Compare also with [Su84]. A voltage space on a connected graph X is a triple (F,G; ξ) where G is a group acting on a nonempty set F (from the right) and ξ : π → G is a homomorphism from the groupoid π = π(X) to G. The group G is called the voltage group, F is the abstract ﬁbre and ξW is the voltage of a reduced walk W . Since the product of reduced walks is mapped by the homomorphism ξ to the product of voltages, it follows that any trivial walk carries the trivial voltage and, consequently, inverse walks carry inverse voltages. In order to deﬁne a voltage space on a graph it is obviously enough to specify the voltages on walks of length 1. Even more, we only need to specify the voltages on darts ξ : D → G subject to the condition

−1 ξx−1 = (ξx) .

Therefore, ξ is determined by the values on an orientation D+ ⊂ D. By induction one can then extend the assignment of voltages to all walks, not just to the reduced ones. Clearly, homotopic walks carry the same voltage. Note that a voltage associated with a semiedge is always an involution. If the voltage group acts faithfully, we speak of a monodromy voltage space; by taking the permutation representation of a faithful action we get the permutation voltage space as its canonical representative. If the voltage group acts regularly, we speak of a regular voltage space. In particular, if G acts on F = G by right translation, we speak of the Cayley voltage space (ordinary voltages in the terminology of Gross and Tucker [GT87]). Obviously, Gu := ξ(πu) is a group, called the local voltage group at the vertex u. Moreover, let W : u → v be a walk. Then the inner automorphism

# −1 W (g) = ξW gξW of G takes Gu onto Gv. A voltage space is called locally transitive whenever a (and hence all) local voltage group acts transitively on the abstract ﬁbre. 32 Chapter 2. Coverings and liftings of automorphisms

We say that a voltage space (F,G; ξ) on a graph X is associated with a covering projection p : X˜ → X if there exists a labelling ` : V˜ → F of the vertex set V˜ of X˜ such that, for each vertex u of X, the restrictions

ù = `|fibu : fibu → F are bijections and moreover,

(`, ξ):(V˜ , π) → (F,G) is a morphism of actions. In other words, for every (reduced) walk W : u → v of X and for eachu ˜ ∈ ﬁbu we have

`(˜u · W ) = ù˜ · ξW . (For simplicity, we use the same dot sign for the action of π and the ˜ ˜ action of G.) Two coverings p1 : X1 → X and p2 : X2 → X are said ˜ ˜ to be isomorphic if there exists a graph isomorphism Φ: X1 → X2 such that p1 = p2Φ. Voltage spaces associated with isomorphic covering projections are called equivalent. Note that the action of ξW ˆ on F “represents” the bijection W :u ˜ 7→ u˜ · W from fibu to fibv introduced as the monodromy action in Proposition 2.2: indeed, its permutation representation W ∈ SymL F is precisely the permutation ξW ∈ SymR F . Furthermore, from the connectedness, one can show u u that (ù, ξ) : (fibu, π ) → (F,G ) is an epimorphism of actions with ù being a bijection. Hence Gu acts on F “in the same way modulo rela- u belling” as π does on fibu, and thus both actions have “the same local monodromy”. In fact, it is the local group which is responsible for the structure of the covering. In particular, a voltage space associated with a covering projection defined on a connected graph is necessarily locally transitive, and as in the classical case [Sk86], a covering projection is regular if and only if Gu has a normal stabilizer. Each covering projection of graphs gives rise to an associated voltage space, for instance to the permutation voltage space (cf. [GT77]). Theorem 2.7. Let p : X˜ → X be a covering projection between connected graphs. Then there exists a locally transitive permutation voltage space associated with p. Conversely, every locally transitive voltage space (F,G; ξ) on a graph X derives a covering projection X˜ → X whose associated voltage space 2.2. Voltage spaces 33

is (F,G; ξ). Such a covering is determined up to isomorphism, and if X is connected then so is X˜.

We denote by Xξ → X a covering projection derived from a voltage space (F,G; ξ) on a graph X.

Proof. We set F = fibb, where b is a fixed vertex. By Proposition 2.1(3) each fibre fibv has the same cardinality. Take any labeling `v : F → fibv. For each walk W : u → v, we set the voltage ξW to be the permutation ˆ W ∈ SymR F representing the bijection W . That means `(˜u)·ξW = `(˜v) if and only ifu ˜ · W =v ˜. Now the voltage group G ≤ SymR F is generated by the voltages ξW , where W ranges through all walks in ˜ X. Since X is connected, for any two verticesu ˜1, u˜2 ∈ fibb there is a ˜ (reduced) walk W :u ˜1 → u˜2. It follows that there is a u-based closed ˜ walk W = p(W ) such thatu ˜1 · W =u ˜2. Hence, `(˜u1) · ξW = `(˜u2) and u G is transitive on fibu. The labelling ` and the voltage assignment ξ is defined such that for eachu ˜ ∈ fibu we have `(˜u · W ) = ù˜ · ξW . Hence (`, ξ):(V˜ , π) → (F,G) is a morphism of actions. Conversely, given locally transitive permutation voltage assignment (F, G, ξ) on X = (D,V ; I, λ) we define the derived covering graph X˜ = ˜ ˜ ˜ ˜ (D, V, I, λ) as follows. For u ∈ V or x ∈ D we set fibu = {u} × F ˜ ˜ and fibx = {x} × F . Hence V = V × F and D = D × F . We set ˜ ˜ I(xi) = (Ix)i and λ(xi) = λ(x)j, where j = i · ξx. The mapping ˜ p : xi 7→ x takes surjectively D onto D and it is locally bijective. Let ˜ I(x) = u. We have p(I(xi)) = p(ui) = u = I(x) = I(p(xi)). Further, ˜ p(λ(xi)) = p(λ(x)j) = λ(x) = λ(p(xi)). It follows that p is a covering ˜ X → X. Moreover, the natural projection ` : ui 7→ i determines a morphism of actions. Indeed, the equality `(˜u · W ) = ù˜ · ξW holds by definition of λ˜ for walks of length one, and it extends (by induction) to walks of any length. If X is connected and (F, G, ξ) is locally transitive then X˜ is connected. Indeed, any two fibres fibu, fibv are joined by a lift of a walk W joining u to v in X. Since (F, G, ξ) is locally transitive for any i, j ∈ F , u there is an element ξW in the local group G with W ∈ πu, such that i · ξW = j. The unique lift of W with initial vertex ui joins ui to uj. Consequently, X˜ is connected. The isomorphism problem is solved in Theorem 2.12. 2 34 Chapter 2. Coverings and liftings of automorphisms

The lift X˜, as well as the covering p : X˜ → X determined by a voltage space (F, G, ξ) will be called the derived graph, and the derived covering, respectively. In particular, if the covering projection p is n- fold we may assume that fibu = {u1, . . . , un} for each vertex u of X and that the label of the vertex ui is i ∈ {1, . . . , n}. Then the permutation voltage ξx ∈ Sn of a dart x joining the initial vertex u to the terminal −1 vertex v = Ix is defined by the following rule: ξx(i) = j if and only if there is a dart x˜ ∈ fibx joining ui to vj. With a regular covering projection we can associate a Cayley voltage space (G, G; ξ) as follows. First label the elements of every vertex fibre by elements of G ∼= CT(p) so that the left action of CT(p) is viewed as the left translation on itself. Then the voltage action of G is the right translation on itself, that is, W (g) = gξW (cf. [Gr74]). We have the following theorem (compare with [GT87, pp. 57–71]). Theorem 2.8. Let p be a regular covering projection between connected graphs. Then there exists an associated locally transitive Cayley voltage space. Conversely, every locally transitive regular (in particular, Cayley) voltage space (F,G; ξ) on a graph X derives a regular covering projection X˜ → X whose associated voltage space is (F,G; ξ). Such a covering is determined up to isomorphism, and if X is connected then so is X˜.

We denote by X ×ξ G → X a regular covering projection derived from a Cayley voltage space (G, G; ξ) on a graph X. Proof. We present only the constructive part of the proof. First recall a regular action of G on F is equivalent to a Cayley action of G on itself by right translation. Hence we may assume that the voltage space is a Cayley voltage space (G, G; ξ) is defined on a graph X = (D,V ; I, λ). ˜ ˜ ˜ ˜ ˜ ˜ Define X = (D, V ; I, λ) by setting D = D × G = {xk | x ∈ D, k ∈ G}, ˜ ˜ ˜ V = V × G = {vk | v ∈ V, k ∈ G}, I(xk) = I(x)k and λ(xk) = ˜ (λx)kξ(x). Then the first coordinate projection X → X is a covering projection associated with (G, G; ξ) and the graph X˜ is the derived graph determined by the Cayley voltage space (G, G; ξ). The covering isomorphism problem is settled in the next section, see Theorem 2.14. 2 2.2. Voltage spaces 35

011 111

110 010

101 001

000 100 ↓

010- 100- 0 001- 1

Figure 2.3: Q3 covers D(3)

Example 2.3. (Cayley graphs are regular covers of monopoles). Let us call a graph with one vertex and arbitrary number of darts a monopole. Recall the deﬁnition of the Cayley graph. Given a group G and a symmetric generating (multi)set S = S−1 of G the Cayley graph Cay(G, S) is the graph (G×S,G; IS, λS), where IS = pr1 is the projection onto the −1 ﬁrst factor and the involution on darts is given by λS(g, s) = (gs, s ). (Note that S being possibly a multiset allows Cayley graphs to have repeated generators.) Consider the monopole X = (S, v; I, λ) where I is the constant function and λ(s) = s−1. It is easy to see that the projection onto the second factor pr2 : G × S → S extends to a regular covering projection Cay(G, S) → X. Conversely, let (G, G; ξ) be an arbitrary Cayley voltage space on a monopole X = (D, v; I, λ). Then the derived covering graph is the Cayley graph Cay(G, S) where S = {ξx | x ∈ D} is the corresponding 36 Chapter 2. Coverings and liftings of automorphisms

symmetric (multi)set of generators for the group G. This shows that Cayley graphs are nothing but regular covers of monopoles. Summing up, by allowing semiedges we have obtained a characterization of Cayley graphs which overcomes the trouble encoun- tered in the classical approach that not all Cayley graphs were regular coverings of bouquets of circles [GT87, pp. 68–69, 272].

Observe that the voltage group can be larger than really necessary. The following proposition shows that any given voltage space can be replaced by an equivalent voltage space where the local groups do not depend on the vertex and coincide with the whole voltage group.

Proposition 2.9. Let (F,G; ξ) be a voltage space associated with a covering projection p : X˜ → X of connected graphs, and let b ∈ X be a base vertex. Then there exists an equivalent voltage space (F,G; ξ0) all of whose local voltage groups are the same and the local group at b has not changed.

Proof. For each vertex u ∈ X choose a preferred walk W u : u → b, where W b is the trivial walk at b. This can easily be done, for instance, by specifying a spanning tree T for X and choose W u : u → b as a walk in T . Let W : u → v be an arbitrary walk in X. Deﬁne 0 −1 0 ξW = (ξW u ) ξW ξW v . Then, the new voltage ξ of each preferred walk 0 b is trivial and ξS = ξS for every closed walk S ∈ π . Now, modify the 0 labelling of ﬁbres according to the rule `t = `t · ξW u , t ∈ X. It is easy to see that the new voltage space has all the claimed properties. 2

Voltage space constructed in the proof of Proposition 2.9 will be called T -reduced. Its characteristic feature is that all the tree-darts are assigned by a trivial voltage. Since any two vertices can be joined by a walk carrying the trivial voltage, no walk can have voltage outside Gb. For if a walk W : u → v had a voltage not in Gb, then the closed walk S = (W u)−1WW v at b 0 0 b would have its voltage ξS = ξW 6∈ G , which is absurd. Hence, the local group Gb can be chosen as the new voltage group without aﬀecting the covering. Therefore, assuming that a voltage space (F,G; ξ) satisﬁes Covering isomorphism problem 37

the condition Gb = G does not loss of any generality, but can sometimes considerably simplify the notation. Hence we have the following corollary.

Corollary 2.10. Let p : X˜ → X be a (regular) covering of connected graphs. Then there is an associated permutation (Cayley) voltage space (F,G; ξ) such that G = Gu for any vertex u of X.

Exercises

2.2.1. Prove that generalized Petersen graphs GP (n, k) can be deﬁned by means of a Cayley voltage space on a 2-vertex graph. 2.2.2. Find all regular quotients of the 3-dimensional cube and determine the respective Cayley voltage spaces. 2.2.3. Construct all 4-fold coverings over a bouquet of two cycles and distinguish them up to isomorphism. 2.2.4. Prove that a Cayley voltage space (G, G; ξ) on a connected graph X determines a derived covering X˜ → X with X˜ connected if and only if every local group Gb = G. 2.2.5. A Cayley voltage space (Z2, Z2; ξ) with ξ(x) = 1 ∈ (Z2, +) determines a canonical double cover X˜ over X. Show that X˜ is connected if and only if X is not bipartite. 2.2.6. Let X be a connected graph with v vertices and e edges. Let k = k e − v + 1, G = Z2 and T be a spanning tree of X. Let x1, x2, . . . , xk be −1 a set of cotree darts such that xj 6= xi for any i 6= j. Let ξ(xi) ∈ G forms a generating set for G.

(1) Prove that the 2k cover over X˜ deﬁned by the Cayley voltage space (G, G, ξ) is connected.

(2) Describe the kernel of the associated epimorphism W → ξW from the fundamental group π(X) → G. 2.2.7. * Prove Theorem 2.8.

2.3 Covering isomorphism problem

We deﬁne a lifting of any graph morphism as follows: For a given morphism f : Y → X of graphs and a covering projection p : X˜ → X, if there exists a morphism f˜ : Y → X˜ such that the diagram 38 Chapter 2. Coverings and liftings of automorphisms

˜ ? X f˜ p f Y / X commutes, then f˜ is called a lifting of f. One may ask naturally when the morphism f can be lifted? The lifting problem shall be concerned with the following two direc- tions in this book. Firstly, it is related to the isomorphism problem of the covering projections of a given graph. Secondly, given covering projection p between two graphs we will investigate which automorphisms of the base graph lift along p. In this section we concentrate to the covering isomorphism problem while the lifting automorphism problem will be discussed in the following sections. Theorem 2.11. [Lifting Criterion] Let p : X˜ → X be a covering projection between connected graphs and let f : Y → X be a morphism of graphs. Let Y be connected. Then, the morphism f has a lift f˜ : ˜ Y → X if and only if for any vertex y0 ∈ Y , there exists a vertex ˜ x˜0 ∈ ﬁbf(y0) such that f∗(π1(Y, y0)) ⊂ p∗(π1(X, x˜0)).

ρ g(y) X˜ x˜0

˜ f p ? y σ f ◦ σ f(y) Y f - X y0 σ2 x0 f ◦ σ2

Figure 2.4: Lifting of f 2.3. Lifting and covering isomorphism problems 39

Proof. (⇒) is easy: If f lifts to f˜, then ˜ ˜ ˜ f∗(π1(Y, y0)) = (p◦f)∗(π1(Y, y0)) = (p∗ ◦f∗)(π1(Y, y0)) ⊂ p∗(π1(X, x˜0)). (⇐): Suppose that ˜ f∗(π1(Y, y0)) ⊂ p∗(π1(X, x˜0)). ˜ ˜ Then a lift f :(Y, y0) → (X, x˜0) of f can be defined as follows: For any vertex y ∈ Y , choose a walk σ from y0 to y. Then f ◦ σ is a walk from the vertex x0 to a vertex x = f(y). Now, by the “Unique walk lifting theorem” (see Proposition 2.1), the walk f ◦ σ lifts uniquely to a walk starting from the vertex x˜0. Say ρ for such lifting walk. Now, define f˜(y) as the end vertex of ρ. To show that f˜(y) is well defined, we need to prove that show f˜(y) is independent of a choice of a walk σ from from y0 to y. To do this, let σ2 be another walk from from y0 to −1 −1 y. Then, σ · σ2 is a closed walk at y0 and [σ · σ2 ] ∈ π1(Y, y0). By the −1 hypothesis, a closed walk (f∗[σ]) · (f∗([σ2]) in (X, x0) can be lifted to ˜ a closed walk in X based atx ˜0. Hence, the liftings of f ◦ σ and f ◦ σ2 have the same end vertex. Hence, the lifting f˜(y) is well defined. 2

Let Γ be a group of automorphisms of the graph X. Two coverings e pi : Xi → X, i = 1, 2, are said to be isomorphic with respect to Γ if there e e exist a graph isomorphism Φ : X1 → X2 and a graph automorphism γ ∈ Γ such that the diagram

e Φ - e X1 X2

p1 p2 ? γ ? X - X commutes. Or, equivalently, if there exists a γ ∈ Γ such that γp1 can be lifted to an isomorphism Φ. Such a lifting Φ is called a covering isomorphism with respect to Γ. Note that for any group Γ of automorphisms of G, the covering isomorphic relation with respect to Γ on the set of coverings of G is an equivalence relation. As a labelled version of the base graph, the group Γ is often taken as the trivial group. 40 Chapter 2. Coverings and liftings of automorphisms

When Γ is trivial, we simply say that two covering projections are e e isomorphic. In particular, when X1 = X2 with p1 = p2 Φ is a covering transformation. The number β(X) = |E(X)|−|V (X)|+1 is called the Betti number of X. Let T be a spanning tree of a connected graph X. Recall that a permutation voltage assignment φ is said to be T -reduced if φ assigns the identity to the arcs of T . Let CT (X; r) denote the set of T -reduced permutation voltage spaces ({1, 2, . . . , r}, Sym(r), φ), on X. In order to find an algebraic property when two r-fold coverings p : Xφ → X and q : Xψ → X are isomorphic, we first observe that an isomorphism Φ: Xφ → Xψ of the coverings takes the fibre over a vertex v onto −1 −1 the fibre over v. Thus, Φ|p−1(v) : p (v) → q (v) is a permutation on the r vertices {v1, v2, . . . , vr} for each v ∈ V (X) and it can be considered as a permutation on the index set F = {1, 2, . . . , r}. Now, we define f : V (X) → Sym(r) by f(v) = Φ|p−1(v) for v ∈ V (X) so that Φ(u, h) = (u, f(u)(h)) for each vertex (u, h) of the covering graph Xφ. Now, for a dart uv of X, if (u, h) is adjacent to (v, k) in Xφ, then φ(uv)(h) = k and Φ(u, h) = (u, f(u)(h)) is adjacent to Φ(v, k) = (v, f(v)(k)) in Xψ. Thus, we have ψ(uv)f(u) = f(v)φ(uv), or ψ(uv) = f(v)φ(uv)f(u)−1 for all uv ∈ D(X). The converse is left to the reader as an exercise.

