Graphs, Coverings Between Graphs and Applications

Home , Covering graph

Graphs, coverings between graphs and applications

Contents

1 Introduction 2

2 Graphs 2

3 Action of the fundamental groupoid 4

4 Lifting automorphisms, regular coverings 6

5 Voltage spaces 10

6 Some applications of coverings 13

7 Flows 16

8 Solution of the lifting problem in terms of voltages 18

9 Coverings of the Petersen graph 20

10 Lifting graph automorphisms, case of abelian CT (p) 21

11 History 24 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 un- bounded number number of vertices (points). 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 proba- bilistic method in graph theory and finally, employing symmetries of graphs one can use finite groups to construct highly symmetrical graphs 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 with 1-dimensional topological objects which are usually embedded to surfaces or to a more complex CW-complexes. The aim of this introductory lecture 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 graph coverings to solve several classical problems. 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. A significant part of the following text is based on a paper by Malniˇc,Nedela and Skoviera:ˇ Lifting graph automorphisms by voltage assignments (European J. Comb. 21, 2000, 927-947).

2 Graphs

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 and λ is an involutory permutation of D. 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 having a given vertex u as their common initial vertex is denoted by Du. The cardinality of Du is the valency of the vertex u. The orbits of λ are called edges; thus each dart determines 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. 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

2 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 fI = I0f 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 by graph theorists. 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 −1 −1 −1 −1 walk 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 2.1 The subset π(X, u) ⊆ π(X) of all reduced closed walks 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.

Proof We prove the second part. Since X is connected for any two vertices u, v there is a walk W : u → v. Then the mapping S 7→ W −1SW takes a closed u-based walk S ∈ πu onto a closed v-based walk. It is easily seen that it preserve the homotopy classes, composition of classes and u −1 −1 identity is mapped onto an identity. Moreover, if S1,S2 ∈ π then W S1W = W S2W implies −1 u v −1 S1 = S2. Hence S 7→ W SW gives a monomorphism π → π . However, S 7→ WSW gives a monomorphism πv → πu. Thus πu =∼ πv. The group π(X) =∼ π(X, u) will be called the fundamental group of X at u. We would like to warn the reader that, in contrast to the classical case of graphs without semiedges, the fundamental group need not be a free group. Nevertheless, it is a free product of cyclic groups, each of which is isomorphic to ZZ or ZZ2.

Lemma 2.2 Let X be a ﬁnite connected graph with e edges, s ≤ e semiedges and v vertices. Then the fundamental group π(X) is a group of rank e−v +1. More precisely, it is a free product of e − s − v + 1 copies of ZZ and s copies ZZ2.

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

3 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. The proof is finished by an observation that each homotopy class of u-based walks can be expressed in terms of generators. We first reduce a closed u-based walk to a minimal representative W of the homotopy class in u π . Let W = ...x1 . . . x2 . . . xk..., where x1, x2, . . . xk form the set of cotree darts listed in the order as they appear in W . Let Wi be the subwalk joining xi to xi+1. By minimality Wi is a −1 path for all i = 1, 2, . . . k − 1 and Wk = PkQk QkRk is a concatenation of four paths, where Qk begins at u. Let Qi be the unique path in T joining u to Wi. The terminal vertex of Qi is met by Wi exactly once. Thus Wi = PiRi where Pi and Ri are, respectively, the subpaths terminating −1 and initiating at the terminal vertex of Qi. Then replacing each Wi by PiQi QiRi we do not −1 Qk−1 −1 change the homotopy class of W . This way we get W ∼ QkRkx1P1Q1 i=1 QiRixi+1Pi+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 bijective correspondence between the set of semiedges of X and the set of free factors in π(X, u) isomorphic to ZZ2. The rest of the terminology not explicitly defined here (but freely used in the sequel) is tacitly adopted from graph theory as well as group theory and topology [28, 34, 14]. We only briefly mention the following. A (right) action of a group G on a set Z is defined by a function σ : Z×G −→ Z, with the common notation σ(z, g) = z·g, such that z·1G = z and z·(gh) = (z·g)·h. We stick to the standard notation Gz = {g ∈ G | z · g = z} to denote the stabilizer of z ∈ Z under the action of G. The left action is defined in a similar manner. 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 3.3. By SymL Z and SymR Z we denote the left and the right symmetric group on a (nonempty) set Z, respectively. The right symmetric group on the set {0, 1, . . . , n − 1} is commonly denoted by Sn. Suppose that an action of A – a group or a fundamental groupoid – on some set Z is “represented” by a subgroup of the symmetric group SymR Z. We then consistently use the “bar notation” a to denote the permutation which corresponds to a ∈ A. Finally, we call an isomorphism f : A −→A0 of two such algebraic structures consistent with the respective actions if there exists an isomorphism of these actions which has the form (−, f); more precisely, there exists a bijection φ of the corresponding sets such that φ(x · a) = φx · fa.

3 Action of the fundamental groupoid

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 vertexu ˜ ∈ X˜, the set D˜u˜ is bijectively mapped onto Dpu˜. −1 The graph X is usualy referred to as the base graph and X˜ as the covering graph. By fibu = p u −1 and fibx = p x we denote the fibre over u ∈ V and x ∈ D, respectively.

Example 3.1 Figure 1 shows a covering Q3 −→ st3, where Qn denotes the n-cube and stn the n-semistar, that is a graph on one vertex and n semiedges.