Theorem 2.12. Let X be a graph and φ, ψ be permutation voltage assignments deﬁned on X. Two r-fold coverings p : Xφ → X and q : Xψ → X are isomorphic if and only if there exists a function f : V (X) → Sym(r) such that ψ(uv) = f(v)φ(uv)f(u)−1 for each arc uv. Moreover, if φ and ψ are T -reduced (with respect to some spanning tree T ) then the coverings p and q are isomorphic if and only if there exists a permutation σ ∈ Sym(r) such that ψ(uv) = σφ(uv)σ−1 for each arc uv not in T .

Given graph X choose a spanning tree T and an orientation D+. Set β = β(X). Denote Iso (X; r) the number of r-fold coverings of a graph X and by Isoc (X; r) denote the number of connected ones. Let β be β(X). Then a T -reduced permutation voltage assignment can be considered as a β-tuple of permutations in Sym(r) by labeling the 2.3. Lifting and covering isomorphism problems 41

positive darts in D(X) − D(T ) as e1, e2, . . . , eβ, and the set CT (X; r) can be identiﬁed as

CT (X; r) = Sym(r) × Sym(r) × · · · × Sym(r), (β factors).

We deﬁne an Sym(r)-action on the set CT (X; r) by simultaneous coordinatewise conjugacy: For any g ∈ Sym(r) and any (σ1, . . . , σβ) ∈ CT (X; r),

−1 −1 −1 g(σ1, σ2, . . . , σβ) = (gσ1g , gσ2g , . . . , gσβg ).

It follows from Theorem 2.12 that two T -reduced permutation voltage assignments φ and ψ in CT (X; r) yield isomorphic coverings of X if and only if they belong to the same orbit under the Sym(r)-action. Hence, one can have the following theorem from the orbit-counting theorem commonly known as Burnside’s Lemma.

Theorem 2.13. The number of r-fold coverings of X is

X ¡ ¢β−1 `2 `r Iso (X; r) = `1! 2 `2! ··· r `r! .

`1+2`2+···+r`r=r

An algebraic characterization of two isomorphic graph coverings given in Theorem 2.12 can be rephrased for regular coverings as follows [HKL96, Sk86].

Theorem 2.14. [HKL96, Sk86] Two connected regular coverings Xξ1 → X and Xξ2 → X derived from T -reduced Cayley voltage assignments (A, A, ξi), i = 1, 2 on X are isomorphic if and only if there exists an automorphism σ ∈ Aut(A) such that σ ◦ ξ1 = ξ2.

Exercises

2.3.1. Compute the number of 4-fold coverings over a 3-dipole and the number of regular 4-fold coverings and construct all the regular ones. 2.3.2. Under what condition two semi-regular subgroups of the automorphism group determine isomorphic projections onto the respective quotients? 42 Chapter 2. Coverings and liftings of automorphisms

2.4 Lifting automorphism problem - classical approach

Structural properties of various mathematical objects are to a large extent reﬂected by their automorphisms. It is therefore natural to compare related objects through their automorphism groups. Unfor- tunately, the automorphism group of an object is not a functorial invariant in general. This excludes relatively easy comparison of the respective automorphism groups via morphisms. Much better chances to get relevant results occur when the general problem is relaxed. Usu- ally, one considers a morphism m : Y → X satisfying some additional properties and asks whether associated with an automorphism f of X there exists an automorphism f˜ of Y such that the diagram

f˜ Y - Y m m ? f ? X - X commutes. This is the essence of the problem of lifting automorphisms. Its most typical, well studied, but somewhat more general instance is lifting continuous mappings along topological covering projections. Nevertheless, speciﬁc questions about lifting automorphisms received less attention. The automorphism lifting problem has recently appeared in the context of graph coverings and maps on surfaces. Due to a discrete nature of these objects, their covering projections are usually treated combinatorially rather than topologically. We are thus led to examine the lifting problem from this point of view as well. We have seen that for a covering projection p : X˜ → X, any walk in X can be lifted to a walk in the covering graph X˜. Also every covering transformation is a lifted automorphism ϕ of the identity automorphism of X which means p ◦ ϕ = idX ◦ p. Let p : X˜ → X be a covering projection of graphs and let f be an automorphism of X. We say that f lifts if there exists a morphism f˜ of X˜, called a lift of f, such that the diagram 2.4. Lifting automorphism problem - classical approach 43

f˜ Xe - Xe p p ? f ? X - X commutates. When f˜is a lift of f, we say f˜projects onto f. Later it will be shown in that if f lifts and f has a ﬁnite order in Aut(X), so does f −1 and hence a lift of an automorphism should be an automorphism. More generally, let A ≤ Aut(X). Then the lifts of all automorphisms in A form a group, the lift of A, often denoted by A˜ ≤ Aut(X˜). In particular, the lift of the trivial group is the covering transformation group CT(p). ˜ There is an associated group epimorphism pA : A → A with CT(p) as its kernel. The set of all lifts Lft(f) of a given f ∈ Aut(X) is a coset of CT(p) in A˜. By Proposition 2.5, every covering transformation f˜ ∈ Lft(id) is uniquely determined by the image of a single vertex of X˜. In fact, this is true for all lifts of an automorphism of the base graph X. Proposition 2.15. Let p : X˜ → X be a covering projection of connected graphs and let f ∈ Aut(X) have a lift. Then each f˜ ∈ Lft(f) is uniquely determined by the image of a single vertex (as well as by the image of a single dart) of X˜. Proof. Let f˜ andg ˜ be two lifts taking a vertexx ˜ to the same vertex y˜. Then f˜g˜−1 belongs to CT(p) and ﬁxes x˜. It follows that f˜g˜−1 = 1 implying f˜ =g ˜. 2

Every lifted automorphism in a covering projection p : X˜ → X preserves the fibres over darts and also the fibres over vertices setwise. Thus the fibres form a system of blocks of imprimitivity of the action of the group of all lifts. ˜ Example 2.4. (A regular quotient Kn → Kn/Zn) Let X = Cay(Zn; Zn− ∼ {0}) = Kn and let X = (D,V ; I, λ) be a monopole defined by D = Zn − {0}, V = {u}, I : i 7→ u for i = 1, . . . , n − 1 and λ(i) = −i. ˜ Setting p(j, i) = i a covering p : X → X from Kn onto the monopole is ∼ n−1 defined. We have CT(p) = Zn. Set m = b 2 c. The monopole graph 44 Chapter 2. Coverings and liftings of automorphisms

n−1 X consists of b 2 c = m loops, and one semi-edge if n is even. The pairs of darts forming loops form a system of blocks, and for each loop there is an automorphism interchanging the two darts. It follows that |Aut(X)| = 2m ·m!. However |Lft(p)| ≤ |CT(p)|·|Aut(X)| = n2m ·m! < n! for every n ≥ 4. Employing the following Lemma 2.16 one can prove ∗ that Lft(p) is the Frobenius group which is isomorphic to Zn o Zn. Hence the size of the automorphism of a covering graph may be much larger then the group of of lifts.

Example 2.5. (Canonical double coverings:) Let X = (D,V ; I, λ) be a graph. Its canonical double covering is defined by setting D˜ = ˜ ˜ ˜ D × Z2, V = V × Z2, I(xi) = (I(x))i and λ(xi) = λ(x)i+1. It can be easily seen that the mapping xi 7→ x determines a regular covering ˜ p : X → X. The nontrivial element of CT(p) takes xi 7→ xi+1, and every ˜ automorphism ψ ∈ Aut(X) lifts to two automorphisms ψj, j ∈ Z2, ˜ ˜ defined by ψj(xi) = ψ(x)j+i, for i ∈ Z2. Hence, |Aut(X)| ≥ 2|Aut(X)|. If the equality holds, we shall call the graph X stable. Thus stability ˜ means Lft(p) = Aut(X). The octahedral graph O3 is an example of an unstable graph: Let u, v be two non-adjacent vertices in O3. Then ˜ α = (u1, v1) is an automorphism of Aut(X) interchanging the lifts u1 and v1 but fixing u0 and v0. Hence it does not preserve the fibres over u and v. One can prove that complete graphs are stable. On the other hand, the dodecahedral graph is unstable, as well as many other cubic arc-transitive graphs. Direct computation or checking the list by Conder [CD02] shows that the dodecahedral graph is 2-transitive while its canonical double cover is a 3-transitive cubic graph.

In case of imprimitive action of a group G on a graph X, group theorists use to form a (simple) quotient graph X whose vertices are blocks of imprimitivity, and they investigate the induced action of G on X. Recall that the kernel K of the action is formed by elements of G ﬁxing each of the blocks set-wise is a normal subgroup K C G. In particular, CT(p) is a normal subgroup of the group of all lifts along ˜ a covering p : X → X. Two blocks B1, B2 are adjacent if there is an edge uv in X, where u ∈ B1 and v ∈ B2. The mapping v 7→ Bv, where 2.4. Lifting automorphism problem - classical approach 45

v ∈ Bv, extends to a covering X → X if and only if the vertices of B form an independent set. The following lemma establishes the mentioned facts in case of regular coverings explicitly.

Lemma 2.16. Let p : X˜ → X be a regular covering between connected graphs and let A = Aut(X). Then the normalizer NAut(X˜)(CT(p)) of CT(p) projects. In particular, the normalizer NAut(X˜)(CT(p)) coincides with the group Ae of all lifts.

Proof. Let A = CT (p). Then the covering p : X˜ → X is equivalent to ˜ ˜ ˜ X → X/A; x 7→ [x]A. For each f ∈ NAut(X˜)(CT(p)), the projection ˜ f :[x]A → [fx]A is well-deﬁned. Hence, all automorphisms in the normalizer NAut(X˜)(CT(p)) are lifts of automorphisms in X. The inverse inclusion is clear. 2

Example 2.6. (Groups of ﬁbre preserving automorphisms:) In a reg- ˜ ular covering p : X → X, the normalizer NAut(X˜)(CT(p)) is the group of all lifts of Aut(X). However, NAut(X˜)(CT(p)) may not coincide with the group of ﬁbre preserving automorphisms. For instance, take the complete bipartite graph Kn,n, n > 2, and 0 represent its vertices as Zn ×Z2 so that V = {j | j ∈ Zn}∪{j | j ∈ Zn} is the bipartition. Then one can decompose the edges of the complete bipartite graph Kn,n into n prefect matchings M0,M1,...,Mn−1, where 0 Mi is formed by the n edges joining a vertex j to a vertex (j + i) for i, j ∈ Zn. Coloring the edges of Mi by i ∈ Zn induces a regular covering projection p of Kn,n onto an n-valent dipole D(n) with CT(p) generated 0 0 by j 7→ j + 1 and j 7→ (j + 1) which is isomorphic to Zn. Note the automorphism group G = Aut(D(n)) is isomorphic Sn ×Z2 whose order is 2 · n!, hence

|Lft(Aut(X))| = |NAut(X˜)(CT(p))| = n|Aut(X)| = 2n · n!. ∼ On the other hand, the automorphism group of Kn,n is Aut(Kn,n) = Sn o Z2, the wreath product, which contains all permutations of vertices of each partite set and the interchanging of two partite sets and its 2 whose order is 2 · (n!) . Moreover, all automorphisms of Kn,n are ﬁbre 46 Chapter 2. Coverings and liftings of automorphisms

preserving and hence the group of ﬁbre preserving automorphisms is 2 the full group Sn o Z2. Hence there are 2 · (n!) ﬁbre preserving automorphisms and this number is grater than |NAut(X˜)(CT(p))| = 2n · n! for n > 2.

Remark: The above results related with the semi-regularity of CT(p) can be alternatively explained from the point of view of Theorem 1.7 by considering the monodromy action of the fundamental group πu on the ﬁbre ﬁbu.

The next three results, much in the spirit of classical theory [Ma91], are aimed at a smooth transition to a combinatorial treatment of lifting automorphisms. The ﬁrst one states that, brieﬂy speaking, an automorphism of the base graph has a lift if and only if this automorphism is consistent with the actions of the fundamental groups.

f˜W˜ f˜u˜ - f˜˜b f˜ f˜ p p - X˜ u˜ W˜ ˜b ? ? fW p p fu - p fb f ? f ? ? X - u W b

Figure 2.5: Proof of Theorem 2.17

Theorem 2.17. Let p : X˜ → X be a covering projection of connected graphs and let f ∈ Aut(X). Then f lifts to an f˜ ∈ Aut(X˜) if and only 2.4. Lifting automorphism problem - classical approach 47

if, for an arbitrarily chosen base vertex b ∈ X, there exists a bijection φ : ﬁbb → ﬁbfb so that the pair

b fb (φ, f∗) : (fibb, π ) → (fibfb, π ) becomes an isomorphism of the fundamental group acting spaces and ˜ f|fibb = φ. Moreover, there is a one-to-one correspondence f˜ ↔ φ between Lft(f) and all functions φ for which (φ, f∗) is such an isomorphism, with the relation

f˜u˜ = φ(˜u · W ) · fW −1 for any W : pu˜ → b in X.

That is, f˜u˜ is the terminal vertex of the lifting of the walk fW −1 which is initiated at vertex φ(˜u · W ). ˜ Proof. (⇒) : Let f be a lift of f, and let vertices b ∈ X andu ˜ ∈ ﬁbb be given. We ﬁrst show that for an arbitrary walk W : pu˜ → b, we have

f˜(˜u · W ) = f˜u˜ · fW. (2.1)

Indeed, let W˜ :u ˜ → u˜ ·W be a lifting of W . Then f˜W˜ : f˜u˜ → f˜(˜u·W ) projects to fW , implying (2.1). In particular, (2.1) holds true for each b ˜ u˜ ∈ fibb and W ∈ π . Since φ := f|fibb and f are bijections, we have the required isomorphism of actions. (⇐:) Conversely, let (φ, f∗) be such an isomorphism. Define the required lift f˜ on vertices of X˜ as follows. Let u˜ be an arbitrary vertex of X˜ and let u = pu˜. Choose an arbitrary walk W : u → b and set

f˜u˜ = φ(˜u · W ) · fW −1.

We have to show several things. First of all, this mapping is well deﬁned, that is, it does not depend on the choice of W . Indeed, let −1 W1,W2 : u → b be any two walks. Thenu ˜ · W1 = (˜u · W2) · W2 W1. −1 −1 Therefore, φ(˜u · W1) = φ(˜u · W2) · f(W2 W1) = φ(˜u · W2) · fW2 · fW1, −1 −1 and hence φ(˜u · W1) · fW1 = φ(˜u · W2) · fW2 . Next, we see from the deﬁnition of f˜ that

pf˜u˜ = p(φ(˜u · W ) · fW −1) = p(φ(˜u · W )) · fW −1 = fb · fW −1 = f(b · W −1) = fpu.˜ 48 Chapter 2. Coverings and liftings of automorphisms

We verify that it is a bijection. To show that it is onto, let v˜ be an arbitrary vertex of X˜ and choose W : pv˜ → fb arbitrarily. It is easily checked that the vertex φ−1(˜v · W ) · f −1W −1 is mapped to v˜. To show −1 ˜ ˜ −1 the injectivity, let φ(˜u1 · W1) · fW1 = fu˜1 = fu˜2 = φ(˜u2 · W2) · fW2 . ˜ ˜ Then pf(˜u1) = fpu˜1 = fpu˜2 = pf(˜u2) implying p(˜u1) = p(˜u2). It follows thatu ˜1,u ˜2 are in the same fibre and by the first part of the ˜ ˜ proof we may assume W1 = W2 = W . Consequently, fu˜1 = fu˜2 implies −1 −1 φ(˜u1 · W ) · fW = φ(˜u2 · W ) · fW . Since φ is a bijection, it follows u˜1 =u ˜2. Having f˜ defined on vertices it is immediate that the mapping extends to darts. We conclude that f˜ as above is a lift of f. This shows that taking the restriction Lft(f) → Lft(f)|fibb defines a function onto the set of all such φ for which (φ, f∗) is an isomorphism of fundamental group actions. The injectivity of this function follows from Proposi- tion 2.15. 2

It is a consequence of Theorem 2.17 that the problem of whether an automorphism f has a lift can be reduced to the problem of whether there is an automorphism between the fibres fibb and fibfb considered as the spaces of actions of fundamental groups. This in turn can be tested just by comparing how the automorphism maps a stabilizer under the action of the fundamental group. In this case, a stabilizer π˜b of the b ˜ ˜b action (fibb, π ) at b is the image of π by the group homomorphism ˜b b p∗ : π → π . In addition, it enables us to give a more explicit formula expressing how an arbitrary lifted automorphism acts on the vertex fibres. Theorem 2.18. Let p : X˜ → X be a covering projection of connected graphs, and let f ∈ Aut(X) and b be a vertex of X. Then there exists an isomorphism of actions b fb (φ, f∗) : (fibb, π ) → (fibfb, π ) if and only if f∗ maps the stabilizer π˜b of an arbitrarily chosen base ˜ fb point b ∈ fibb isomorphically onto some stabilizer πv˜ ≤ π . In this case we have v˜ = φ˜b. It gives a one-to-one correspondence v˜ ↔ φ between the set {v˜ ∈ fibfb | f∗π˜b = πv˜} and all such admissible isomorphisms φ. In particular, the number of all liftings of f equals the cardinality |{v˜ ∈ fibfb | f∗π˜b = πv˜}|. 2.4. Lifting automorphism problem - classical approach 49

Proof. (⇒) is clear. (⇐:) Let fπb = πfb. Eachu ˜ ∈ ﬁb can be written as u˜ = ˜b · S for ˜b v˜ b some S ∈ πb because X is connected. Deﬁne φ by setting φu˜ =v ˜ · fS. It is easy to check that (φ, f∗) is the required isomorphism of actions. The claimed one-to-one correspondence should also be clear since φ is completely determined by the image of one point. 2

As an immediate corollary of the above two theorems we have the following observation. There exists a covering transformation mapping u˜1 ∈ fibu tou ˜2 ∈ fibu if and only if the stabilizers coincide: πu˜1 = πu˜2 . From the fact that a covering projection is regular if and only if CT(p) acts transitively, a regular covering can be characterized by means of stabilizers under the action of the fundamental group. Corollary 2.19. A covering projection of connected graphs p : X˜ → X is regular if and only if, for an arbitrarily chosen vertex b of X, all the b stabilizers under the action of π on fibb coincide, that is to say, the stabilizer is a normal subgroup. b ˜ Recall that a stabilizer of the action (fibb, π ) at b is the image of ˜b ˜b b π by the group homomorphism p∗ : π → π .