Consider an arbitrary dart x ∈ X with its initial vertex u. By deﬁnition, for each vertex u˜ ∈ ﬁbu there exists a unique dartx ˜ 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 atu ˜. Consequently, by induction

4 7 ...•...... c...... a ...... 7 ..... 7 ...... b ...... 7 ...... c3 ...... a6 ...... b ...... 5 ...... 3•...... 5....•...... •...6 ...... a3 ...... c.6 ...... c5 ...... a5...... b3...... b6 ...... b ...... •...... a...... c ...... c ...... b...1 .....a. 2...... 2 b4 ...... c1 ...... a.....4 1••...... •...4 ...... 2 ...... a1 . b2 ..... c4 ...... b0...... a0 ...... c0 ....•...... 2.4 0

Figure 1: A covering projection Q3 −→ st3, given by xi −→ x. we obtain the following “Unique walk lifting theorem”. (For the classical topological variant, see [28, pp. 151].)

Proposition 3.2 Let p : X˜ −→ X be a covering projection of graphs and let W : u −→ v be an arbitrary walk in X. Then:

(a) For every vertex u˜ ∈ ﬁbu there is a unique walk W˜ which projects to W and has u˜ as the initial vertex.

(b) Homotopic walks lift to homotopic walks.

Proof (a) We use induction by 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 W˜ 1 with an initial vertexu ˜ and a terminal vertexv ˜. Then W lifts to the unique walk W˜ 1x˜, wherex ˜ is the unique lift of x based at v. (b) 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˜. Then xx−1 lifts to a uniquev ˜-based closed walkx ˜x˜−1. It follows that P˜ and Q˜ are homotopic. (c) 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 pairwise different, otherwise we get a pair of lifts ofx ˜−1 based at the same vertex contradicting −1 part (a). Hence |fibu| ≤ |fibv|. Replacing W by W we get |fibv| ≤ |fibu|, hence |fibu| = |fibv|. By induction the statement extends to a pair of vertices at any distance.

Note that this theorem 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 3.2 that the cardinality |ﬁbu| of a covering projection p : X˜ → X, where X is

5 connected, does not depend on the vertex u. If |ﬁbu| = n is an integer, 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 3.3 Let p : X˜ −→ X be a covering projection of graphs. Then there exists a right action of π on the vertex set of X˜ deﬁned by

u˜ · W =v, ˜ where W : pu˜ −→ pv˜ and v˜ is the endvertex of the unique walk W˜ over W starting at u˜. Moreover,

Wˆ :u ˜ 7→ u˜ · W is a bijection ﬁbpu˜ −→ ﬁbpv˜.

In view of Proposition 3.2(b), the action of the fundamental groupoid can, in fact, be extended u to an action of all walks. In particular, the 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 denote by u u πu˜ = πu˜ the stabilizer ofu ˜ ∈ fibu under the action of π . Clearly, the actions of the fundamental groups at two distinct vertices are isomorphic. Indeed, if Wˆ is the bijection as in Proposition 3.3 −1 u u v and if W∗S = W SW , S ∈ π , then (W,Wˆ ∗) : (fibu, π ) −→ (fibv, π ) is an isomorphism of actions. Problems 1. Let p : X˜ → X be a covering of graphs. Prove that the mapping p∗ : π(˜u, X˜) → π(u, X) defined by W˜ 7→ pW˜ = W (W˜ a closedu ˜-based walk) is a monomorphism of groups. 2∗. What sort of p induces an inclusion of a trivial group into the fundamental group? Give an algorithm constructing the respective covering graph.

4 Lifting automorphisms, regular coverings

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. Unfortunately, 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. Usually, one considers a morphism m : Y −→ X satisfying some additonal properties and asks whether associated with an automorphism f of X there exists an automorphism f˜ of Y such that the following diagram f˜ Y −−−−→ Y     my ym 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

6 covering projections. Nevertheless, speciﬁc questions about automorphisms received minimal 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 usally treated combinatorially rather than topologically. We are thus led to examine the lifting problem from this point of view as well. 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 an automorphism f˜ of X˜, a lift of f, such that the following diagram

f˜ X˜ −−−−→ X˜     py yp f X −−−−→ X is commutative. Note that if f lifts, then so does f −1. 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 known to be the group of covering transformations and is denoted by 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˜. As an immediate consequence of the unique walk lifting we have the following observation.

Proposition 4.1 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 Let f˜ takesx ˜ ontox ˜. We want to show f˜(˜y) =y ˜ for any darty ˜ in D˜. Since X˜ is connected, there exists a walk W˜ with initial dartx ˜ and terminal darty ˜. By the assumption the projection W = p(W˜ ) is ﬁxed by the projection of f˜. Hence f˜(W˜ ) = W˜ , and consequently, f˜(˜y) =y ˜.

Again, the corresponding topological version of Proposition 4.1 may fail to be true. The next proposition is a trivial but important consequence of Proposition 4.1.

Proposition 4.2 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 takingx ˜ onto the same darty ˜. Then f˜g˜−1 belongs to CT (p) and ﬁxesx ˜. It follows that f˜g˜−1 = 1 implying f˜ =g ˜.

The fact that CT(p) acts semiregularly if X˜ is connected is particularly nice. Therefore, for the rest of this section, as well as for most of the paper, we shall assume the graphs X and X˜ 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. Thus, we define the covering projection to be regular if the action of CT(p) is regular on each fibre. There is an alternative approach to regular coverings of graphs. Given connected graph X˜ = (D,˜ V˜ ; I,˜ λ˜) let A ≤ Aut(X˜) be a group of automorphisms acting semiregularly on the sets D˜, V˜ of darts and vertices of X˜. Set X = (D,V ; I, λ), where D, V are the sets of orbits in the action of A on D˜, V˜ , respectively, set I[x]A = [Ix˜ ]A and

7 λ[x]A = [λ˜(x)]A, where [x]A denotes the orbit containing x ∈ D˜. The above deﬁned graph X will be called a regular quotient of X˜ by A. Since an automorphism f ∈ A commutes with both the incidence function I˜ and dart-reversing involution λ˜, the deﬁnitions of I and λ are correct.

Lemma 4.3 A mapping p : X˜ → X taking x 7→ [x]A is a regular covering with CT (p) = A.A regular covering p between connected graphs is equivalent to a covering X˜ → X where X is a regular quotient of X˜ by some A ≤ Aut(X˜), A =∼ CT (p).

Proof Free action of A on the vertices and darts of X˜ implies that p is a covering X˜ → X. The ˜ elements of A permute the darts in fib[x] = [x] for each x ∈ D. By definition A is transitive on [x]A. Conversely, if p : X˜ → X is regular then A = CT (p) acts regularly on fibres over vertices and darts. 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.

Lifted automorphisms preserve the ﬁbres over darts and over vertices. Thus the ﬁbres form a system of blocks of imprimitivity of the action of group of lifts. The following lemma will turn to be useful.

Lemma 4.4 Let p : X˜ → X be a regular covering between connected graphs with A = CT (p) ≤ Aut(X˜) = G. Then the normalizer NG(A) projects.

Proof The covering pX˜ → X is equivalent to x 7→ [x]A. Since f˜ ∈ NG(A) the projection f :[x]A → [fx˜ ]A is well-defined. The next three results, much in the spirit of classical theory [28], 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.

Theorem 4.5 Let p : X˜ −→ X be a covering projection of connected graphs and let f be an automorphism of X. Then f lifts to some f˜ ∈ Aut X˜ if and only if, for an arbitrarily chosen base vertex b ∈ X, there exists an isomorphism of actions

b fb (φ, f) : (fibb, π ) −→ (fibfb, π ) ˜ of the fundamental groups such that f|fibb = φ. Moreover, there is a bijective correspondence between Lft(f) and all functions φ for which (φ, f) is such an isomorphism, and f˜u˜ = φ(˜u · W ) · fW −1,W : pu˜ −→ b.

Proof. Let f˜ be a lift of f. We first show that for an arbitrary walk W : u −→ v in X and u˜ ∈ fibu we have f˜(˜u · W ) = f˜u˜ · fW. (1) Indeed, let W˜ :u ˜ −→ u˜ · W . Then f˜W˜ : f˜u˜ −→ f˜(˜u · W ) projects to fW , implying (1). In b ˜ particular, (1) holds true for eachu ˜ ∈ fibb and S ∈ 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. Letu ˜ 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.

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

We verify that it is a bijection. To show that it is onto, letv ˜ 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 −1 ˜ ˜ −1 mapped tov ˜. To show that it is 1-to-1, 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, −1 −1 f˜u˜1 = f˜u˜2 implies φ(˜u1 · W ) · fW = φ(˜u2 · W ) · fW . Since φ is a bijection, it followsu ˜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 groups. That this function is 1-to-1 follows from Proposition 4.2.

It is a consequence of Theorem 4.5 that the problem of whether some automorphism has a lift can be reduced to the problem of whether this automorphism gives rise to an isomorphism 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 addition, it enables us to give a more explicit formula expressing how an arbitrary lifted automorphism acts on the vertex ﬁbres.

Theorem 4.6 Let p : X˜ −→ X be a covering projection of connected graphs, and let f be an automorphism of X. Then:

(a) 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 point b ∈ fibb isomor- fb phically onto some stabilizer πv˜ ≤ π . In this case we have that v˜ = φ˜b, and there is a bijective correspondence between all choices of such a vertex v˜ and all such isomorphisms.

˜ fb −1 (b) Choosing a base point d ∈ ﬁbfb and Q ∈ π such that Q πd˜Q = fπ˜b, all such bijections φ = φP are given by b φP (˜b · S) = d˜· P f(S), (S ∈ π ), (2) fb where P belongs to the coset N(πd˜)Q of the normalizer of πd˜ within π . Moreover, φP 0 = φP 0 if and only if P ∈ πd˜P .

Proof. It is clear that if (φ, f) is an isomorphism of actions, then the condition holds. For the converse, let fπb = πfb. Eachu ˜ ∈ ﬁb can be written asu ˜ = ˜b · S for some S ∈ πb because ˜b v˜ b 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 bijective correspondence should also be clear since φ is completely determined by the image of one point. This proves part (a). To prove part (b), letv ˜ = d˜· P be any point satisfying the condition as in part (a). Then −1 −1 −1 P πd˜P = πd˜·P = Q πd˜Q, that is PQ ∈ N(πd˜). The last statement is clear as well.

9 As an immediate corollary of the above two theorems we have the following observation.

There exists a covering transformation mappingu ˜1 ∈ ﬁbu tou ˜2 ∈ ﬁbu if and only if πu˜1 = πu˜2 . Since a covering projection is regular if and only if CT(p) acts transitively, it follows that regular coverings can be characterized by means of stabilizers under the action of the fundamental group.

Corollary 4.7 A covering projection of connected graphs p : X˜ −→ X is regular if and only if, b for an arbitrarily chosen vertex b of X, all the stabilizers under the action of π on ﬁbb coincide, that is to say, the stabilizer is a normal subgroup.

Problems 1. Find all regular quotients of the underlying graph of 3-dimensional cube.