It follows from Lemma 2.16 that if a group of automorphisms A lifts along a regular covering projection p then it lifts to an extension of CT(p) by A. Naturally, one can ask that under what condition the lifted group is isomorphic to a split extension CT(p) o A. This problem is investigated in [MNS00] and [Ma98]. It is a hard problem and will be discussed later again. A special case is given as follows. Proposition 2.20. Let p : X˜ → X be a regular covering projection of graphs. Let A ≤ Aut(X)v be a subgroup of the stabilizer of a vertex v. If A lifts then its lifting is a split extension of CT(p).

Exercises

2.4.1. Determine the lift of f ∈ Aut(st3) of order 3 along the covering described in Example 2.1. Describe the group of lifts. 2.4.2. Prove that if X = Kn is the complete graph and p : X˜ → X is the canonical double covering then Lft(p) = Aut(X˜). 50 Chapter 2. Coverings and liftings of automorphisms

2.5 Lifting of graph automorphisms in terms of voltages

A construction of a certain type of symmetrical graphs or maps is one of interesting topics in algebraic or topological graph theory. In most cases, it can be possible by using a lifting of automorphisms if the symmetry is described in terms of an automorphism. One of classical examples is a construction of inﬁnite families of cubic 5-arc-transitive graphs given by D.Z.˘ Djokovic, ”Automorphisms of graphs and coverings,” JCTB 16(1974), 243-247, or construction of highly symmetrical maps introduced in Biggs [Bi74]. A lot of recently published papers (see for instance [Feng, Kwak, Marusic, Malnic]) employs a lifting of automorphism to construct several types of symmetric graphs. Theo- retical backgrounds were developed in a paper by Malnic, Nedela and Skoviera [MNS00].

Let p : X˜ → X be a covering projection of connected graphs. By Theorem 2.17, an automorphism f ∈ Aut(X) has a lift f˜ if and only if, for an arbitrarily chosen base vertex b ∈ X, there exists a bijection φ : ﬁbb → ﬁbfb so that the pair

b fb (φ, f∗) : (fibb, π ) → (fibfb, π ) becomes an isomorphism of the fundamental group acting spaces and ˜ f|fibb = φ. Let (F,G; ξ) be a voltage space associated with the covering p : X˜ → X. As remarked in Section 2.2, the action of the fundamental b group π on fibb is “the same, modulo relabelling” as the action of the local group Gb on F . Can we transfer the isomorphism problem between the fibres as acting spaces of fundamental groups to a similar one by just considering the local voltage groups? The answer is yes, provided that certain additional requirements are imposed on the voltage space. This is especially relevant in the context of lifting groups rather than individual automorphisms. Let A be a group of automorphisms of X. We say that the voltage space (F,G; ξ) is locally A-invariant at a vertex v if, for every f ∈ A and for every walk W ∈ πv, we have

ξW = 1 =⇒ ξfW = 1. (2.2) 2.5. Liftings: in terms of voltages 51

We remark that we can speak of local invariance without specifying the vertex, if a local invariance at a vertex implies the same at all vertices. A voltage space is locally f-invariant for an individual automorphism f of X if it is locally invariant with respect to the cyclic group hfi generated by f. This is equivalent to requiring that condition (2.2) be satisfied by both f and f −1. If f has finite order or if X is a finite graph, then it is even sufficient to check (2.2) for f alone. Assume that the voltage space is locally A-invariant. Then for each f ∈ A, its inverse f −1 must also satisfy condition (2.2). Hence, a local A-invariance is equivalent to the requirement that for each f ∈ A there u u fu exists an induced isomorphism f# : G → G of local voltage groups such that the diagram

u f∗ πu - πfu

ξ ξ u ? f# ? Gu - Gfu

u u commutates; in other words, f#(ξW ) = ξf uW for any W ∈ π . It is clear that in order to transfer the isomorphism problem between the fibres as acting spaces of fundamental groups to local voltage groups, we must require a local invariance in advance. If the voltage space is, say, a monodromy one, then the existence of a lift implies a local invariance. However, this needs not be true in general. Theorem 2.21. Let (F,G; ξ) be a monodromy voltage space associated with a covering p : X˜ → X of connected graphs, and let f be an automorphism of X. Then the following statements are equivalent: (1) f has a lift, (2) the voltage space (F,G; ξ) is locally f-invariant, b b fb (3) the mapping f# : G → G taking ξW 7→ ξfW is a well-defined isomorphism of the local voltage groups as acting groups on F . Proof. (1) ⇒ (2). Let f have a lift f˜. By Theorem 2.17 the restric- ˜ u tion φ = f|u gives an isomorphism (φ, f∗) of the actions (fibu, π ) → fu (fibfu, π ). In particular, since (F,G; ξ) is an associated voltage space, 52 Chapter 2. Coverings and liftings of automorphisms

u ξ(W ) = 1 implies that W ∈ π acts on fibu as the identity. Since (φ, f∗) is an isomorphism of the actions, fW acts on fibfu as the identity. Since (F,G; ξ) is a monodromy voltage space, it follows that ξ(fW ) = 1 proving (2). u To show (2) ⇒ (3), it is enough to show that f#(ξW ) = ξfW for u u W ∈ π is well-defined. Let W1 and W2 be closed walks in π such that ξ = ξ . We want to prove ξ = ξ . Then 1 = ξ ξ−1 = W1 W2 fW1 fW2 W1 W2 u ξ −1 ∈ G . By the local f-invariance W1W2

−1 1 = ξ −1 = ξ −1 = ξf(W )ξ −1 = ξf(W )ξ f(W1W2 ) f(W1)f(W2 ) 1 f(W2 ) 1 f(W2) and we are done. (3) ⇒ (1). By Theorem 2.17, f has a lift if and only if f is consistent with the actions of fundamental groups. We need to find a bijection φ : fibu → fibfu such that φ(˜u · W ) = φ(˜u) · fW . Since (F,G; ξ) is an associated voltage space there are bijections ù : fibu → F and u u `fu : fibfu → F determining morphisms of actions (fibu, π ) → (F,G ), fu fu u (fibfu, π ) → (F,G ), respectively. By assumption, f# exists and it u u fu gives an isomorphism of actions (ϕ, f#):(F,G ) → (F,G ). We set −1 u u fu φ = `fu ϕù. Since f# is consistent with the actions of G and G , f is consistent with the actions of the respective fundamental groups on the fibres. That is, the diagram

(φ, f u) u ∗ - fu (ﬁbu, π ) (ﬁbfu, π )

(`u, ξ) (`fu, ξ)

? u ? (ϕ, f#) (F,Gu) - (F,Gfu) commutates. 2

For a regular voltage space associated with a regular covering, we can immediately infer that an automorphism lifts if and only if every closed walk of the trivial voltage is mapped to a closed walk of the 2.5. Liftings: in terms of voltages 53

same voltage. This result was previously observed in context of lifting automorphisms of regular maps in [GS93], [NS97] and in context of enumeration of coverings in [HKL96, HKL96].

Corollary 2.22. Let the covering as well as the voltage space be regular. Then the following three statements are equivalent:

(1) f has a lift, (2) the voltage space (F,G; ξ) is locally f-invariant, b b fb (3) the mapping f# : G → G taking ξW 7→ ξfW is a well-deﬁned isomorphism of the local voltage groups with respective actions on F .

Moreover, if (G, G, ξ) is a Cayley voltage space then Gb = G = Gfb and b f# is an automorphism of G. Note that in Condition (3) in Corollary 2.22, we could erase the requirement on consistency of the isomorphism ξW 7→ ξfW , since by regularity any group isomorphism is consistent with the (right) action of G on F . In particular, if (G, G, ξ) is a Cayley voltage space then group automorphisms are by deﬁnition consistent with the action of G on itself given by right multiplication. In what follows we show that it does not hold for locally transitive monodromy voltage spaces.

Example 2.7. Let X be a two-vertex graph with a singe edge joining them and two loops at each vertex. Formally let u, v be vertices and + let D = {xu, yu, xv, yv, z} be an orientation of X, I(xu) = I(yu) = −1 −1 −1 −1 I(xu ) = I(yu ) = u, I(xv) = I(yv) = I(xv ) = I(yv ) = v. Deﬁne a permutation voltage space on X by setting F = {1, 2, 3, 4}, ξxu = (1, 2),

ξyu = (3, 4), ξxv = (1, 3)(2, 4), ξxu = (1, 4)(2, 3) and ξz = 1. It is easy to see that the local groups Gu = Gv = h(1, 2), (1, 3)(2, 4)i is the dihedral −1 group of order 8. The mapping f : xu 7→ yv, yu 7→ xv, z 7→ z extends to a graph automorphism interchanging the vertices u, v. Then f# takes (1, 2) 7→ (1, 4)(2, 3) and (1, 3)(2, 4) 7→ (3, 4). Since it maps generators h(1, 2), (1, 3)(2, 4)i onto generators h(1, 4)(2, 3), (3, 4)i and the deﬁning relations are preserved, it extends to a group automorphism of D4 which is clearly not consistent with the action. Actually, it can be seen directly (inspecting the derived graph) that f has no lift. 54 Chapter 2. Coverings and liftings of automorphisms

The next corollary constitutes the last stage of combinatorialization of the lifting problem by reducing it to a system of equations in the symmetric group Sn, where n is the size of the ﬁbre.

Corollary 2.23. Let (F, SymR F ; ξ) be a locally transitive permutation voltage space on a graph X. Then, an automorphism f of X has a lift if and only if the system of equations

ξS · τ = τ · ξfS in SymR F has a solution τ, where S runs through a generating set for πb. Moreover, there is a one-to-one correspondence between all solutions and the lifts of f.

b b Proof. By Theorem 2.21(3) the mapping f# : ξW 7→ ξf#W , W ∈ π , is a group isomorphism Gb → Gfb consistent with the permutation actions. This happens if and only if there is a permutation ϕ : F → F b b such that ϕ(x · g) = ϕ(x) · f#(g) ∀g ∈ G . Representing the bijection b ϕ as an element τ of SymR F and taking g = ξW for some W ∈ π we get ξW · τ = τ · ξf#W . In particular, it holds for W forming the fundamental cycles with respect to some spanning tree. Conversely, u having the solution of ξS · τ = τ · ξfS, for the generators of G the u consistent isomorphism f# is determined by the images of generators u −1 f#(S) = ξfS = τ ξSτ. To see the last part of the statement assume that 1 · ξS = i for some u −1 ˜ element S of π . Let j = 1 · ξfS = 1 · τ ξSτ. We have f(ui) = vj for ˜ each ui ∈ ﬁbu, i = 1, 2, . . . , n. This way f is determined on ﬁbu. Then by Theorem 2.21 f˜ is determined everywhere. Thus a solution τ of the above system of identities determines precisely one lift. 2

Example 2.8. (Connected regular covers of the semistar st3 on which Aut(st3) can lift:) Take the semistar st3 as the base graph and denote a, b, c the three semiedges. Clearly, Aut(st3) = Sym(a, b, c) is the symmetric group of degree 3, which is generated by two cycles f = (a b c) and g = (b c). The fundamental group πu = ha, b, c | a2 = b2 = 2 c i = Z2 ∗ Z2 ∗ Z2. The system of equations for lifting Aut(st3) follows: ξa ·τ = τ ·ξb, ξb ·τ = τ ·ξc, ξc ·τ = τ ·ξa for lifting f, and ξa ·ω = ω ·ξa, ξb · ω = ω · ξc, ξc · ω = ω · ξb for lifting g. 2.5. Liftings: in terms of voltages 55

Indeed, a connected regular cover of the semistar st3 on which Aut(st3) can lift is isomorphic to a cubic Cayley graph based on a group G generated by 3-involutions and admitting group automorphisms per- muting the generators following the action of Sym(a, b, c). (See Exam- ple 2.3). The least examples include K4, K3,3, Q3 and GP (8, 3). However, the Petersen graph does not cover st3 at all, since such a covering would produce a 3-edge-colouring.

Example 2.9. (All n-fold coverings of the Petersen graph P on which Aut(P ) lifts:) Let V (P ) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} being the vertex set

0 1

8 6 9 5 2

4 3

Figure 2.6: The Petersen graph. labelled as in Figure 2.6. Since every covering projection of the Petersen graph P can be described by means of a permutation voltage space, it is suﬃcient to characterize all the permutation voltage assignments on the darts of P such that the derived covering has the required property. By Proposition 2.9 we may assume that the darts of an arbitrarily chosen spanning tree carry the identity permutation. Consequently, every covering projection P˜ → P is completely described by a 6-tuple of permutations ξ = (ξ0, ξ1, . . . , ξ5), where ξi is the voltage carried by the dart xi joining i to i + 1, for i = 0, 1,..., 5 with arithmetic mod 6. Since Aut(P ) is generated by the automorphisms f = (012345)(876), g = (03)(12)(45)(78), h = (01845)(29376), 56 Chapter 2. Coverings and liftings of automorphisms

Corollary 2.23 gives us three systems of equations ξSτ = τξfS, ξSη = 9 ηξgS and ξSω = ωξhS in Sn, where S ∈ π ranges over all elements of the fundamental group at the vertex 9. The fundamental group π9 is generated by the following six closed walks (for brevity indicated just as cyclic sequences of vertices) W0 = (96018), W1 = (98127), W2 = (97236), W3 = (96348), W4 = (98457) and W5 = (97506). Thus the above system of equations reduce to the following 18 equations in 9 variables ξ0, . . . , ξ5, τ, η, and ω, where every generator of π and every generator of Aut(P ) gives rise to one equation:

−1 −1 −1 −1 ξ0τ = τξ1 ξ0ω = ωξ1 ξ0η = ηξ2 ξ1 ξ3 −1 ξ1τ = τξ2 ξ1ω = ωξ0 ξ1η = ηξ3 −1 ξ2τ = τξ3 ξ5ω = ωξ2 ξ2η = ηξ2 −1 −1 −1 −1 ξ3τ = τξ4 ξ2ω = ωξ5 ξ3η = ηξ2 ξ4 ξ3 −1 ξ4τ = τξ5 ξ4ω = ωξ3 ξ4η = ηξ3ξ4ξ5 −1 ξ5τ = τξ0 ξ3ω = ωξ4 ξ5η = ηξ0ξ1ξ2. From the ﬁrst column it easily follows that the required six voltages −i i ξi can be expressed as ξi = τ ντ , i = 0, 1,..., 5, where ν and τ are permutations satisfying the identity ντ 6 = τ 6ν. Thus f lifts if and only if the covering projection is determined by the 6-tuple

−1 −2 2 −3 3 −4 4 −5 5 (ξ0, ξ1, . . . , ξ5) = (ν, τ ντ, τ ντ , τ ντ , τ ντ , τ ντ ), where ν and τ satisfy the identity ντ 6 = τ 6ν. For instance, any choice of ν combined with any choice of τ of order 2, 3 or 6 is good. By specifying τ we actually prescribe how f should lift. For the automorphism group Aut(P ) = hf, g, hi we get: the group Aut(P ) lifts if and only if the cover of P is determined by the 6-tuple of voltages

(ν, τ −1ντ, τ −2ντ 2, τ −3ντ 3, τ −4ντ 4, τ −5ντ 5), where ν and τ are two of the four generators of G = hν, τ, η, ω;(R),... i and (R) is the system of 13 identities arising from the above system of equations if we replace the ﬁrst column by the identity ντ 6 = τ 6ν and −i i substitute all ξi by τ ντ in all the other equations. The above permutation group G = hν, τ, η, ω;(R),... i can be viewed as an image of a universal 4-generator group G˜ generated by four abstract generators ν, τ, η, ω and satisfying the set of identities (R). Each 2.5. Liftings: in terms of voltages 57

‘admissible voltage group’ for the investigated lifting problem is a quotient of G˜. If the abelianization of G˜ is inﬁnite then there are inﬁnitely many solutions of the considered lifting problem.