5 Voltage spaces

The goal of this section is to explain, unify and generalize the classical concepts of ordinary, permutation and relative voltage assignments introduced in the 70’s by Alpert, Biggs, Gross, Tucker and others [14]. In fact, this unified approach works in the general setting of fairly arbitrary topological spaces, see [21]. Compare also with [37]. 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 and ξ : π −→ G is a homomorphism (where, as usual, π = π(X)). 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) . By induction we 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 it follows from the above condition 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 this 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 [14]). Obviously, Gu = ξ(πu) is a group, called the local voltage group at the vertex u. Moreover, let W : u −→ v. Then the inner automorphism # −1 W (g) = ξW gξW of G takes Gu onto Gv. A voltage space is called locally transitive whenever some (and hence any) local voltage group acts transitively on the abstract fibre. 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

10 vertex u, 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 each u˜ ∈ 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.) Voltage spaces associated with isomorphic covering projections are called equivalent. Note that the action of ξW “represents” the bijection Wˆ :u ˜ 7→ u˜ · W introduced in Proposition 3.3: indeed, its permutation representation W ∈ SymL F is precisely the permutation ξW ∈ SymR F . Note further that u u u (ù, ξ) : (fibu, π ) −→ (F,G ) is an epimorphism of actions with ù being a bijection. Hence G u acts on F “in the same way modulo relabelling” 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. (If a covering is considered as a bundle [36], then its structure group corresponds to the local monodromy; we here deal with conjugate local groups within the voltage group instead of with conjugate isomorphisms, incorporated into the definition of a bundle, from the abstract structure group to concrete automorphisms of fibres.) 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 [39], 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. [13]).

Theorem 5.1 Let p : X˜ → X be a covering projection between connected graphs, X ﬁnite. Then there exists an associated locally transitive permutation voltage space associated with p. Conversely, every locally transitive voltage space (F,G; ξ) on a graph X determines a derived covering projection X˜ → X which is determined up to isomorphism, if X is connected then X˜ is connected.

Proof We set F = fibb, where b is a fixed fibre. Since each fibre fibv has the same cardinality, there is a labelling `v : F → fibv. For each walk W : u → v,we set the voltage ξW to be the permutation in W ∈ SymR F representing the bijection Wˆ . That means `(˜u) · ξW = `(˜v) if and only ifu ˜ · W =v ˜. Now the voltage group G ≤ Sym(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 closed u-based walk W = p(W˜ ) such u thatu ˜1 · W =u ˜2. Hence, `(˜u1) · ξW = `(˜u2) and 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. More- over, 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.

11 If X is connected and (F, G, ξ) 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 u i, j ∈ F , 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 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 every 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 −1 its initial vertex u to its terminal vertex v = Ix is defined by the following rule: ξx(i) = j if and only if there is a dartx ˜ ∈ fibx joining ui to vj. With each 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. [11]). We have the following theorem (compare with [14, pp. 57–71]).

Theorem 5.2 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 voltage space (F,G; ξ) on a graph X determines a regular derived covering projection X˜ → X which is determined up to isomorphism.

3 Example 5.3 We revisit Example 3.1. The regular voltage space ({0, 1,..., 7}, ZZ2; ξ) on the 3 semistar st3, where the homomorphism ξ : π −→ ZZ2 is deﬁned by setting ξa = 001, ξb = 010 and ξc = 100, is associated with the covering Q3 −→ st3. By replacing each element of the abstract ﬁbre {0, 1,..., 7} by its 3-digit binary code, the Cayley voltage space, associated with the same covering, is obtained. Another equivalent regular voltage space is ({0, 1,..., 7},S8; ψ), where ψa = (01)(23)(45)(67), ψb = (02)(13)(46)(57) and ψc = (04)(15)(26)(37).

Example 5.4 Let us call a graph with one vertex and arbitrary number of darts a monopole. Given a group G and a symmetric generating (multi)set S = S−1 of G we deﬁne the Cayley graph Cay(G, S) to be the graph (G×S, G; IS, λS), where IS = pr1 is the projection onto the ﬁrst factor −1 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 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 encountered in the classical approach that not all Cayley graphs were regular coverings of bouquets of circles [14, pp. 68–69].

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 5.5 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 equal and the local group at b has not changed.

12 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 for X.) Let W : u −→ v 0 −1 be an arbitrary walk in X. Set ξW = (ξW u ) ξW ξW v . As a result, the new voltage of each 0 b preferred walk is trivial and ξS = ξS for every closed walk S ∈ π . Now, modify the labelling of 0 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.

Since any two vertices can be joined by a walk carrying the trivial voltage, no walk can have voltage outside Gb. For if W : u −→ v had a voltage not in Gb, then the closed walk u −1 v 0 0 b S = (W ) WW at 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 the condition Gb = G does not mean any loss of generality, but can sometimes considerably simplify the notation. Problems 1. 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. A Cayley voltage space (ZZ2, ZZ2; ξ) with ξ(x) = 1 ∈ (ZZ2, +) determines a canonical double cover X˜ over X. Show that X˜ is connected if and only if X is not bipartite. k 3. Let X be a connected graph with v vertices and e edges. Let k = e − v + 1, G = ZZ2 and −1 T be a spanning tree of X. Let x1, x2, . . . , xk be 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. (a) Prove that the 2k cover over X˜ deﬁned by the Cayley voltage space (G, G, ξ) is connected.

(b) Describe the kernel of the associated epimorphism from the fundamental group π(X) → G.

4.* Prove Theorem 5.2.

6 Some applications of coverings

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

Proposition 6.1 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 incident to edges colored by i. Set G = hτ1, τ2, . . . , τki and let X 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. 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 then v · τi1 τi2 . . . τim τim+1 is a pendant vertex, 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 at least a sequence made of m involutions τi to map v onto v. Thus the length d of C is at least 2m + 1.

13 In fact one can prove that the Cayley graph X constructed above regularly covers the tree ∼ Tk(m) with CT (p) = StabG(v). Spectrum of a graph and of its cover 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 exists a square matrix M = (di,j) of order n 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.

Lemma 6.2 The characteristic polynomial of M divides the characteristic polynomial of X.

Lemma 6.3 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˜.

Large graphs of given degree d and diameter 2 A vertex/diameter problem for diameter two is to find for a given number d the largest order v(d) of graph of degree d and diameter two. An easy observation gives an upper bound v(d) ≤ d2 + 1. It is well-known that the equality can be achieved only for d = 1, 2, 3, 7, 57. The extremal graphs (called Moore graphs) are known for d = 1, 2, 3 and 7, namely, K2, the pentagon, the Petersen graph and the Hoffman-Singleton graph. The problem of existence of a 57-valent Moore graph is 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 6.4 Let q be a prime power of the form s = 4s + 1 and let GF (q) be the Galois field of 0 order q. Let X be a two-vertex graph, whose vertices u, u are joined by q darts xg, g ∈ GF (q), and 0 there are s loops attached to both u,u . Let l1, l2, . . . , ls be darts representing loops incident to u −1 0 0 0 0 (li 6= lj ), let l1, l2, . . . , ls be darts representing loops incident to u . Let h be a primitive element + + + 2 of GF (q) and GF (q) be the additive group of the field. Set G = GF (q) ×GF (q) , ξxg = (g, g ), 2i 0 2i−1 ξ(li) = (0, h ) and ξ(li) = (0, h ). This way a Cayley voltage space (G, G, ξ) is defined on X. 2 8 1 2 The derived graph has 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 6.5 Let X˜ be the derived graph given by the above deﬁned Cayley voltage space on X. Then X˜ has diameter two, it is vertex-transitive.

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. It will be proved later that an automorphism of the base graph transposing the two vertices lifts. ˜ 0 Knowing that X is vertex-transitive, it has a diameter two 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). 0 2 A path P : u(0,0) → u(x,y) is of length two if and only if for a given (x, y) 6= (g, g ) at least one of the 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, . . . 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 (g1, g2) of the equation 2 2 (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,

14 2i we can always find i such that y = x(g1 + g2 + 2i). If x = 0 then either y = ±h and u(0,y) is reachable by a path of length 1, or y = ±h2i−1 and it can be expressed as a sum of the form ±h2i ± h2j implying that it is reachable by a path of length two. As curiosity we note that the least graph X˜ comes from the field GF (5) and it is the Hoffman- Singleton graph, see Fig. 2.

(0,0)

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

(2,4)

(3,4)

(4,1)

Figure 2: Hoﬀman-Singleton graph determined as a regular cover over a two-vertex graph with voltages in ZZ5 × ZZ5 (this description was discovered by Siagiov´a)ˇ

Problems. 1. Using graph coverings prove that for every k ≥ 2 and for every g ≥ 3 there are graphs of girth exactly g. 2. Prove that every cubic graph is a regular quotient of a cubic Cayley graph. 3. Determine the smallest cubic Cayley graph covering the Petersen graph and the dodeca- hedral graph. 4. Prove that each regular graph of even valency is a regular quotient of a Cayley graph. 5. Show that not every equitable partition of a graph comes from a graph covering. 6. Prove that an equitable partition is induced by a graph covering if and only if the associated matrix M = (di,j) is symmetric. 7.a) Define the generalised Petersen graph GP (n, k) by a Cayley voltage space on a two-vertex graph. 7.b)∗ Which of the GP (n, k) can be described as a regular covers over a monopole? 8. Using the description of Siagiovádepictedˇ on Fig. 2 compute alternatively the eigenvalues of the Hoffman-Singleton graph. 9.∗ Prove or disprove the following statement: Let X˜ → X be a graph covering, X be without semiedges. Prove or disprove: The sets of eigenvalues of X and X˜ coincide.

15 10.∗ 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. 11.∗∗ 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.

7 Flows

Voltage assignments valued in abelian groups can be viewed as flows. Usually it is required that in all, or in most vertices the Kirchhoff law is satisfied. Let G be an abelian groupP and X = (D,V ; I, λ) be a connected graph. A Cayley voltage space (G, G, ξ) on X such that ξ = 0 x∈Dv x for every vertex v ∈ V will be called a circulation, or a G-circulation. Generally, for A ⊆ V denote by DA the set of darts with initial vertices in A and by DA,A the set of darts such that x 6= λ(x) and Ix = Iλ(x) ∈ A. Finally, by DA,A¯ we denote the set of darts with initial vertices in x such ¯ that either x = λ(x) or Iλ(x) ∈ V \ A = A. It follows that the edges underlying DA,A¯ form an edge-cut separating A.A bridge is either a link e in a graph X without semiedges such that X −e is disconnected, or a single semiedge in X such that X − e is a graph without semiedges.

LemmaP 7.1 Let X be a connected graph without and (G, G, ξ) be a circulation on X. Let A ⊆ V . Then x∈D(A,A¯) ξx = 0. In particular, if {x, λx} is a bridge then ξx = ξλx = 0. P P P Proof x∈D(A,A¯) ξx = x∈D(A,V ) ξx − x∈D(A,A) ξx = 0 − 0 = 0.

A G-circulation on X such that ξx 6= 0 for every x ∈ D is called a (nowhere-zero) G-ﬂow. By the above lemma graphs with bridges admits no circulation. The following classical results were proved by Tutte.

Theorem 7.2 (Tutte 1954) If G and H are finite abelian groups of the same order then X admits a G-flow if and only if X admits a H-flow.

A ZZ-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 7.3 (Tutte 1950) A graph X admits a k-ﬂow if and only if it admits a ZZk-ﬂow.

In view of Theorems 7.2,7.3 to decide whether given graph X there is a G-flow on X, it is sufficient to find the least k for for which X admits a k-flow. Three Tutte’s flow conjectures related to existence of k-flows follow. Five-Flow Conjecture. Every bridgeless graph without semiedges has a 5-flow. Four-flow Conjecture. Every bridgeless graph without semiedges not containing the Pe- tersen graph as a minor has a 4-flow. Three-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 7.4 (Seymour 1981) Every bridgeless graph without semiedges has a 6-ﬂow.