Example 2.10. (Continuation of Example 2.9) By taking ξi = (12) ∈ S2 for i = 0, 1,..., 5, we get the generalized Petersen graph GP (10, 3) as a cover of the Petersen graph such that the full automorphism group lifts. The solutions of the above system of equations are τ = ω = η = 1 in S2. If we set ξ0 = ξ2 = ξ4 = (12)(34), ξ1 = ξ3 = ξ5 = (13)(24) in the group S4, another interesting example is obtained. In this case, the solutions of the above system of equations are τ = (14), ω = (1243) and η = (14)(23). Thus the derived covering graph is a 3-arc-transitive cubic graph and has 40 vertices. In fact, the resulting graph can be viewed as a canonical double covering D˜ of the dodecahedron D. Since ˜ |Aut D| > |Aut D × Z2| it follows that the dodecahedron is unstable (see [MS93] for the deﬁnition and [NS96] for further information and results). This is the ﬁrst symmetrical example of its kind, as D is sharply 2-arc-transitive.

Example 2.11. (Irregular covers of Kn on which Aut(Kn) lifts:) Let the vertex set of complete graph X = Kn be V = {0, 1, 2, . . . , n − 1} and dart-set be D = (V × V ) \{(j, j)|j ∈ V } as expected. Choose the spanning tree T of Kn to be the star containing edges {(0, j), (j, 0)} for j = 1, 2, . . . , n − 1. We deﬁne a T -reduced permutation voltage space (F, H, ξ) on X, where F = Zn − {0} and H ⊆ Sym(F ), by setting ξ(i,j) = (i, j) for each cotree dart (i, j). To show that ξ is locally G = Aut(X) invariant observe that G = Sym(V ) is generated by all transpositions (i, j), where i, j ∈ F , together with a single transposition (0, 1). Given transposition (i, j)(i, j ∈ F ) the system of equations from Corollary 2.23

(i, j) · τ(i,j) = τ(i,j) · (i, j) (i, k) · τ(i,j) = τ(i,j) · (j, k) for all k 6= 0 (j, k) · τ(i,j) = τ(i,j) · (i, k) for all k 6= 0 (k, `) · τ(i,j) = τ(i,j) · (k, `) for all k, ` 6= 0 58 Chapter 2. Coverings and liftings of automorphisms

has a solution τ(i,j) = (i, j). Similarly for an automorphism (0, 1) ∈ Aut(X) the associated system of equations

(k, 1) · τ(0,1) = τ(i,j) · (k, 1) for all k 6= 0, 1 (i, j) · τ(0,1) = τ(0,1) · (i, j) for all i, j 6= 0, 1 has a trivial solution τ(0,1) = 1. Hence by Corollary 2.23 the automorphism group of Kn lifts. The action of the permutation voltage group on F is obviously irregular since the voltage assignments are not regular permutations. In fact, the covering transformation group CT(p) = 1 is trivial. One can see it as follows. The voltage group H is generated by all the transpositions in Sym(F ) hence H = Sym(F ) acts transi- ∼ tively on F with the point stabilizer isomorphic of S(1) = Sn−2. Since CT(p) is given by the action on a single ﬁbre and by Theorem 1.7 ∼ CT(p) = NH (S(1))/S(1). However, conjugation by any permutation outside S1 does not preserve S1 thus NH (S(1)) = S(1) and CT(p) = 1. On the other hand, the covering graphs are vertex-transitive. One can construct them as ﬁrst subdividing each edge by introducing a vertex of degree two and then taking the line graph of the subdivision of Kn.

Exercises

2.5.1. With the above notation prove that the following permutation voltage assignment in S3 deﬁned on K5 is locally invariant: ξ(1, 2) = ξ(2, 1) = ξ(3, 4) = ξ(4, 3) = (1, 2), ξ(1, 3) = ξ(3, 1) = ξ(2, 4) = ξ(4, 2) = (1, 3) and ξ(1, 4) = ξ(4, 1) = ξ(2, 3) = ξ(3, 2) = (2, 3). Compute CT(p). 2.5.2. Show that in general the existence of a lift of an automorphism does not imply the local invariance. 2.5.3. The Heawood graph is the point-line incidence graph of the Fano plane. Is the Heawood graph a regular cover over st3. If yes, describe the associated Cayley voltage space. Can one construct the Heawood graph from a Aut(st3)-locally invariant voltage space over st3? 2.5.4. Prove that the lifted graph X˜ in the Example 3.1 is vertex-transitive. 2.5.5. Using the lifting condition prove that if k2 ≡ ±1(mod n) then the generalised Petersen graph GP (n, k) is vertex-transitive. 2.5.6. For all n ≤ 5 describe n-folded coverings of K4 such that the whole automorphism group lifts. Lifting problem, case of abelian CT(p) 59

3id 4

id id x2 - 2 R 5 x3 x1 id id 1 x4 6

Figure 2.7: K3,3 with T -reduced voltages

2.5.7. Find a solution of the lifting problem in terms of group presentations (as simple as possible) in case A = AutX and X ranges through the one-skeletons of the ﬁve Platonic solids.

2.6 Lifting problem, case of abelian CT(p)

Let X be a connected graph and let T be a spanning tree of X. Let

+ D = {x1, . . . , xβ, xβ+1, . . . , xβ+t} be an orientation of X, where x1, . . . , xβ are the cotree darts and + xβ+1, . . . , xβ+t are the tree darts, so that β + t = |D |. Any automorphism f of X can be represented by a β × (β + t) matrix C = CT (f), whose rows correspond to the β cotree darts in D+ and columns cor- + respond to the darts in D . Any cotree dart xi determines a unique elementary cycle Ci in T +xi which is directed consistently with xi. Let εi,1 εi,2 εi,β+t fCi = (x1 , x2 , . . . , xβ+t ) be an image of the cycle Ci considered as a β + t vector. The exponent εi,j is 1 if xi appears in fCi, it is −1 −1 if xi appears in fCi and it is 0 otherwise. The entries of CT (f) are the exponents εi,j. The associated reduced matrix MT (f) is a β × β matrix formed by the ﬁrst β columns of CT (f) corresponding to the cotree darts.

Example 2.12. Let X = K3,3 be the complete bipartite graph with the bipartition V = {1, 3, 5} ∪ {2, 4, 6}. Choose the spanning tree T 60 Chapter 2. Coverings and liftings of automorphisms

to be the path T = 123456 as illustrated by dark lines in Figure 2.7. We orient the cotree darts by setting x1 = 14, x2 = 25, x3 = 36 and x4 = 61. The corresponding elementary cycles, deﬁned as cyclic permutations of vertices, are: C1 = (1432), C2 = (2543), C3 = (3654) and C4 = (123456). (They are the generators of the homology group H1(X).) Let r and ` be the automorphisms of X given by permutations of vertices as follows: r = (1 2 3 4 5 6), ` = (2 4 6) as cycles. We have r(C1) = (2543), r(C2) = (3654), r(C3) = (4165) and r(C4) = (123456). Similarly, `(C1) = (1634), `(C2) = (4563), `(C3) = (3256) and `(C4) = (143652). One easily gets     0 1 0 0 −1 0 −1 −1  0 0 1 0   0 0 −1 0  M (r) =   M (`) =   . T  −1 0 0 −1  T  0 1 −1 0  0 0 0 1 1 −1 1 0

Note that the i-th rows of MT (r) and MT (`) represent the elements r(Ci) and `(Ci), respectively in the homology group H1(X).

A square matrix is called unimodular if its determinant is ±1. The set of n×n unimodular integer matrices forms a group under the matrix multiplication, called the unimodular (matrix) group and denoted by SL±(n, Z).

Proposition 2.24. Let X be a connected graph with a spanning tree T . Then the mapping f 7→ MT (f) is a homomorphism from Aut(X) to the unimodular group SL±(β, Z), deﬁning a matrix representation of Aut(X).

Proof. For graphs without semiedges the proof is done in [Sir01]. We extend it onto graphs with semiedges. Let X be a graph, f ∈ Aut(X) and let MT (f) be a matrix representation of f with respect to a spanning tree T and an orientation D+. First, we observe that

(1) a semiedge is always a cotree edge, that is, a semiedge does not belong to any spanning tree T ,

(2) each semiedge x determines an elementary cycle Cj(x) oriented consistently with the orientation of x, 2.6. Liftings: case of abelian CT(p) 61

(3) the image of a semiedge under a graph automorphism f is again a semiedge. By (1) and (2) we can reorder the cotree edges and elementary cycles of X such that the ﬁrst s rows and the ﬁrst s columns correspond to the set of semiedges of X. Recall that a transposition of two rows (or columns) of a matrix preserves the determinant up to the sign ±1. The condition (3) implies· that the¸ matrix M = MT (f) can be then expressed S 0 as a block matrix , where S is an s × s permutation matrix 0 C representing the action of f on the semiedges, C is (β − s) × (β − s) matrix representing action of f on elementary cycles induced by the links and loops. Since the matrix C represents the action of f on a subgraph X0 ⊆ X induced by the sets of links and loops of X, it is unimodular. The matrix S is a permutation matrix, and hence it is unimodular. Since both C and S are also unimodular, M is unimodular as well. 2

Given an abelian group A, the Cartesian product Aβ can be considered as a β-dimensional Z-module. If K is a submodule then the β set of integer (column) vectors ~a = (a1, a2, . . . , aβ) ∈ Z such that a1x1 + a2x2 + ··· + aβxβ = 0 ∈ A for each ~x = (x1, x2, . . . , xβ) ∈ K will be called an orthogonal complement of K and will be denoted K⊥. If K = hxi we write x⊥ instead of hxi⊥. If (A, A, ξ) is a Cayley voltage space on X reduced with respect to T , we denote by  

ξC1  ξ  ~  C2  ξ = (ξC , ξC , . . . , ξC ) =  .  1 2 β  .  ξ Cβ β×1 the β dimensional column vector of voltages on the elementary cycles. Call ξ~ the matrix representation of a voltage ξ~ with respect to a spanning tree T . Note that for each Ci we may form a unique −1 u generator Si = WiCiWi in the fundamental group π , where Wi is the unique path in T (possibly trivial) joining u to Ci. Since ξ is T -reduced, ξSi = ξCi for all i = 1, . . . , β. Since A is abelian, 62 Chapter 2. Coverings and liftings of automorphisms

ξfSi = ξfCi for all i = 1, . . . , β and any automorphism f. Note that ~ MT (f)ξ = (ξf#C1 , ξf#C2 , . . . , ξf#Cβ ). Since the local voltage group is u u P A = ξ(π ) = { i aiξSi | ai ∈ Z}, it follows that Q ~⊥ ξW = 0 for W = i aiSi if and only if ~a = (a1, a2, . . . , aβ) ∈ ξ , the orthogonal complement of ξ.~ Moreover, since A is abelian for any 0 Q P W in π(X) there is W = a S such that ξ = ξ 0 = a ξ = P i i i W W i i Si i aiξCi .

Theorem 2.25. Let X be a connected graph with cycle rank β and a spanning tree T and let (A, A, ξ) be a T -reduced locally transitive Cayley voltage space on X with an abelian group A. Then the following statements are equivalent.

(1) An automorphism f of X lifts, (2) (A, A, ξ) is locally f-invariant, (3) ξ~⊥ = (Mξ~)⊥, that is, the orthogonal complements of the vectors ~ ~ ξ and Mξ are identical, where M = MT (f) is the matrix representation of f with respect to the spanning tree T .

Moreover, if A = Zn is cyclic then the above statements are equivalent to

(4) there exists an eigenvalue α coprime to n such that Mξ~ = αξ~.

Proof. By Theorem 2.21 (1) ⇔ (2). (2) ⇒ (3) If (2) holds then by Theorem 2.21(3) the automorphism u u fu f induces an isomorphism f# of the local groups A → A given by ξW 7→ ξfW . Since the voltage space is T -reduced and locally transitive, it follows Au = A for each u ∈ V . Thus f u = f : A → A is a group P # # ~ β ~ automorphism. Let 0 = ~a · ξ = i=1 aiξi, where ~a · ξ means the dot product. Then Ã ! Xβ Xβ Xβ Xβ ~ 0 = f# aiξi = aif#ξi = aif#ξSi = aiξfSi = ~a · Mξ. i=1 i=1 i=1 i=1 2.6. Liftings: case of abelian CT(p) 63

It shows (ξ~)⊥ ⊆ (Mξ~)⊥. To prove (Mξ~)⊥ ⊆ (ξ~)⊥ we do a similar P ~ β −1 calculation. Assuming 0 = ~a · Mξ = i=1 aiξfSi and using (f#) we have Ã ! Xβ Xβ −1 −1 0 = (f#) aiξf(Si) = ai(f#) ξf(Si) i=1 i=1 Xβ Xβ −1 ~ = ai(f )#ξf(Si) = aiξSi = ~a · ξ. (2.3) i=1 i=1 (3) ⇒ (2) Suppose that the orthogonal complements are identical. Assume ξW = 0 for an u-based closed cycle W . Since W is expressible in terms of the β generators S ,S ,...,S and since A is abelian, we have P 1 2 β β ~ 0 = ξW = i=1 aiξSi = ~a · ξ for some integer vector ~a = (a1, a2, . . . , aβ), where ai = bi − ci for i = 1, . . . , β and bi, ci are the number of times the cotree darts x , x−1, respectively, appears in W . By assumption P i i ~ β 0 = ~a · Mξ = i=1 aiξfSi = ξfW . It follows that (A, A, ξ) is locally f-invariant. ∼ To show the last statement, let A = Zn. Assume any of the conditions (1), (2) and (3) holds. Since every group automorphism of the cyclic group Zn is of the form h 7→ αh for some α coprime to n, we ~ ~ ~ ~ ~ ~ ⊥ ~ ⊥ have Mξ = f#ξ = αξ. Conversely, if Mξ = αξ, it holds (ξ) = (Mξ) clearly showing (3). 2

Example 2.13. (The canonical double covering of a nonbipartite graph) For every nonbipartite graph X one can easily deﬁne a Cayley voltage space (Z2, Z2, ξ) which is Aut(X)-locally invariant by setting ξx = 1 for every dart x. Indeed, ξW = 0 if and only if |W | is even. If |W | is even then |f#W | is even for any f ∈ Aut(X). The existence of an odd cycle implies that (Z2, Z2, ξ) is locally transitive, and so the derived graph X˜ is connected. The covering graph X˜ is called the canonical double covering of X [NS96]. It is a minimal bipartite cover over X in the following sense: If Y is a bipartite cover of X then it covers X˜.

Example 2.14. (Inﬁnitely many connected coverings over X on which Aut(X) can lift:) Let X be a connected graph. Choose an arbitrary 64 Chapter 2. Coverings and liftings of automorphisms

spanning tree T of X and let n > 0 be the number of cotree links and n m loops, and let m be the number of semiedges. Set A = Zk ⊕ Z2 for any positive integer k > 1 and n > 0. With a preferred orientation on X, define a voltage ξ in such a way that the darts of the chosen spanning tree receive the trivial voltage, the voltages on darts of semiedges generate the Z2-part of A, and the voltages on darts of links or loops n of cotree edges generate the Zk-part of A. However, the Z-modules Zk m and Z2 are special ones. In these Z-modules, any two generating sets of size n, m respectively, have the same orthogonal complement consisting of integer vectors whose entries are divisible by k or 2, respectively. Since every f ∈ Aut(X) maps the links, loops and semiedges to links, loops and semiedges respectively, it follows that the vectors ξ~ = ~ (ξC1 , ξC2 , . . . , ξCβ ) and MT (f)ξ = (ξf#C1 , ξf#C2 , . . . , ξf#Cβ ) have the same orthogonal complement. Hence, by Theorem 2.25(3) Aut(X) lifts. Since n > 0, one can form an infinite family of regular covers over X by taking the free parameter k to be any integer. Another infinite family of coverings p : X˜ → X such that Aut(X) lifts along p can be obtained by iterating the above construction.

We summarize the above construction as follows.

Proposition 2.26. Let X be a connected graph with at least one cycle. Then there are inﬁnitely many ﬁnite connected regular covers over X on which Aut(X) lifts.

Remark: The construction in Example 2.14 generalizes to the case involving an arbitrary coeﬃcient ring, thereby extending homological coverings of Biggs [Bi74, Bi84] and Surowski [Su88] to graphs with semiedges. The name homological coverings come from the following consideration. Given a connected graph X and its spanning tree T we may form a T -reduced voltage assignment ξ : D → πu(X) taking values in the fundamental group by setting trivial voltages on the tree darts and taking the respective generator of the fundamental group induced by x for any cotree dart x. The reader might observed that the covering derived from this Cayley voltage assignment is a universal covering. Replacing the fundamental group by its abelianisation we get another (homological) voltage assignment. However, the universal 2.6. Liftings: case of abelian CT(p) 65

abelian cover may still be infinite. Factoring the infinite cyclic factors modulo a chosen integer k, we get a homological cover reduced modulo k which is finite for every k.