The proofs of the above theorems can be found in Diestel [7, Chapter 6]. The ﬂow-colouring duality will be discussed later. The following observation is easy.

16 Lemma 7.5 Let p : X˜ → X be a covering of graphs. If X admits a G-ﬂow then X˜ admits a G-ﬂow as well.

Proof. If f : D → G is the flow-function then for eachx ˜ ∈ fibx set f˜(x) = f(x). In general a G-flow does not project along a covering projection.

Lemma 7.6 A cubic graph has a 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 7.2 there is a ZZ2 × ZZ2-flow f on X. Let x,y and z be darts based at a vertex u. Assuming f(x) = f(y) we get f(z) = f(x) + f(y) = 0, a contradiction. Hence f is a regular 3-edge-colouring. Vice-versa, assume the edges of X are coloured by 1, 2 and 3. We set f(x) = f(x−1) = c ∈ {1, 2, 3} if the edge {x, x−1} is coloured by c. Setting a = (1, 0), b = (0, 1) and c = (1, 1) we get a ZZ2 × ZZ2-flow. Hence X has 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. The following theorem is related with the well-known folklore conjecture establishing that every Cayley graph is Hamiltonian. Restricting 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.

Theorem 7.7 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 G and set of generators S. The generating set S can be of two types, either S consists of three involutions, or S = {r, r−1, `}, where `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 G = hr, `i. Let N C G be the minimal normal subgroup of G. If r∈ / N we form a quotient X¯ = X/[N] which is a Cayley graph given by G¯ = G/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 G implies T = ZZp is cyclic for some prime p, p 6= 2 because r is of odd order. If ` ∈ N then G = N and ` = 1, since N is odd. Hence G = 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 G and of N contradicting the minimality.

17 Problems 1. Prove the following (Parity lemma). Let C be an edge-cut in a 3-edge-coloured cubic graph. Then for each colour i = 1, 2, 3 the number of edges in C coloured by i has the same parity as |C|. 2. Find an example of cubic hamiltonian graph covering a non-hamiltonian one. Describe the projection of a hamiltonian cycle. 3. Prove that cubic Cayley graphs with up to 120 vertices have a 4-ﬂow.

8 Solution of the lifting problem in terms of voltages

Let p : X˜ −→ X be a covering projection of connected graphs. The question whether an automorphism f of X has a lift is by Theorem 4.5 equivalent to the problem whether f is consistent with the actions of fundamental groups πb and πfb. Let (F,G; ξ) be a voltage space associated with the above covering. As remarked in Section 5, u the action of the fundamental group π on ﬁbu is “the same, modulo relabelling” as the action of the local group Gu on F . Can we transfer the isomorphism problem involving 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 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. (3)

We remark that we can speak of local invariance without specifying the vertex, as local invariance at some 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 group hfi generated by f. This is equivalent to requiring that condition (3) be satisfied by both f and f −1. If f has finite order it is even sufficient to check (3) for f alone. Assume that the voltage space is locally A-invariant. Since with each f ∈ A its inverse f −1 must also satisfy condition (3), local A-invariance is equivalent to the requirement that for each f ∈ A there exists an induced isomorphism f #u : Gu −→ Gfu of local voltage groups such that the following diagram f πu −−−−→ πfu     ξy yξ

f #u Gu −−−−→ Gfu

#u u is commutative; in other words, f (ξW ) = ξfW where W ∈ π . It is clear that in order to transfer the isomorphism problem involving fundamental groups to local groups we must require local invariance in the ﬁrst place. If the voltage space is, say, a monodromy one, then the existence of a lift implies local invariance. However, this need not be true in general.

Theorem 8.1 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 three statements are equivalent:

18 (a) f has a lift, (b) (F,G; ξ) is locally f-invariant,

# b fb (c) the mapping f b : G −→ G taking ξW 7→ ξfW is a well-deﬁned isomorphism of the local voltage groups consistent with respective actions on F .

Proof. Let f has a lift f˜. By Theorem 4.5 the restriction φ = f˜|u gives an isomorphism (φ, f) of u fu the actions (fibu, π ) → (fibfu, π ). In particular, since (F,G; ξ) is an associated voltage space, u ξ(W ) = 1 implies that W ∈ π acts on fibu as identity. Since (φ, f) is an isomorphism of the actions fW acts on fibfu as identity. Since (F,G; ξ) is a monodromy voltage space, it follows that ξ(fW ) = 1 proving (b). #u u In order to prove (b) ⇒ (c) it is enough to show that f (ξW ) = ξfW where W ∈ π is u well-defined. Let W1 and W2 be closed walks in π such that ξ(W1) = ξ(W2). We want to prove −1 u ξfW1 = ξfW2 . Then 1 = ξW1 ξ = ξ −1 ∈ G . By the local f-invariance W2 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. (b) ⇒ (c). By Theorem 4.5, 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 `fu : fibfu → F u u fu fu determining morphisms of actions (fibu, π ) → (F,G ), (fibfu, π ) → (F,G ), respectively. By assumption, f #u exists and it gives an isomorphism of actions (ϕ, f #u ):(F,Gu) → (F,Gfu). We −1 #u u fu set 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. For regular voltage spaces associated with a regular covering we can immediately infer that an automorphism lifts if and only if every closed walk carrying the trivial voltage is mapped to a closed walk with the same property. A special form of this result was previously observed by Gvozdjak and Siráˇn[15]ˇ and Nedela and Skovieraˇ [30]. Corollary 8.2 Let the covering as well as the voltage space be regular. Then f lifts if and only if the voltage space is locally f-invariant. 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. This approach has already been used by Malniˇcand Maruˇsiˇcin [23]. The method is further illustrated in Section 9.