Example 2.15. (All cyclic coverings of K3,3 on which an automorphism group G = hr, ì lifts:) With the notation introduced in Exam- ple 2.12 we want to describe all Cayley voltage spaces (Zn, Zn, ξ) on X = K3,3 such that the group of automorphisms G = hr, ì lifts, with r = (1 2 3 4 5 6), ` = (2 4 6). Let R := MT (r) and L := MT (`) be the matrix representations of r and `, respectively derived in Exam- ple 2.12. By Theorem 2.25 we need to solve the system of congruences ~ ~ ~ ~ Rξ = αξ, Lξ = βξ in variables ξ1, ξ2, ξ3, ξ4, α, β modulo n, where α, β are coprimes to n. First four congruences from Rξ~ = αξ~ have a solution ξ~ = c(1, α, α2, −α3 − 1), where the parameters are related by 3 c(α + 1)(α − 1) ≡ 0(mod n). Since we require (Zn, Zn, ξ) to be locally transitive, c is coprime to n. It follows that we may assume c = 1. ~ ~ The system Lξ = βξ consists of congruences −(ξ1 + ξ3 + ξ4) = βξ1, −ξ3 = βξ2, ξ2 − ξ3 = βξ3 and ξ1 − ξ2 + ξ3 = βξ4. Inserting ξ2 = α 2 and ξ3 = α in the second equation we get β = −α which allows us to get rid of the parameter β. The third equation is equivalent with α2 − α + 1 ≡ 0(mod n). The fourth equation translates into 2 ξ4 = (−α)(α − α + 1) ≡ 0(mod n). The first equation reduces to the same congruence α2 − α + 1 ≡ 0(mod n) as the second one. Observe that multiplying both sides of α2 − α + 1 ≡ 0 by α2 − 1 we get the congruence (α3 +1)(α−1) ≡ 0, required for L(r)ξ~ = αξ~. We summarize the above computation as follows. The group G = hr, ì lifts if and only if there is α in Zn such that α is coprime to n and α2 − α + 1 ≡ 0(mod n). In such case the T - reduced voltage assignment is (up to a multiplicative constant) defined by ξ~ = (1, α, α2, 0). Few small solutions of the congruence are α = 1, 2, 3, 4 when n = 2, 3, 7, 13, respectively. The group G = hr, ì ≤ Aut(K3,3) defined above is transitive on darts and it is the orientation preserving automorphism group of a hexagonal embedding of K3,3 in the torus. It follows that the lifts are arc-transitive cubic graphs. The above family is described in [FK03] in another way. 66 Chapter 2. Coverings and liftings of automorphisms

Example 2.16. (Cyclic coverings of a dipole on which hρ, λi lifts:) Recall that a k-dipole is a graph whose dart-set is D = Zk × Z2, the vertex set is V = Z2, the incidence function is the projection on the second coordinate and the dart-reversing involution takes (i, j) 7→ (i, j+ 1). Fix orientation as (i, j) 7→ (i, j +1). Denote by ρ the automorphism D → D taking (i, j) 7→ (i + 1, j) and by λ the automorphism taking (i, j) 7→ (−i, j+1). Obviously ρ is a cyclic permutation of darts of order k ﬁxing both vertices, while λ can be viewed as a 180 degree rotation round an axis ﬁxing the central point of the edge {(0, 0), (0, 1)}. We choose a spanning tree T such that it contains the unique tree edge {(0, 0), (0, 1)}. The matrices representing the automorphisms ρ and λ with respect to T are obtained as   −1 1 0 0 ... 0 0    −1 0 1 0 ... 0 0     −1 0 0 1 ... 0 0   ......  M (ρ) =  ......  (2.4) T  . . . . ··· . .     −1 0 0 0 ... 1 0   −1 0 0 0 ... 0 1  −1 0 0 0 ... 0 0 (k−1)×(k−1)

  0 0 0 ... 0 0 −1    0 0 0 ... 0 −1 0     0 0 0 ... −1 0 0   ......  M (λ) =  ......  . (2.5) T  . . . ··· . . .     0 0 −1 ... 0 0 0   0 −1 0 ... 0 0 0  −1 0 0 ... 0 0 0 (k−1)×(k−1)

We want to describe all the locally transitive T -reduced Cayley voltage spaces (Zn, Zn, ξ) such that the group of automorphisms G = hρ, τi lifts. Let ξ = (ξ1, ξ2, . . . , ξk−1) be the vector of voltages in such a voltage space, more precisely ξ(i, 0) = ξi, for i = 1, 2, . . . , k − 1. By ∗ Theorem 2.25, ρ lifts if and only if there exists α ∈ Zn such that 2.6. Liftings: case of abelian CT(p) 67

~ ~ MT (ρ)ξ = αξ. We write it as a system of equations in Zn:

−ξ1 + ξ2 = αξ1 −ξ1 + ξ3 = αξ2 −ξ1 + ξ4 = αξ3 ... −ξ1 + ξk−1 = αξk−2 −ξ1 = αξk−1

2 j−1 This system has a solution: ξj = ξ1(1 + α + α + ··· + α ) for j = 1, 2, . . . , k − 1, where ξ1 and α are parameters, α is coprime to n satisfying

2 k−1 ξ1(1 + α + α + ··· + α ) = 0 (mod n).

Since we assume that our voltage space is locally transitive, we may set ξ1 = 1. We summarize it as follows: The automorphism ρ : D → D taking (i, j) 7→ (i+1, j) on the dipole lifts if and only if the T -reduced voltage-assignment is (up to multiplicative constant) deﬁned by

2 j−1 ξj = 1 + α + α + ··· + α for j = 1, 2, . . . , k − 1, where α is coprime to n satisfying

1 + α + α2 + ··· + αk−1 = 0 (mod n).

∗ Again, by Theorem 2.25, λ lifts if and only if there exists β ∈ Zn ~ ~ such that MT (λ)ξ = βξ. Rewriting it as a system of equations in Zn, we get

−(1 + α + α2 + ··· + αk−j) = β(1 + α + α2 + ··· + αj−1), for j = 1, 2, . . . , k − 1. In particular, for j = 1 and for j = k − 1 we get

−(1 + α + α2 + ··· + αk−2) = β, and −1 = β(1 + α + α2 + ··· + αk−2). 68 Chapter 2. Coverings and liftings of automorphisms

Thus β2 = 1. Similarly, inserting from the second equation (j = 2) into the ﬁrst one (j = 1) we get β(1 + α) − αk−2 = β. Using β2 = 1 and α is coprime to n, we derive αk−3 = β. Inserting third equation into the ﬁrst one we deduce

β(1 + α + α2) − αk−3 − αk−2 = β.

This gives β(1 + α + α2) − β − αβ = β. Finally we derive α2 = 1. Assume ﬁrst that k is odd. By αk−3 = β we get β = 1. But if β = 1 our system of equations reduces to just one equation 1 + α + α2 + ··· + k−2 2 k−1 α = −1, which by α = 1 gives (1 + α) 2 = −1. Multiplying by α k−1 we get (1 + α) 2 = −α which implies α = 1. Hence we have a unique solution ξi = i for i = 1, 2, . . . , k − 1. Assume k is even. By αk−3 = β we get α = β. Our system of equations reduces to 1 + α + α2 + ··· + αk−2 = −α, which hold true for α satisfying α2 = 1. Finally, 1 + α + α2 + ··· + αk−1 = 0 (mod n) k reduces to (1 + α) 2 = 0 (mod n). We summarize the above computations as follows. G = hρ, λi lifts if and only if the T -reduced voltage-assignment ξ (up to multiplicative constant) satisﬁes the following conditions:

(1) If k is odd then k = 0(mod n) and ξj = j for j = 1, 2, . . . , k − 1, (2) if k is even then there exists α such that α2 = 1(mod n) and k (1 + α) 2 = 0 (mod n), for each such an α there is a locally G- invariant voltage assignment deﬁned by ξ2j = j(1 + α) and by ξ2j−1 = (j − 1)(1 + α) + 1.

It follows that a k-dipole of odd valency admits (up to multiplication ∗ in Zn) exactly one T -reduced voltage assignment in a cyclic group Zn. On the other hand, if k is even there are at least two essentially diﬀerent admissible voltages corresponding to α = ±1. The group G = hr, `i is dihedral and it is the group of orientation preserving automorphisms of the spherical embedding of k-dipole. Note that if α = 1 and k = n then the derived graph is Kn,n. Lifting problem, case of elementary abelian CT(p) 69

Exercises

2.6.1. Prove that K4 has only one non-trivial regular cover with cyclic CT(p) such that the full automorphism group lifts. ∼ 2.6.2. Prove that K3,3 has a non-trivial regular cover with cyclic CT(p) = Zn such that the full automorphism lifts if and only if n = 3. Describe the associated Cayley voltage space. 2.6.3. * Is there a regular cover of K3,3 with abelian CT(p) such that the full automorphism group lifts and CT(p) is generated by two or three generators? 2.6.4. * Prove that there are infinitely many 5-transitive cubic graphs. Hint: Let a vertex set V be the set of all involutory permutations in S6 which are either transpositions or can be expressed as a product of three disjoint transpositions. Two such permutations of different cycle structure are joined by a link if they commute. Show that the above defined graph X is a cubic 5-transitive graph. 2.6.5. * Describe all the cyclic covers over the octahedral graph and over the cube Q3 such that the full automorphism group lifts. 2.6.6. * Employing the computation in Example 2.9 describe all the cyclic covers over the Petersen graph such that the full automorphism group lifts. 2.6.7. * Prove Proposition 2.24. 2.6.8. *A totally unimodular matrix is an integer matrix for which every square non-singular submatrix is also unimodular. Prove: For a matrix M, if

(1) all its entries are either 0, −1 or +1; (2) any column has at most two nonzero entries; and (3) the column with two nonzero entries have entries with opposite sign,

then the matrix M is totally unimodular (but the converse is not true). 2.6.9. Prove that any permutation matrix is totally unimodular.

2.7 Lifting problem, case of elementary abelian CT(p)

Let X be a graph and let f ∈ Aut(X). As a special case of the previous section, let the voltage group A be an elementary abelian group, say 70 Chapter 2. Coverings and liftings of automorphisms

n A = Zp = Zp × Zp × · · · × Zp considered as the additive group of the n- dimensional vector space over the ﬁeld GF (p). With the notation in the previous section, we aim to construct all connected regular elementary abelian coverings over X on which f can be lifted. As before, let T be a spanning tree of X and let (A, A, ξ) be a locally transitive, f-locally invariant and T -reduced Cayley voltage space on X. Then the covering graph X˜ derived from (A, A, ξ) is connected, and + n hence the voltages φ(D ) generate the voltage group Zp . Therefore, we have Lemma 2.27. n ≤ β, where β is the number of cotree edges of X. ~ As before, we denote by ξ = (ξC1 , ξC2 , . . . , ξCβ ) the β-dimensional column vector of voltages on the elementary cycles. Then, we have ~ ~ Mξ = MT (f)ξ = (ξf#C1 , ξf#C2 , . . . , ξf#Cβ ). β β Lemma 2.28. Let Φ: Z → Zp be the epimorphism reducing the factors of the product modulo p. Then ξ~⊥ = (Mξ~)⊥ if and only if Φ(ξ~⊥) = Φ((Mξ~)⊥). Proof. (⇒) is clear. β (⇐) : Let ~a = (a1, a2, . . . , aβ) ∈ Z be any vector and let Φ(~a) = ~r = (r1, r2, . . . , rβ) so that ai = pqi + ri for some integers qi, 0 ≤ ri < p for all i. Since A is an elementary abelian p-group, each nontrivial element Pβ Pβ has order p. Hence i=1 aiξi = 0 is equivalent with i=1 riξi = 0. Now, the suﬃciency is clear. 2

In view of Lemma 2.28, we may consider the orthogonal complement of ξ~ as a subspace of the vector space Zβ. Identify the (column) voltage p   − ξC1 −  − ξ −  ξ~ = (ξ , ξ , . . . , ξ ) with a (β × n) matrix [ξ~] =  C2  C1 C2 Cβ  ···  − ξ − Cβ β×n n ~ with rows ξCi ∈ A = Zp . Call [ξ] the matrix representation of a voltage ξ~ with respect to a spanning tree T . Also, one can consider the matrix ~ n β representation [ξ] as a linear transformation from Zp to Zp . We denote by C([ξ~]) and R([ξ~]) the column and row spaces of the matrix [ξ~], re- ~ spectively. Then R([ξ]) = hξC1 , ξC2 , . . . , ξCβ i = A and the orthogonal 2.7. Liftings: case of elementary abelian CT(p) 71

complement ξ~⊥ is nothing but the null space N ([ξ~]t) of the transposed matrix [ξ~]t of [ξ~], that is,

~⊥ β ~ ~ t ~ t ξ = {~a = (a1, a2, . . . , aβ) ∈ Zp | ~a · ξ = [ξ] ~a = 0} = N ([ξ] ).

The next lemma is obvious. ~ Lemma 2.29. Let M = MT (f) and [ξ] be the matrix representations of an automorphism f ∈ Aut(X) and a voltage ξ,~ respectively. Then the ~ ~ product M[ξ] is the matrix representation of f#ξ = (ξf#C1 , ξf#C2 , . . . , ξf#Cβ ). Moreover, we have

~ ⊥ ~ t β ~ t (f#ξ) = N ((M[ξ]) ) = {~a ∈ Zp | (M[ξ]) ~a = 0}.

β ~ ~ t * Zp = C(M[ξ]) ⊕ N ((M[ξ]) ) ~ ~ [f#ξ] = M[ξ] 6 6 (M[ξ~])t M = MT (f) ~ [ξ] - Zn β ~ ~ t p Zp = C([ξ]) ⊕ N ([ξ] ) [ξ~]t ~ ~ Figure 2.8: Matrix representations of voltages ξ and f#ξ

For a relation between the matrix representations of a voltages ξ~ ~ and f#ξ = (ξf#C1 , ξf#C2 , . . . , ξf#Cβ ), see Figure 2.8. Theorem 2.30. Let X be a connected graph with cycle rank β, T be a spanning tree and let (A, A, ξ) be a T -reduced locally transitive Cayley n voltage space on X with an elementary abelian group A = Zp . Then the following statements are equivalent for an f ∈ Aut(X).

(1) f lifts to an automorphism of the derived graph Xξ, (2) ξ~⊥ = (Mξ~)⊥, that is, the orthogonal complements of the T -reduced ~ ~ vectors ξ and Mξ, where M = MT (f), are identical. 72 Chapter 2. Coverings and liftings of automorphisms

(3) N ([ξ~]t) = N ([ξ~]tM t). (4) R([ξ~]t) = R([ξ~]tM t), or equivalently C([ξ~]) = C(M[ξ~]). ~ t t β (5) The null space N ([ξ] ) is an M -invariant subspace of Zp . ~ β (6) The column space C([ξ]) is an M-invariant subspace of Zp . Proof. By Theorem 2.25 (1) ⇔ (2). (2) ⇔ (3) holds because ξ~⊥ = N ([ξ~]t) and (Mξ~)⊥ = N ([ξ~]tM t). (3) ⇔ (4) because the row space and the null space are always mutually orthogonal; and C(A) = R(At) for any matrix A. (3) ⇔ (5) Since f is automorphism, the matrix M t is a linear auto- β t ~ t t ~ t morphism on Zp . And M always maps N ([ξ] M ) to N ([ξ] ). β (4) ⇔ (6) because the matrix M is a linear automorphism on Zp and it always maps C([ξ~]) to C(M[ξ~]). 2

Remark: To construct all connected elementary abelian coverings over X on which f ∈ Aut(X) can be lifted, we employ the condition (6) in Theorem 2.30. The locally transitive, f-locally invariant and n n T -reduced Cayley voltage spaces (Zp , Zp , ξ) on X associated with the desired coverings can be determined using the following algorithm:

(1) Find the matrix representation M = MT (f) which is a (β × β)- matrix. (2) Find all invariant subspaces W of M.

(3) For each M-invariant subspace W , choose a basis α = {a1, a2,..., an} β with ai ∈ Zp for the subspace W . (4) Now, deﬁne the voltages on the cotree darts by the rows of the | | | matrix  a1 a2 ··· an  . | | | β×n

e Recall that two covering projections pi : Xi → X, i = 1, 2, are e e isomorphic if there exists a graph isomorphism Φ : X1 → X2 such that p2 ◦ Φ = p1. 2.7. Liftings: case of elementary abelian CT(p) 73

Theorem 2.31. Let X be a graph and let M be a β × β matrix representing an automorphism of X. Given an M-invariant subspace W (with respect to left action), the covering graph projection derived n n from the Cayley voltage assignment (Zp , Zp , ξ) described in Remark in page 72, where n = dim W , does not depend on the choice of the basis for W . If W1 and W2 are two such subspaces, then the respective covering ˜ projections Xi → X are isomorphic if and only if there is an automorphism g of X such that W2 = M(g)W1.

n Proof. By Theorem 2.14, there exists a group automorphism M on Zp such that W2 = MW1. Moreover, it is proved in [MMP04, Prop. 6.3(b)] and [MP06] that M = M(g) is a representation of an automorphism of X. 2

By Remark in page 72 and Theorem 2.31, the problem of determining of all equivalence classes of f-locally invariant Cayley voltage spaces r r (Zp, Zp, ξ) deﬁned on X reduces to the following classical problem in linear algebra: Problem: Given an invertible matrix M of order β, determine all M-invariant subspaces of the β-dimensional vector space over the Galois ﬁeld GF (p). The Jordan canonical form of M = M(f) over GF (p) determines all minimal M-invariant subspaces which are necessarily cyclic. Each M-invariant subspace can be constructed as a direct sum of the minimal invariant ones. For a concrete matrix and a given prime p of a reasonable size, the invariant subspaces can be computed using a com- puter program. However, to prove statements on all elementary abelian coverings over a given graph one needs a more detailed understanding of the structure of common invariant subspaces with respect to a set of matrices. For convenience, let us recall some facts, for more details the reader is referred to [Ja53]. Invariant subspaces of one matrix: Let M be an invertible (n×n) matrix over F = GF (p) representing a linear transformation ~x 7→ n n1 n2 nk M~x on the vector space F . Denote by κM (x) = f1(x) f2(x) ··· fk(x) s1 s2 sk and mM (x) = f1(x) f2(x) ··· fk(x) the characteristic and minimal 74 Chapter 2. Coverings and liftings of automorphisms

polynomials of M respectively, where fj(x) are pairwise distinct polynomials irreducible over F. Then the vector space Fn can be decomposed as n s1 s2 sk F = N (f1(M) ) ⊕ N (f2(M) ) ⊕ · · · ⊕ N (fk(M) ). Moreover, all M-invariant subspaces can be obtained by first determin- sj ing the invariant subspaces of N (fj(M) ) for each j = 1, 2, . . . , k, and sj then taking their all possible direct sums. The subspace N (fj(M) ) has dimension djnj, where dj = deg fj(x) is the degree of fj(x). Its minimal M-invariant subspaces are cyclic of the form hx, Mx, . . . , M dj −1xi, where x ∈ N (fj(M)). Moreover, each of such minimal cyclic invariant subspaces in N (fj(M)) defines an increasing sequence of nested invariant subspaces of length at most sj and at least one of such sequences has length sj. If nj > sj then a variety of pairwise disjoint minimal sj cyclic subspaces exists in N (fj(M) ), while if nj = sj then a minimal sj M-invariant subspace in N (fj(M) ) exists uniquely. In particular, if sj nj = sj = 1 then N (fj(M) ) itself is a minimal M-invariant subspace. Consequently, if κM (x) = mM (x) with nj = sj = 1, then N (fj(M)), j = 1, 2, . . . , k are the only minimal M-invariant subspaces, and all the others are direct sums of these minimal ones. Common invariant subspaces of more than two matrices: Let Mi ∈ GL(n, F)(i = 1, 2, . . . , `) be invertible matrices such that G = hM1,M2,...,Mì is a finite subgroup of GL(n, F). As concerns finding G-invariant subspaces, we recall the following consequence of a theorem by Maschke (see [AB90, page 116]). Theorem 2.32. (Maschke) Let G ≤ GL(n, F) be a group of linear transformations over a finite field F of characteristic p and let gcd(|G|, p) = 1. Then the space Fn decomposes uniquely into a direct sum of minimal G-invariant subspaces and every G-invariant subspace is a direct sum the minimal ones. A minimal G-invariant subspace may not be a minimal invariant subspace for neither of the generators Mi. However, for each generator Mi of G, an Mi-invariant subspace is a direct sum of the minimal Mi- invariant subspaces. In case when Maschke theorem holds, we have to identify minimal common invariant subspaces with respect to the set of matrices Mi, i = 1, 2, . . . , `. The remaining case gcd(p, |G|) 6= 1 could be technically more difficult to analyze. 2.7. Liftings: case of elementary abelian CT(p) 75

An algorithm for determining the G-invariant subspaces requires ﬁrst to identify the minimal polynomials of each of Mi and to factorize them into irreducible factors. There is an exhaustive literature on this mi topic, see for instance [LN84]. Since mMi (x) divides x − 1, where mi mi is the order of Mi, it is worth to ﬁnd the irreducible factors of x − 1 over Zp. Such method is described in [LN84, Theorem 2.45].

Example 2.17. (All elementary abelian coverings of K3,3 on which an automorphism group G = hr, ì lifts:) In Example 2.15 we have investigated locally G-invariant voltage spaces defined on K3,3 with cyclic voltage group, where G is a group generated by the automorphisms r = (1, 2, 3, 4, 5, 6) and ` = (2, 4, 6). In other words, we have classi- fied the regular coverings ϕ of K3,3 with cyclic covering transformation group CT(ϕ) on which G lifts. In this example we classify the regular covers of K3,3 with elementary abelian covering transformation group on which G lifts. The characteristic and the minimal polynomials for the matrix R = MT (r) are

2 κR(x) = mR(x) = (x − 1)(x + 1)(x − x + 1).

As concerns the matrix L = MT (`), its characteristic and the minimal polynomials are

2 2 2 κL(x) = (x + x + 1) , mL(x) = x + x + 1, respectively. Since the orders of R and L are 6 and 3, and |G| = |hR,Li| = 18 = 2·32, we ﬁrst assume p > 3. Then by Maschke theorem the group G = hR,Li is completely reducible, which means that the 4- dimensional vector space V over GF (p) can be decomposed into a direct sum of minimal G-invariant subspaces and every G-invariant subspace is a sum of the minimal ones. A routine calculation shows that the quadratic polynomials (x2 − x + 1) and (x2 + x + 1) decompose into products of linear factors if and only if p ≡ 1(mod 3). Moreover, if p ≡ 1(mod 3) then

x2 − x + 1 = (x − λ)(x + λ2) and x2 + x + 1 = (x + λ)(x − λ2), where λ3 ≡ −1(mod p). 76 Chapter 2. Coverings and liftings of automorphisms

Case of p ≡ −1(mod 3). Since there are no 1-dimensional L- invariant subspaces, V is indecomposable or V = U1 ⊕ U2 decomposes into two minimal G-invariant subspaces of dimension 2. A good can- t t didate to test is an R-invariant space U1 = h(1, 1, 1, −2) , (1, −1, 1, 0) i t t generated by two eigenvectors ~v1 = (1, 1, 1, −2) and ~v2 = (1, −1, 1, 0) corresponding to eigenvalues ±1 of R. A direct computation shows that LU1 = U1, hence U1 is a G-invariant subspace. The complementary t t G-invariant subspace was identiﬁed as U2 = h(1, 0, −1, 0) , (0, 1, 1, 0) i. There are no other non-trivial G-invariant subspaces.

Case of p ≡ 1(mod 3). In this case there are two 1-dimensional L-invariant spaces W1(λ) and W2(λ) corresponding to eigenvectors 2 t 2 t ~w1 = (1, −λ , −λ, 0) and ~w2 = (1, λ, λ , 0) , respectively. A direct computation proves that the same vectors are eigenvectors for R as well. Thus W1(λ) and W2(λ) are G-invariant and there are no other 1-dimensional G invariant subspaces, since the R-invariant subspaces h~v1i, h~v2i corresponding to the eigenvalues ±1 are not preserved by L. On the other hand, the 2-dimensional R-invariant space U1 = t t h(1, 1, 1, −2) , (1, −1, 1, 0) i = h~v1i ⊕ h~v2i is L-invariant as well (as can be shown by a direct computation). Hence the decomposition onto minimal G-invariant subspaces reads as follows: V = U1 ⊕ W1(λ) ⊕ W2(λ).

All G-invariant subspaces together with the corresponding G-invariant voltages are given in Table 2.1.

The cases of p = 2 and 3 are left to the reader.

Determining isomorphism classes. On the set of all the G- invariant subspaces of V, we may employ Theorem 2.31 to determine the isomorphism classes of coverings. To do this, we need to investigate an action of Aut(K3,3) on the invariant subspaces. Since G acts regularly on arcs, we observe that the stabilizer of the arc 12 is h(3, 5), (4, 6)i. It follows that Aut(K3,3) decomposes onto left cosets as Aut(K3,3) = G ∪ (4, 6)G ∪ (3, 5)G ∪ (3, 5)(4, 6)G. Set α = (3, 5) and β = (4, 6). It is suﬃcient to investigate actions of the generators on the set of invariant subspaces. Their associated T -reduced matrices A = MT (α) 2.7. Liftings: case of elementary abelian CT(p) 77

and B = MT (β) are     1 −1 0 0 0 0 −1 −1  0 −1 0 0   0 1 −1 0  A =   and B =   ,  0 0 −1 0   0 0 −1 0  0 1 1 1 −1 0 1 0 respectively. The action on a column space U is deﬁned by left multiplication on a matrix [U] whose columns form a base of U. Technically the question whether M[U] = [U] transfers to a problem of solving n systems of linear equations, where n is the dimension of U. Since both matrices MT (g) and MT (g) are regular, we may deal with the subspaces of dimension one, two and three separately. Direct computations show that AV1(λ) 6= V2(λ), BV1(λ) 6= V2(λ), AV2(λ) 6= V1(λ) and BV2(λ) 6= V1(λ). As concerns dimension 2 we have     1 1 0 2  1 −1   −1 1  AU = A   =   . 1  1 1   −1 −1  −2 0 0 0

Then AU1 = U2 if and only if the following two systems of linear equations have solutions:         0 2 · ¸ 1 0 2 · ¸ 0  −1 1  x  0   −1 1  x  1    =   ,   =   .  −1 −1  y  −1   −1 −1  y  1  0 0 0 0 0 0 The solutions do exist, they are x = y = 1/2 and x = −1, y = 0, respectively. This shows that the voltage assignments attached with U1 and U2 determine isomorphic coverings. Finally, we compute C = ~ ~ A[U1⊕V1(λ)] and form the three systems of equations C~x = b1, C~x = b2 ~ ~ ~ ~ and C~x = b3, where [b1, b2, b3] = [U1 ⊕ V2(λ)]. Similarly as above, A takes U1 ⊕ V1(λ) → U1 ⊕ V2(λ) if and only if each of the above three systems of linear equations has a solution. In particular, for ~ t b1 = [1, 1, 1, −2] ∈ U1 we get the following system of equations     0 2 1 + λ2   1 x  −1 1 λ2   1     y  =   ,  −1 −1 λ   1  z 0 0 −λ2 − λ −2 78 Chapter 2. Coverings and liftings of automorphisms

which has no solution for any p ≡ 1(mod 3). We summarize the results as follows. ˜ Let X be a connected p-elementary abelian covering over K3,3 such that the ﬁbre-preserving group acts regularly on arcs. Then for p > 3, all such coverings over K3,3 are described in Table 2.1 by means of the associated voltage spaces. Except the dimension two, diﬀerent voltage assignments describe nonisomorphic coverings. In particular, there are two non-isomorphic coverings if p ≡ −1(mod 3), and six if p ≡ 1(mod 3).

Exercises

2.7.1. Let K3,3 be the complete bipartite graph with the bipartition V = {1, 3, 5} ∪ {2, 4, 6}. Determine equivalence classes of G-invariant elementary abelian voltage spaces on K3,3 for the singular primes p = 2, 3, where G is generated by the automorphisms r = (1, 2, 3, 4, 5, 6) and ` = (2, 4, 6). 2.7.2. Determine equivalence classes of Aut(X)-invariant elementary abelian voltage spaces on K3,3. 2.7.3. Determine equivalence classes of Aut(X)-invariant elementary abelian voltage spaces on X for X ∈ {D(3),K4,Q3, O3}. 2.7. Liftings: case of elementary abelian CT(p) 79

dim Invariant [ξ ]t [ξ ]t [ξ ]t [ξ ]t condition subspace C1 C2 C3 C4 p ≡ 1 (3) 1 V (λ) [1] [−λ2] [−λ] [0] 1 λ3 ≡ −1(p) 2 p ≡ 1 (3) 1 V2(λ) [1] [λ] [λ ] [0] 3 · ¸ · ¸ · ¸ · ¸ λ ≡ −1(p) 1 1 1 −2 2 U none 1 1 −1 1 0 · ¸ · ¸ · ¸ · ¸ 1 0 −1 0 2 U none 2 0 1 1 0         1 1 1 −2 p ≡ 1 (3) 3 U ⊕ V (λ)  1   −1   1   0  1 1 λ3 ≡ −1(p) 1 −λ2 −λ 0         1 1 1 −2 p ≡ 1 (3) 3 U ⊕ V (λ)  1   −1   1   0  1 2 λ3 ≡ −1(p) 1 λ λ2 0         1 0 0 0  0   1   0   0  4 V         none  0   0   1   0  0 0 0 1

Table 2.1: G-invariant voltage spaces, Case p > 3. (Here a ≡ b(n) means a = b(mod n)) 80 Chapter 2. Coverings and liftings of automorphisms Chapter 3

Applications

3.1 Enumeration of Graph Coverings

We note that the same graph X˜ can cover the base graph X in several diﬀerent ways. We are interested in enumerating covering projections not the covering graphs and the number being computed always refer to nonisomorphic coverings. In what follows we shall continue the discussion started in Secion ??. Let Iso (X; r) denote the number of r-fold coverings of a graph X and let Isoc (X; r) denote the number of connected ones. Let β be the Betti number β(X). Recall a T -reduced permutation voltage assignment can be considered as a β-tuple of permutations in Sym(r) by labeling the positive arcs in D(X) − D(T ) as e1, e2, . . . , eβ, and the set all r fold coverings CT (X; r) can be identiﬁed as

CT (X; r) = Sym(r) × Sym(r) × · · · × Sym(r), (β factors).

The number is determined CT (X; r) is determined in Theorem 2.13.

Next, we discuss a computation of the number Isoc (X; r) of connected r-fold coverings of X. It is well known (see for example [Ma91]) in topology that there exists a one-to-one correspondence between the connected r-fold coverings of X and the conjugacy classes of index r ˜ subgroups in the fundamental group of X. Indeed, Let p1 : X1 → X

81 82 Chapter 3. Applications

β r = 1 r = 2 r = 3 r = 4 r = 5 1 1 2 3 5 7 2 1 4 11 43 161 3 1 8 49 681 14721 4 1 16 251 14491 1730861 5 1 32 1393 336465 207388305 6 1 64 8051 7997683 24883501301

Table 3.1: The number Iso (X; r) for small r and small β

˜ and p2 : X2 → X be two isomorphic coverings. That means that there is a graph isomorphism Φ such that p1 = Φ◦p2. Since Φ is ﬁbre preserving mapping consistent with the right action of the fundametal group ∗ ˜ ∗ ˜ π(X, x0) it follows that the images p1π(X1, x˜0) and p2π(X2, Φ(˜x0)) are ˜ ˜ conjugate in π(X, x0). Conversely, if π(X1, x˜0) = π(X1, y˜0) are conjugate by g, wherex ˜0 andy ˜0 belong to a ﬁbre over x0 then the conjugation by g gives rise to an isomorphism between coverings. This way the problem of enumeration of connected graph coverings over X with the Betti number β translate to a group theoretical problem of counting conjugacy classes of subgroups of given index.

The number αF(β)(r) of index r subgroups of a free group of rank β was determined by Hall [Ha49] as follows.

Theorem 3.1. (Hall 1949) Let F(β) be the free group generated by β elements and let αF(β)(r) be the number of index r subgroups in F(β). Then,

Xr−1 β−1 β−1 αF(β)(r) = r(r!) − ((r − j)!) αF(β)(j) with αF(β)(1) = 1. j=1

Proof. We present a combinatorial proof. First, recall that Fβ = π1(Bβ, ∗) is a free group generated by β elements and all n-fold (connected or disconnected) coverings of a bouquet Bβ of β cycles (a one vertex graph with β loops and no semiedges) are determined by permutation voltages in

Sn × · · · × Sn, (β times). 3.1. Enumeration of Graph Coverings 83

(i) Given any voltage σ = (σ1, . . . , σβ), let the orbit of 1 under the action of Gσ = hσ1, . . . , σβi on F = {1, 2, . . . , r} be

O = {1, b2, . . . , bt},

σ and let VO be the corresponding orbit in the covering Bβ −→ Bβ. Let V be the n-element set of vertices and α be a labelling α : F → V taking 1 7→ v1, where v1 ∈ VO is ﬁxed. Each σ and each σ α gives a subgroup π1(Bβ , v1) of Fβ of index t. (ii) How many 1-similarity classes of σ are there? In other words, how many labellings α : F → V taking 1 → v1 and preserving O set-wise does exist? There are (n − 1) ··· (n − t + 1) choices of labels in O and there are (n − t)! choices of labels for vertices in V \ VO. Note that all σ1, . . . , σβ in σ should have the same letters for the orbit of 1, but free for the letters which do not in the orbit of 1. Hence, in total, there are (n − 1) ··· (n − t + 1)[(n − t)!]β = (n − 1)![(n − t)!]β−1 permutations in the 1-similar class of σ.

(iii) Let SFβ (t) denote the number of subgroups of Fβ of index t. Since each subgroup of index t is induced by the same number of β-tuples of permutations (in its 1-similar class) among (n!)β β permutations in (Sn) , we have, Xn β β−1 (n!) = (n − 1)![(n − t)!] SFβ (t). t=1 Divide by (n − 1)! to get Xn−1 β−1 β−1 n(n!) = [(n − t)!] SFβ (t) + SFβ (n). t=1 2

Liskovets [Li71] computed the number Isoc (X; r) of conjugacy classes of subgroups of index r of F(β) in terms of the numbers αF(β)(m) for m|r. 84 Chapter 3. Applications

Theorem 3.2. The number Isoc (X; r) is equal to the number of conjugacy classes of index r subgroups in the free group generated by β elements. It is given by the formula X X ³ ´ 1 r (β−1)m+1 Isoc (X; r) = αF(β)(m) µ d , r r md m|r d| m where µ is the number-theoretic M¨obiusfunction.

Proof. By Theorem 1.8 we have 1 X X Isoc(X; r) = Epi(K, Z ), r ` `| r K

Inserting in the above formula we get the required result. 2

Later, Hofmeister [Ho98] and Kwak and Lee [KL96] independently found another combinatorial enumeration formulae for the number of connected r-fold coverings of a graph.

Let CT (X; H) denote the set of T -reduced H-voltage assignments of X. Any regular r-fold covering of X is isomorphic to an derived covering p : X ×ξ H → X for a group H of order r and for a T - reduced Cayley voltage assignment ξ in CT (X; H). Moreover, if the graph X ×ξ H is connected, then the group H is isomorphic to the covering transformation group. We now, let Iso (X; H) denote the number of regular H-coverings, and Isoc (X; H) the number of the ones that are connected. Similarly, 3.1. Enumeration of Graph Coverings 85

β r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 1 1 1 1 1 1 1 2 1 3 7 26 97 624 3 1 7 41 604 13753 504243 4 1 15 235 14120 1712845 371515454 5 1 31 1361 334576 207009649 268530771271 6 1 63 7987 7987616 24875000437 193466859054994

Table 3.2: The number Isoc (X; r) for small r and small β we let IsoR(X; r) denote the number of regular r-fold coverings regard- less of the group H involved, and of course IsocR(X; r) denotes the number of the ones that are connected. It is not difficult to show that the components of any regular covering X ×ξ H → X are isomorphic as coverings of X, and any two isomorphic connected regular coverings of X must have isomorphic covering transformation groups. Moreover, any two regular coverings of X of the same folding number are isomorphic if and only if their components are isomorphic as coverings. Notice that each component of an H-covering of X is an S-covering of X for some subgroup S of H. The following theorem of Kwak et al. [KCL98] lists some basic counting properties about regular coverings. X Theorem 3.3. (1) For any r ∈ N, IsoR(X; r) = IsocR(X; d). d|r X (2) For any r ∈ N, IsocR(X; r) = Isoc (X; H), where the sum is H over all nonisomorphic groups of order r. X (3) For any finite group H, Iso (X; H) = Isoc (X; S), where the S sum is over all nonisomorphic subgroups of H. (4) For any finite groups H and K with (|H|, |K|) = 1, Iso (X; H ⊕ K) = Iso (X; H) Iso (X; K), and Isoc (X; H ⊕ K) = Isoc (X; H) Isoc (X; K), 86 Chapter 3. Applications

where H ⊕ K is the direct sum of two groups H and K. To complete the enumeration of the regular coverings, we need to ﬁnd some computational formulae for Isoc (X; H). Note that the set CT (X; H) of normalized H-voltage assignments of X can be identiﬁed as CT (X; H) = H × H × · · · × H, (β factors). Let

r X(H; r) = {(g1, g2, . . . , gr) ∈ H | {g1, g2, . . . , gr} generates H}.