Corollary 8.3 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 in SymR F

ξS · τ = τ · ξfS, where S runs through a generating set for πb, has a solution. Moreover, there is a bijective correspondence between all solutions and the restrictions Lft(f)|ﬁbb . # b Proof. By Theorem 8.1(c) the mapping f b : ξW 7→ ξfW , 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 such that ϕ(x · g) = ϕ(x) · f #b (g). Representing the bijection ϕ as an b element τ of SymR F and taking g = ξW for some W ∈ π we get ξW · τ = τ · ξfW . In particular, it holds for W forming the fundamental cycles with respect to some spanning tree. Conversely, u #u having the solution of ξS · τ = τ · ξfS, for the generators of G the consistent isomorphism f #u −1 is determined by the images of generators f (S) = ξfS = τ ξSτ.

19 9 Coverings of the Petersen graph

Let P be the Petersen graph, with V (P ) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} being the vertex-set la- belled as in Figure 3. We characterize all covering projections P˜ −→ P such that the automorphism group Aut P lifts.

0 1 ...... •• ...... •• .. 68 ...... 9 ...... 5•...... ••...... 2 ...... • ...... 7 ...... ••...... 4 3

Figure 3: The Petersen graph.

Since every covering projection of P can be described by means of a permutation voltage space, it is suﬃcient to characterize all the assignments of permutations to the darts of P such that the derived covering has the required property. By [39, 14] 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 (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) and h = (01845)(29376), Corollary 8.3 gives us three systems of equations ξSτ = τξfS, ξSη = ηξgS 9 and ξSω = ωξhS, 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 systems of equations reduce to the following 9 18 equations in 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 ξi can be expressed as −i i 6 6 ξi = τ ντ , i = 0, 1,..., 5, where ν and τ are permutations satisfying the identity ντ = τ ν.

20 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 6 6 −i i identity ντ = τ ν and substitute all ξi by τ ντ in all the other equations.

Example 9.1 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 (see remarks after Corollary ??) D˜ of the dodecahedron D. Since |Aut D˜| > |Aut D × ZZ2| it follows that the dodecahedron is unstable (see [27] for the deﬁnition and [29] for further information and results). This is the ﬁrst symmetrical example of its kind, as D is sharply 2-arc-transitive.

Problems 1. Prove that the lifted graph X˜ in the Example 6.4 is vertex-transitive. 2. Show that if k2 ≡ ±1 mod n then the generalised Petersen graph GP (n, k) is vertex- transitive.

3. For all n ≤ 5 describe n-folded coverings of K4 such that the whole automorphism group lifts.

10 Lifting graph automorphisms, case of abelian CT (p)

Let X be a connected graph and let T be a spanning tree of X. Let r and t be the number of cotree and tree edges, respectively. An orientation of X is a decomposition D = D+ ∪ D− such that D+ as well as D− takes from each edge {x, x−1) exactly one dart. It follows that that the intersection D+ ∩ D− is formed by the semiedges of X. For simplicity in what follows we assume that X has no semiedges. Any automorphism A of X can be represented by an r × (r + t) + matrix C = CT (A), whose rows correspond to the r cotree darts in D and columns correspond + to darts in D . Any cotree dart xi determines a unique elementary simple cycle Ci in T + xi ²i,1 ²i,2 ²i,r+t which is directed consistently with xi. Let CiA = x1 , x2 , . . . , xr+t be an image of the cycle −1 Ci considered as an r + t vector. The exponent ²i,j is 1 if xi appears in CiA, it is −1 if xi appears in CiA and it is 0 otherwise. The entries of CT (A) are the exponents ²i,j. The associated reduced matrix LT (A) is an r × r matrix formed by the ﬁrst r columns of CT (A).

21 Proposition 10.1 Let X be a connected graph with a spanning tree T . Then the mapping A 7→ LT (A) is a homomorphism into the unitary group U(r, Z). Given abelian group H the product Hr can be considered as an r-dimensional Z-module. If K is a submodule then the set of integer vectors ~y = (y1, y2, . . . , yr) such that y1a1+y2a2+···+yrar = 0 ∈ H for each ~a = (a1, a2, . . . , ar) ∈ K will be called an orthogonal complement of K. If (H, H, ξ) ~ is a Cayley voltage space on X reduced with respect to T we denote by ξ = (ξC1 , ξC2 , . . . , ξCr ) be the r dimensional vector of voltages on the elementary cycles. Note that for each Ci we may form −1 u a unique generator Si = WiCiWi in 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, . . . , r. Since H is abelian, ξASi = ξACi for all i = 1, . . . , r and any automorphism A. Theorem 10.2 Let X be a connected graph with cycle rank r, T be a spanning tree and let (H, H, ξ) be a T -reduced locally transitive Cayley voltage space on X. Then the following statements are equivalent (a) an automorphism A lifts into an automorphism of the derived graph, (b) (H, H, ξ) is locally A-invariant,