A β-tuple g = (g1, g2, . . . , gβ) yields a connected covering if and only if g ∈ X(H; β). Under the coordinatewise Aut(H)-action on the set X(H; β), any two elements in X(H; β) belong to the same orbit if and only if they yield isomorphic connected H-coverings, by Theorem ??. Since the Aut(H)-action on X(H; β) is free, one can obtain the following theorem from Burnside’s Lemma. Theorem 3.4. For any ﬁnite group H, |X(H; β)| Isoc (X; H) = . |Aut(H)|

To complete the enumeration of IsoR(X; r) and IsocR(X; r), we need to determine |Aut(H)| and |X(H; β)|. For any finite abelian group H, Kwak et al. [KCL98] computed |Aut(H)| and |X(H; β)|. Now, we introduce a formula to compute the number |X(H; β)| for any finite group H in terms of the Möbiusfunction defined on the subgroup lattice of H. The Möbiusfunction assigns an integer µ(K) to each subgroup K of H by the recursive formula ½ X 1 if K = H, µ(H) = δ = K,H 0 if K < H. H≥K Jones [Jo95, Jo99] used such function to count the normal subgroups of a surface group or a crystallographic group, and applied it to count certain covering surfaces. We see that X β |H| = |CT (X; H)| = |X(K; β)|. K≤H 3.1. Enumeration of Graph Coverings 87

Now, it comes from the M¨obiusinversion that X X β |X(H; β)| = µ(K)|CT (X; K)| = µ(K)|K| . K≤H K≤H

The following theorem comes from Theorem 3.4.

Theorem 3.5. For any ﬁnite group H, 1 X Isoc (X; H) = µ(K)|K|β. |Aut(H)| K≤H

The cyclic group H = Zr has a unique subgroup Zd for each d dividing r, and has no other subgroups. The M¨obiusfunction on the subgroup is µ(Zd) = µ(r/d) (the number-theoretic M¨obiusfunction) and |Aut(Zr)| = ϕ(r) (the Euler ϕ-function), so we have 1 X ³r ´ Isoc (X; Z ) = µ dβ. r ϕ(r) d d|r

s1 s2 s` Theorem 3.6. For any r = p1 p2 ··· p` (a prime factorization), the number of connected Zr-coverings of X is   0 if β = 0,  Y` β Isoc (X; Zr) = (β−1)(s −1) p − 1  p i i if β ≥ 1,  i p − 1 i=1 i and the total number of Zr-coverings of X is   1 if β = 0,   Y`  (si + 1) if β = 1, Iso (X; Zr) =  i=1 Ã !  Y` β si(β−1)  (pi − 1)(pi − 1)  1 + β−1 if β ≥ 2 . i=1 (pi − 1)(pi − 1)

For the dihedral group Dr of order 2r, a similar computation gives the following [KCL98]: 88 Chapter 3. Applications

Theorem 3.7. For any r ≥ 3, the number of connected Dr-coverings of X is

¡ ¢ Y` pβ−1 − 1 Isoc (X; D ) = 2β − 1 p(mi−1)(β−2) i , r i p − 1 i=1 i

m1 m` where p1 ··· p` is the prime decomposition of r. Even though we have a general computational formula for Isoc (X; H), if the lattice of subgroups of H is complicated, it is not easy to eval- uate its exact value. However, if H is abelian, one can derive another enumeration formula for Isoc (X; H), one which does not involve the lattice structure of the subgroups of H. Our next result follows from Theorem 3.4.

Theorem 3.8. Let t1, . . . , t` and s1, . . . , s` be natural numbers with ` s1 > ··· > s`. Then, the number of connected ⊕h=1thZpsh -coverings of X is Yt pβ−i+1 − 1

` f(β,ti,si) i=1 Isoc (X; ⊕h=1thZpsh ) = p , Y` Ytj ptj −h+1 − 1 j=1 h=1 where t = t1 + ··· + t`, p is prime and Ã !   X` X`−1 X` f(β, ti, si) = (β − t) ti(si − 1) + ti  tj(si − sj − 1) . i=1 i=1 j=i+1

Now, one can compute the number Iso (X; H) for any ﬁnite abelian group H by using Theorems 3.3((c),(d)) and 3.8 repeatedly if necessary. For some abelian groups H and small β, the numbers Isoc (X; H) and Iso (X; H) are listed in Table 3.3.

Corollary 3.9. For any ` ≥ 1, the number of connected `Zp-coverings of X is

(pβ − 1)(pβ−1 − 1) ··· (pβ−`+1 − 1) Isoc (X; `Z ) = , p (p` − 1)(p`−1 − 1) ··· (p − 1) 3.1. Enumeration of Graph Coverings 89

Isoc Iso β (p, q) Zp3 ⊕ Zp Zq2 Zp3 ⊕ Zp ⊕ Zq2 Zp3 ⊕ Zp Zq2 Zp3 ⊕ Zp ⊕ Zq2 1 (2, 3) 0 1 0 4 3 12 2 (2, 5) 6 30 180 32 37 1184 3 (3, 5) 1404 775 1088100 2757 807 2224899 4 (3, 7) 126360 137200 1695792000 161451 137601 22215819051 Table 3.3: The number Isoc (X; H) and Iso (X; H) for some H and small β

β r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 1 1 1 1 1 1 1 1 1 2 1 3 4 7 6 12 8 15 3 1 7 13 35 31 91 57 155 4 1 15 40 155 156 600 400 1395 5 1 31 121 651 781 3751 2801 11811 6 1 63 364 2667 3906 22932 19608 97155 7 1 127 1093 10795 19531 138811 137257 788035 8 1 255 3280 43435 97656 836400 960800 6347715 9 1 511 9841 174251 488281 5028751 6725601 50955971 10 1 1023 29524 698027 2441406 30203052 47079208 408345795

β Table 3.4: The number of index r subgroups in ⊕1 Z and the total number of `Zp-coverings of X is

X` (pβ − 1)(pβ−1 − 1) ··· (pβ−h+1 − 1) Iso (X; `Z )=1 + . p (ph − 1)(ph−1 − 1) ··· (p − 1) h=1 (Again, we note that Hofmeister [Ho95a] independently computed the number Isoc (X; `Zp) by an another method.) We note two facts: (1) The number of connected `Zp-coverings of a connected graph X is equal to the number of the `-dimensional subspaces of the β-dimensional vector space over the ﬁeld GF (p) (see [To88, Ch 1.4]). (2) For a connected H-covering p : X˜ → X, the im- ˜ ˜ age p∗(π1(X)) of the fundamental group of the covering graph X is a normal subgroup of the fundamental group π1(X) of the base graph X, ˜ and the quotient group π1(X)/p∗(π1(X)) is isomorphic to H. Now, it is not hard to show that the number X X |X(H; β)| Isoc (X; H) = , |Aut(H)| H H 90 Chapter 3. Applications

β r = 1 2 3 4 5 6 7 8 9 10 11 1 1111111 1 1 1 1 2 1 3 4 7 6 15 8 19 13 21 12 3 1 7 13 35 31 119 57 211 130 259 133 4 1 15 40 155 156 795 400 1955 1210 2805 1464 5 1 31 121 651 781 4991 2801 16771 11011 29047 16105

Table 3.5: The number IsocR(X; r) for small r and small β

β r = 1 2 3 4 5 6 7 8 9 10 11 1 1223242 4 3 4 2 2 1 4 5 11 7 23 9 30 18 31 13 3 1 8 14 43 32 140 58 254 144 298 134 4 1 16 41 171 157 851 401 2126 1251 2977 1465 5 1 32 122 683 782 5144 2802 17452 11133 29860 16106

Table 3.6: The number IsoR(X; r) for small r and small β where H runs over all nonisomorphic abelian groups of order r, is equal to the number of index r subgroups in the free abelian group Z⊕· · ·⊕Z generated by β elements. For small r and small β, these numbers are listed in Table 3.4. If one can have the classiﬁcation of the groups of order r, then IsoR(X; r) and IsocR(X; r) can be computed by Theorem 3.3 (a) and (b). For example,

R 2 Iso (X; p ) = Isoc (X; 2Zp) + Isoc (X; Zp2 ) + Isoc (X; Zp) + 1 (pβ − 1)(pβ−1 − 1) pβ − 1 pβ − 1 = + p(β−1) + + 1. (p2 − 1)(p − 1) p − 1 p − 1

For small r and small β, these numbers are listed in Tables 3.5 and 3.6.

3.2 Some applications of graph coverings

In this section we show couples of examples to demonstrate a usefulness of a covering construction 3.2. Some applications of graph coverings 91

3.2.1 Existence of regular graphs with large girth A classical problem in graph theory reads as follows. Given any two integers k and g, show that there exists a k-valent graph of girth at least g. In what follows we employ graph coverings to construct such graphs.

Proposition 3.10. Given integers k ≥ 2 and g ≥ 3 there exists a simple vertex-transitive k-valent graph of girth at least g.

Proof. Denote by Tk(m) a complete rooted k-valent tree of radius m, whose pendant vertices are incident with k−1 semiedges. Take a regular k-edge-coloring of Tk(m) by colours 1, 2, . . . , k. Let τi, i = 1, 2, . . . , k, be an involution acting on vertices of Tk(m) transposing the vertices joined by edges colored by i. Set G = hτ1, τ2, . . . , τki and let X = Cay(G, {τ1, τ2, . . . , τk}) be the Cayley graph based at G determined by the set of generators τ1, τ2, . . . , τk. Clearly, X is vertex-transitive, simple and k-valent. To estimate the girth of X, let C be a cycle of length d. Then there is a corresponding identity w = τi1 τi2 . . . τid = 1 such that τij 6= τij+1 . If v is a root of Tk(m) then v · τi1 τi2 . . . τim τim+1 is a pendant vertex of Tk(m), and τi1 τi2 . . . τit for 1 ≤ t ≤ m takes v to a vertex at distance t from v. Since v · w = v we need to add in

τi1 τi2 . . . τim τim+1 a sequence made of at least m involutions τi to map v onto v. Thus the length d of C is at least 2m + 1. 2

In fact one can prove that the Cayley graph X constructed above regularly covers the tree Tk(m) with the covering transformation group ∼ CT(p) = StabG(v).

3.2.2 Large graphs of given degree and diameter 2 A vertex/diameter problem is to ﬁnd the largest (possibly regular) graph (in the number of verticies) which has a given degree (vertex-valency) and a given diameter. For simplicity, we restrict our considerations to graphs with diameter 2. For a given number d we need to ﬁnd the number v(d) of vertices of the largest graph having degree d and diameter 2. An easy observation gives that the size of a distance spanning tree cannot exceed v(d) ≤ d2 + 1. 92 Chapter 3. Applications

There is a dual problem to find a minimum number of vertices µ(d) of a graph of girth 5 and valency d. Such a graph contains a complete d-valent tree of depth two, thus we have v(d) ≤ d2 + 1 ≤ µ(d). The graphs for which v(d) = d2 + 1 = µ(d) are called Moore graphs. It is well-known [Bi74] that Moore graphs can exist only for d = 1, 2, 3, 7, 57. Moore graphs are known for d = 1, 2, 3 and 7, namely, they are K2, the pentagon, the Petersen graph and the Hoffman-Singleton graph. The problem of existence of a 57-valent Moore graph is still open. For general d it is hard to find good lower bounds for v(d). The following construction gives a lower bound for an infinite sequence of d.

Example 3.1. [SS05] Let q be a prime power of the form q = 4s+1 and let GF (q) be the Galois field of order q. Let X be a two-vertex graph, 0 whose vertices u, u are joined by q darts xg, g ∈ GF (q), and there are s loops attached to both u,u0. Choose an orientation D+ such that + xg ∈ D for every g ∈ GF (q). Let `1, `2, . . . , `s be darts representing −1 0 0 0 loops incident to u (ì 6= `j ), let `1, `2, . . . , `s be darts representing 0 + + loops incident to u . Choose an orientation D such that xg ∈ D for every g ∈ GF (q). Let h be a primitive element of GF (q) and GF (q)+ be the additive group of the field. Let (G, G, ξ) be a Cayley voltage + + 2 space on X defined by setting G = GF (q) × GF (q) , ξxg = (g, g ), 2i 0 2i−1 ξ(ì) = (0, h ) and ξ(ì) = (0, h ). Then the derived graph has 2 8 1 2 degree d = q + 2s = (3q − 1)/2, and order 2|G| = 2q = 9 (d + 2 ) .

It has some more interesting properties listed in the following lemma.

Lemma 3.11. The derived graph X˜ from the Cayley voltage space (G, G, ξ) in Example 3.1 has diameter 2.

Proof. Since X˜ comes from a Cayley voltage space defined over a 2- vertex base graph, it is either vertex-transitive or the action of the automorphism group has two orbits - the fibres over vertices. Hence X˜ 0 has a diameter 2 if and only if each vertex u(x,y) or u(x,y) is reachable by a path P of distance at most two from the vertex u(0,0) and from 0 the vertex u(0,0). We show it in the first case. The other case is left to the reader. 0 By definition of the derived graph, a path P : u(0,0) → u(x,y) is of length two if and only if for a given (x, y) 6= (g, g2) at least one of the 3.2. Some applications of graph coverings 93

equations (g, g2) ± (0, h2i−1) = (x, y) or (g, g2) ± (0, h2i) = (x, y) has a solution (g, i), g ∈ GF (q)+ and i ∈ {1, 2, . . . , s}. We have x = g and y − x2 = ±h2i or y − x2 = ±h2i−1. Since the set {±h2i|i = 1, 2, . . . , s} ∪ {±h2i−1|i = 1, . . . , s} covers all the non-zero elements of GF (q), there is i such that either the ﬁrst, or the second equation holds. Similarly, P : u(0,0) → u(x,y) is of length two if there is a solution 2 2 (g1, g2) of the equation (g1, g1) − (g2, g2) = (x, y). We have x = g1 − g2 and y = x(g1 + g2). However, replacing gj by gj + i, we get x = g1 − g2 + and y = x(g1 + g2 + 2i). If x 6= 0 then since GF (q) is of odd order, we can always ﬁnd i such that y = x(g1 + g2 + 2i). If x = 0 then either 2i 2i−1 y = ±h and u(0,y) is reachable by a path of length 1, or y = ±h and it can be expressed as a sum of the form ±h2i ± h2j implying that it is reachable by a path of length two. 2

As a curiosity we note that the least graph X˜ comes from the field GF (5) and it is the Hoffman-Singleton graph determined as a regular cover over a two-vertex graph with voltages in Z5 ×Z5 (this description was discovered by Siagiová).Seeˇ Fig. 3.1.

(0,0)

(1,1) (0,4) (0,2)

(2,4)

(3,4)

(4,1)

2 2 Figure 3.1: The Hoﬀman-Singleton graph derived from (Z5, Z5, ξ)

3.2.3 Spectrum of a graph and its covering graphs Let X = (D,V ; I, λ) be a graph without semiedges. A partition of its vertex set V = V1 ∪ V2 ∪ · · · ∪ Vn is called an equitable partition if there 94 Chapter 3. Applications

exists a square matrix M = (di,j) of order n, called a partition matrix such that for every i, j ∈ {1, 2, . . . , n} and for every vertex x ∈ Vi there are exactly di,j darts with initial vertex x terminating at some vertices of Vj. Let V (X) = {v1, v2 . . . , vm} be the vertex set. The adjacency matrix A(X) = (aij) is the m × m matrix with aij = 1 if (vi, vj) ∈ D and aij = 0 otherwise. The characteristic polynomial of X, denoted by Φ(X; λ) = det(λI − A(X)), is the characteristic polynomial of A(X).

Lemma 3.12. Let X = (D,V ; I, λ) be a graph without semiedges but with an equitable partition of vertices and let its partition matrix be M = (di,j). Then the characteristic polynomial of M divides the characteristic polynomial of X.

Proof. Let V = V1 ∪ V2 ∪ · · · ∪ Vn be an equitable partition of V with a square matrix M = (di,j). Let A(X) be the adjacency matrix of X with the vertex ordering following the equitable partition. Then A(X) can be considered as a block matrix with blocks bi,j, where the row sum of bij is equal to di,j. Now, let λ be an eigenvalue of M with an eigenvector x = (x1, x2, . . . , xn). Then one can show that λ becomes an eigenvalue of A with an eigenvector x = (x1, x2,..., xn), which is also a block column with blocks xj = (xj, xj, . . . , xj). A similarity works also for a generalized eigenvector. It implies that all eigenvalues of M are those of A upto multiplicity. It completes the proof. 2

Corollary 3.13. Let p : X˜ → X be a covering of graphs, let X have no ˜ semiedges. Then V = ∪v∈V fibv is an equitable partition. In particular, the characteristic polynomial of X divides the characteristic polynomial of X˜.

Example 3.2. Let X˜ = (D,˜ V,˜ I,˜ λ˜) and X = (D,V ; I, λ) be two graphs and let X˜ → X be a covering projection derived from a locally transitive permutation voltage assignment (F, G, ξ). For each γ ∈ G, let X(ξ,γ) denote the spanning subgraph of the graph X whose di- −1 rected edge set is ξ (γ). Let V = {v1, v2, . . . , vm}. We deﬁne an (anti-lexicographical) order relation ≤ on V˜ = V × F as follows: for ˜ (vi, s), (vj, t) ∈ V ,(vi, s) ≤ (vj, t) if and only if either s < t or s = t and i ≤ j. Let |F | = n and let P (γ) denote the n × n permutation matrix Flows 95

associated with γ-action, i.e., its (s, t)-entry P (γ)st = 1 if γ · s = t and P (γ)st = 0 otherwise. The tensor product A ⊗ B of the matrices A and B is considered as the matrix B having the element bst replaced by the ˜ matrix Abst. Now, one can show that the adjacency matrix A(X) of a covering graph X˜ is X ˜ A(X) = A(X(φ,γ)) ⊗ P (γ).

γ∈Sn

Exercises

3.2.1. Using graph coverings prove that for every k ≥ 2 and for every g ≥ 3 there are graphs of girth exactly g. 3.2.2. Prove that every cubic graph is a quotient of a cubic Cayley graph. 3.2.3. Determine the least cubic Cayley graphs covering the Petersen graph and the dodecahedral graph. 3.2.4. Prove that each regular graph of even valency is a quotient of a Cayley graph. 3.2.5. Show that not every equitable partition of a graph comes from a graph covering. 3.2.6. Prove that an equitable partition is induced by a graph covering if and only if the associated matrix M = (di,j) is symmetric. 3.2.7. Using the description of Siagiovádepictedˇ on Fig. 2 compute the eigenvalues of the Hoffman-Singleton graph. 3.2.8. * Prove or disprove: Let X˜ → X be a graph covering, X be without semiedges. Then the sets of eigenvalues of X and X˜ coincide. 3.2.9. * What approximation of the degree/diameter problem one can can get for d = 4, 5, 6 and diameter 2, using Cayley voltage spaces (G, G, ξ) defined on X with G abelian and X having at most two vertices. 3.2.10. * Giving a construction try to find a lower bound > d2/4 for the degree/diameter= 2 problem for an infinite sequence of d, d 6= (3q − 1)/2, q a prime power ≡ 1 mod 4.

3.3 Flows

Voltage assignments valued in an abelian group can be viewed as ﬂows. Usually it is required that in all, or in most vertices the Kirchhoﬀ law 96 Chapter 3. Applications

is satisfied. Let A be an abelian group and X = (D,V ; I, λ) be a connected graph. An A-circulationPis a Cayley voltage space (A, A, ξ) on X such that the Kirchhoff law ξ = 0 is satisfied for every vertex x∈Dv x v ∈ V . Generally, for U ⊆ V denote by D(U) the set of darts with initial vertices in U. Set D(U,U) = {x ∈ D(U) | x 6= λ(x) and Iλ(x) ∈ U}. Finally, set D(U, U¯) = D(U) \ D(U, U). In particular, D(V, V¯ ) is the set of semiedges. It follows that the edges underlying D(U, U¯) form an edge-cut sep- arating U.A bridge is an edge-cut of size one.

Lemma 3.14. Let XPbe a connected graph and let (A, A, ξ) be a circulation on X. Then x∈D(U,U¯) ξx = 0 for any U ⊆ V . In particular, the total sum of voltages on semiedges is 0 and if {x, λx} is a bridge then ξx = ξλx = 0. P Proof. First note that ξ = 0 because ξ −1 = −ξ for all x. P Px∈D(U,U) x P x x Hence, x∈D(U,U¯) ξx = x∈D(U) ξx − x∈D(U,U) ξx = 0 − 0 = 0. 2

An A-circulation on X such that ξx 6= 0 for every x ∈ D is called a nowhere-zero A-ﬂow on X, or shortly an A-ﬂow. By Lemma 3.14, all graphs having bridges admit no circulation. The following lemma is trivial but essential in its consequences.

Lemma 3.15. Sum of two A-circulations is an A-circulation.

As an application, we get the following lemma.

Lemma 3.16. Let X = (D, V, I, λ) be a graph, Y a maximum forest in X and let A be an abelian group. Let ξ : D(X \ Y ) → A be a voltage assignment such that for each component C of Y the total sum of voltages on the semiedges incident to C is 0. Then ξ extends to exactly one A-circulation ξ∗ : D(X) → A. In particular, the number of A-circulations on X is |A|β(X)−γ(X), where β(X) is the Betti number of X and γ(X) is the number of components of X incident with semiedges.

Proof. We may assume that X is connected and Y is a spanning tree T . Let ξ : D(X \ Y ) → A be given. First, assume that no semiedges 3.3. Flows 97

−1 exist. Let e = {x0, x0 } be a cotree edge. Then there is a unique path x1x2 ··· xm in T such that (x0x1x2 ··· xm) is an elementary cycle. Deﬁne −1 an A-circulation on X by setting ξe(xi) = ξ(x0) and ξe(xi ) = −ξ(x0) for i = 1, 2, . . . , m and ξe(y) = 0 for y 6= xi, i = 0, 1, 2, . . . , m. ∗ P Now, we set ξ = e ξe, where the sum runs through all cotree edges. Then ξ∗ is clearly an A-circulation.

Now, let X have s semiedges, say z1, z2, . . . , zs. We form a new graph 0 X by adding an extra vertex w and s new darts y1, . . . , ys attached to 0 w. We extend λ by setting λ(zi) = yi. Thus X is a graph without semiedges and one can apply the above construction of ξ∗ by taking 0 0 Ps T = T ∪{y1, z1} and a restriction of ξ on D(X \T ). Since i=1 ξyi = 0, Ps ∗ ξy1 = − i=2 ξyi = ξ (y1) as is required. Hence there is at least one extension ξ∗ of ξ. Assume there are two A-circualations ξ0 and ξ00 that are extensions of ξ. Then ξ0 − ξ00 is an A-circulations taking value 0 on cotree darts. Then it is easy to see that ξ0 − ξ00 is everywhere trivial, hence ξ0 = ξ00. Lemma 3.14 gives γ(X) independent linear equations relating the values of ξ on semiedges of X. It follows that we have |A|β(X)−γ(X) choices for ξ. 2

The following classical results were proved by Tutte.

Theorem 3.17. (Tutte 1954) If A and B are finite abelian groups of the same order then X admits an A-flow if and only if X admits a B-flow.

Proof. We prove that the number of B-flows is the same as the number of A-flows. Let F ⊆ E(X) and let β(F ) denote the Betti number of the subgraph of X induced on F and γ(F ) is the number of components containing semiedges. Denote µ = |A| = |B|. By Lemma 3.16 µβ(F )−γ(F ) is the number of A-circulations of the subgraph induced by F . Equivalently, µβ(F )−γ(F ) is the number of A-circulations whose sup- port (the set of edges endowed by a non-trivial flow) is a subset of F . By the inclusion-exclusion principle we get for the number of nowhere- P |E(X)−F | β(F )−γ(F ) zero A-flows f(X, µ) = F ⊆E(X)(−1) µ . It follows that the number of A-flows depends only on µ = |A|, and we are done. 2 98 Chapter 3. Applications

Note that the polynomial f(X, µ) defined in the proof above is for graphs without semiedges known as a flow polynomial of X.A Z-flow on X will be called a k-flow for some integer k if 0 < |ξx| < k for every dart x ∈ D. Tutte (1950) proved the following statement. Theorem 3.18. (Tutte 1950) A graph X admits a k-flow if and only if it admits a Zk-flow.

Proof. Considering values of a nowhere-zero k-flow as elements of Zk we obtain a nowhere-zero Zk-flow. Conversely, a nowhere zero Zk- flowP can be interpreted a voltage assignment ξ : D → Z such that ξ ≡ 0 (mod k). If there are m semiedges, then as above join the x∈Dv x free ends of semiedges with free ends of the semistar stm thus forming a graph X0 without semiedges end extend ξ on X0 (there is a unique 0 way to do it).P Let PY be a componentP of X . Since ξ is a voltage assignment ξx = (ξx +ξλ(x)) = 0. Let us call v∈VP(Y ) x∈DPv {x,λ(x)}∈E the sum δ(ξ) = + ξx, where v runs through all vertices P v∈V (Y ) x∈Dv with positive ξx a deficiency of the voltage assignment ξ. It x∈Dv P follows that if there is vertex u ∈ V (Y ) such that ξx > 0 then Px∈Du there is another vertex v 6= u in V (Y ) such that ξ < 0. In x∈Dv x such case, by connectivity, there is a path x1, . . . , xn joining u to v. 0 0 Then we form a new voltage assignment ξ taking values ξxj = ξxj − k 0 0 and ξ = ξ −1 + k, for j = 1, 2, . . . , n. Otherwise we set ξ = ξ. x−1 x j j P The resulting voltage assignment ξ0 satisfies the property ξ ≡ x∈Dv x 0 (mod k) and moreover δ(ξ0) < δ(ξ). It follows that iterating the above procedure we eventually obtain a voltage assignment ζ with δ(ζ) = 0. By the definition, ζ is a k-flow on Y . Applying the procedure for each connectivity component of X0 we obtain a k-flow ξ on X0. The restriction of ξ onto darts of X gives the required k-flow on X. 2

In view of Theorems 3.17, 3.18, in order to decide whether there is an A-flow on a given graph X, it is sufficient to find the least k for which X admits a k-flow. Three Tutte’s flow conjectures related to the existence of k-flows are as follows:

(1) 5-Flow Conjecture. Every bridgeless graph without semiedges has a 5-ﬂow. 3.3. Flows 99

(2) 4-flow Conjecture. Every bridgeless graph without semiedges not containing the Petersen graph as a minor has a 4-flow. (3) 3-flow Conjecture. Every bridgeless graph without semiedges and without a 3-edge-cut has a 3-flow. In 1981 Seymour proved that the flow number of a bridgeless graph is bounded by 6. Theorem 3.19. (Seymour 1981) Every bridgeless graph without semiedges has a 6-flow. Proof. The proof can be found in Diestel [Di97, Chapter 6]. 2

The following observation is easy. Lemma 3.20. Let p : X˜ → X be a covering of graphs. If X admits an A-ﬂow then X˜ admits an A-ﬂow as well.

Proof. If f : D → A is the ﬂow-function then for each x˜ ∈ ﬁbx set f˜(x) = f(x). 2

In general an A-flow does not project along a covering projection. Lemma 3.21. A cubic graph has a nowhere-zero 4-flow if and only if it is 3-edge-colourable. In particular, Hamiltonian cubic graphs have 4-flows.

Proof. If there is a 4-flow on X then by Theorem 3.17 there is a Z2 ×Z2- flow ξ on X. Let x,y and z be darts based at a vertex u. Assuming ξx = ξy we get ξz = ξx + ξy = 0, a contradiction. Hence f(e) = f({x, x−1} = ξ(x) is a regular 3-edge-colouring. Vice-versa, assume there is a regular colouring f of edges of X. We may assume that the three colours are the nontrivial elements of Z2×Z2. −1 by 1, 2 and 3. For an edge e = {x, x } we set ξx = ξx−1 = f(e). Since −1 for eachP edge e = {x, x } and for each vertex v we have ξx + ξx−1 = 0 and ξ = 0, the mapping ξ is a Z × Z -flow, and consequently, x∈Dv x 2 2 X admits a 4-flow. A Hamiltonian cycle in X is necessarily of even length, hence it can be properly coloured by using two colours. The remaining edges form a perfect matching which can be coloured by the third colour. Thus a cubic Hamiltonian graph is 3-edge-colourable. 2 100 Chapter 3. Applications

The following well-known result is an immediate consequence of Lemma 3.14.

Lemma 3.22. (Parity lemma) Let C be an edge-cut in a 3-edge- coloured cubic graph. Let c1, c2 and c3 denote the number of edges in C coloured by colours 1, 2 and 3, respectively. Then c1 ≡ c2 ≡ c3 ≡ |C|(mod 2).

The following theorem is related with the well-known folklore conjecture establishing that every Cayley graph is Hamiltonian. Restrict- ing to cubic case we wish to prove at least a weaker result proving that every cubic Cayley graph is 3-edge-colourable. The problem can be reduced to cubic Cayley graphs coming from simple groups or groups which are close to simple groups. Here we present the following particular result. Recall that a group A is solvable if it admits a descending subnormal serie A = N0 B N1 B N2 B N3 B ··· B Nm = 1 such that Ni/Ni+1 is abelian for each i = 0, 1, . . . , m − 1. Theorem 3.23. A loopless cubic Cayley graph based on a solvable group is 3-edge-colourable.

Proof. Let X be a cubic Cayley graph determined by a group A and set of generators S. The generating set S can be of two types, either consisting of three involutions, or S = {r, r−1, `} with `2 = 1 and r 6= r−1. In the ﬁrst case the edges (darts) of X are coloured naturally by the three involutions giving a 3-edge-colouring. In the second case we may assume that the order |r| is odd, otherwise we can colour the even 2-factor determined by r by two colours and the remaining edge by the third colour. Our strategy is to describe minimal quotients of X not containing loops, to colour such a quotient and lift the colouring. Let X be a minimal counterexample given by A = hr, `i. Let N C A be the minimal normal subgroup of A. If r∈ / N we form a quotient X¯ = X/[N] which is a Cayley graph given by A¯ = A/N = hrN, `Ni. By the assumption X¯ is not 3-edge-colourable. But |X¯| < |X| contradicting the minimality. Hence r ∈ N. By Aschbacher [Prop.xx] N = T × T × · · · × T , where T are isomorphic simple groups. ∼ Solvability of A implies T = Zp is cyclic for some prime p, p 6= 2 because r is of odd order. If ` ∈ N then A = N and ` = 1, since 3.3. Flows 101

N is odd. Hence A = N = hri and X is formed by a p-cycle with a semi-edge attached to each vertex. This graph is easily seen to be 3-edge-colourable, a contradiction. Hence `∈ / N and N is a subgroup of index two. The generator ` acts by conjugation on N preserving the simple direct factors as blocks. By minimality N = T × T ` or N = T = T `. In each case we may assume T = hri. If `r` = re then re2 = r implying e2 = 1(mod p), giving solutions e = ±1. Then X is a p prism which is easily seen to be 3-edge-coloured, a contradiction. Assume r` ∈/ hri. Let K = h`r`r−1i ≤ N. Since N is abelian and r ∈ N, K is centralised by r. However, (`r`r−1)` = r`r−1` is an inverse element to the generator of K. Hence K is a proper normal subgroup of A and of N contradicting the minimality. 2

Exercises

3.3.1. Prove the Parity lemma (Lemma 3.22) 3.3.2. Find an example of cubic hamiltonian graph covering a non-hamiltonian one. Describe the projection of a hamiltonian cycle. 3.3.3. Let p : X˜ → X. Prove that the chromatic index χ0(X˜) ≤ χ0(X) provided X has no loops, and for the chromatic number χ(X˜) ≤ χ(X) provided X has no loops and no semiedges. 3.3.4. ∗ Prove that cubic Cayley graphs with up to 120 vertices admit a nowhere-zero 4-ﬂow. 3.3.5. Compute the ﬂow polynomial for K4, K3,3 and the Petersen graph. 102 Chapter 3. Applications Bibliography

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t-transitive, 19 isomorphic, 32 regular, 28 abelianization, 7 transformation, 28 action, 7 covering transformation group, covering transformation group, 28 28 faithful, 8 dart, 2 regular, 8 dart-reversing involution, 2 semi-regular, 8 the fundamental group, 27 edge, 2 the fundamental groupoid, 27 equitable partition, 93 transitive, 8 adjacency matrix, 94 fibre abstract, 31 Betti number, 40 flow, 95 block, 9 flow conjecture, 98 bridge, 96 flow polynomial, 98 Burnside Lemma, 13 fundamental cycle, 6 fundamental group, 5 Cayley graph, 13 fundamental groupoid, 4 characteristic polynomial, 94 circulation, 96 Generalized Petersen graph, 16 complete system of imprimitiv- graph, 2 ity, 10 n-cube or Hypercube, 24 constellation, 21 n-semistar, 24 core, 8, 9 automorphism group, 4 covering Cayley, 13 canonical double, 63 coset, 16 derived, 34 covering, 23 graph, 23 Coxeter, 20 homological, 64 derived, 34

109 110 INDEX

dipole, 25 Orbit-Stabilizer Theorem, 12 generalized Petersen, 16, 57 orbital, 18 halved cube, 24 orientation, 3 Heawood, 20, 58 Hoﬀman-Singleton, 93 permutation representation, 8 monopole, 35 primitive, 10 Moore, 92 project, 43 Octahedral, 14 representations Octahedron, 15 permutation, 8 orbital, 18 oriented, 3 semiedge, 3 Pappus, 19 stabilizer, 8 Petersen, 55 unimodular, 60 half-arc-transitive, 18 unimodular group, 60 homology group, 7 unique walk lifting, 25 homotopic, 4 valency, 2 imprimitive, 10 vertex, 2 incidence function, 2 vertex/diameter problem, 91 isomorphic voltage, 31 G-spaces, 9 voltage group, 31 constellations, 22 local, 31 covering, 39 voltage space Cayley, 31 Kirchhoﬀ law, 96 monodromy, 31 permutation, 31 lift, 42 regular, 31 lifting, 38 Lifting Criterion, 38 walk, 4 link, 3 closed, 4 locally A-invariant, 50 reduced, 4 locally f-invariant, 51 loop, 3

Maschke Theorem, 74 morphism, 3

Octahedral graphs., 14