Moreover, if H = ZZn is cyclic then the above conditions are equivalent with the statement: there exists an eigenvalue α coprime to n such that Lξ~ = αξ~. Proof. By Theorem 8.1 (a) ⇔ (b). (b) ⇒ (c) If (b) holds then by Theorem 8.1(c) the automorphism A induces an isomorphism #u u Au A of the local groups H → H given by ξW 7→ ξAW . Since the voltage space is T -reduced and locally transitive Hu = H for each u ∈ V . Thus A#u = A# : H → H is a group automorphism. ~ Pr Let 0 = ξ · ~a = i=1 aiξi. Then Xr Xr Xr Xr # # # ~ 0 = A ( aiξi) = aiA ξi = aiA ξSi = aiξASi = ~a · Lξ. i=1 i=1 i=1 i=1 Hence the orthogonal complements to ξ~ and Lξ~ are identical. (c) ⇒ (b) Conversely, let the orthogonal complements be identical. Assume ξW = 0 for an u-based closed cycle. Since W is expressible in terms of the r generators S1,S2,...,Sr and since Pr ~ W is abelian, we have ξW = i=1 aiξSi = ~a · ξ for some integer vector ~a = (a1, a2, . . . , ar). By ~ Pr ~ assumption 0 = Lξ · L~a = i=1 biξASi = ξAW , where b = L~a. It follows that (H, H, ξ) is locally A-invariant. ∼ Let H = ZZn. Assume any of the conditions (a), (b) and (c) holds. Since in a cyclic group ZZn every group automorphism is of the form h 7→ αh, we have Lξ~ = A#ξ~ = αξ~ for some α coprime to n. Conversely, if Lξ~ = αξ~ then there is an automorphism A#u = A# induced by A which is by deﬁnition consistent with the action of H on itself by right multiplication. Theorem 8.1 completes the proof. Example 10.3 For every nonbipartite graph X one can easily deﬁne a Cayley voltage space (ZZ2, ZZ2, ξ) which is (Aut X)-locally invariant by setting ξx = 1. Indeed, ξW = 0 if and only if |W | is even. If |W | is even then |fW | is even for any f ∈ Aut X. The existence of odd cycles implies that (ZZ2, ZZ2, ξ) is locally transitive, and so the derived graph X˜ is connected. The covering graph X˜ is called the canonical double covering of X [29]. It is a minimal bipartite cover over X in the following sense: If Y is a bipartite cover of X then it covers X˜.

22 Example 10.4 On every graph X one can define many notrivial Cayley voltage spaces (G, G, ξ) which are (Aut X)-locally invariant on the whole set of vertices. Choose an arbitrary spanning tree T of X and let n be the number of cotree links and loops, and let m be the number of n m semiedges. Set G = ZZk ⊕ ZZ2 for some positive integer k. Define the voltages in such a way that the darts of the chosen spanning tree receive the trivial voltage, the voltages on darts of semiedges generate the ZZ2-part of G, and the voltages on darts of links or loops of cotree edges generate the ZZk-part of G. Lemma 10.5 Let X be a connected graph and let p : X˜ → X, where p is the regular covering defined by the associated Cayley votage space (G, G, ξ). Then Aut X lifts. In particular, for every connected graph X containing a cycle there are infinitely many finite regular connected covers such that Aut X lifts.

Proof Let ξ~ be the vector of voltages on the cotree darts with the preferred orientation taken randomly. We employ Theorem 10.2(c). The orthogonal complement of any n + m dimensional vector is by definition trivial. In particular ξ~⊥ = ~∅ = (Lξ~)⊥, for any L. Since X contains a cycle n > 0 and we 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. The above construction generalizes to the case involving an arbitrary coefficient ring, thereby extending homological coverings of Biggs [4, 5] and Surowski [38] to graphs with semiedges.

Example 10.6 Let X be the complete bipartite graph with the bipartition V = {1, 3, 5} ∪ {2, 4, 6}. Choose the spanning tree T to be the path T = 123456. We orient the cotree darts by setting x1 = 14, x2 = 25, x3 = 36 and x4 = 56. The corresponding elementary cycles are: C1 = (1432), C2 = (2543), C3 = (3654) and C4 = (123456). 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). 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). It is easy to compute the matrices L(R) and L(`). We have     0 1 0 0 −1 0 −1 −1  0 0 1 0   0 0 −1 0  L(R) =   L(`) =   . −1 0 0 −1  0 1 −1 0  0 0 0 1 1 −1 1 0

We want to describe all Cayley voltage spaces (ZZn, ZZn, ξ) on X such that the group of automorphisms G = hR, `i lifts. By Theorem 10.2 we need to solve the system of congruences L(R)ξ~ = αξ~, L(`)ξ~ = βξ~ in variables ξ1, ξ2, ξ3, ξ4, α, β modulo n, where α, β are coprimes to n. First four congruences have a solution ξ~ = c(1, α, α2, α3 + 1), where the parameters are related 3 by c(α + 1)(α − 1) ≡ 0 mod n. Since we require (ZZn, ZZn, ξ) to be locally-transitive, c is coprime to n. It follows that we may assume c = 1. The system L(`)ξ~ = βξ~ contains equations −(ξ1 + ξ3 + ξ4) = βξ1, −ξ3 = βξ2, ξ2 − ξ3 = βξ3 2 and ξ1 − ξ2 + ξ3 = βξ4. Inserting 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 3 2 fourth equation translates into ξ4 = α + 1 = 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 summarise the above discussion in the following lemma.

23 Lemma 10.7 With above notation G = hR, `i lifts if and only if there is α in ZZn such that α is coprime to n and α2 − α + 1 ≡ 0 mod n. In such case the T -reduced voltage assignment is deﬁned by ξ~ = (1, α, α2, 0).

Few small solutions are α = 2, 3, 4 modulo n = 3, 7, 13, respectively. Problems 1. Prove that K4 has only one non-trivial cover with cyclic CT (p) such that the full automorphism group lifts.

2. Prove that K3,3 has no non-trivial cover with cyclic CT (p) such that the full automorphism lifts. ∗ 3. Is there a regular cover of K3,3 such that the full automorphism group lifts and CT (p) is generated by less than four generators? 4.∗ A graph is called s-arc-transitive if for any two reduced walks of length s there is a graph automorphism taking one to the other. It is known that a cubic graph X is at most 5-transitive. 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 per- mutatations 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. 5.∗ Describe all the cyclic covers over the octahedral graph and over the cube such that the full automorphism group lifts.

11 History

Combinatorial treatment of graph coverings had its primary incentive in the solution of Hea- wood’s Map Colour Problem due to Ringel, Youngs and others [33]. That coverings underlie the techniques that led to the eventual solution of the problem was recognized by Alpert and Gross [12]. These ideas further crystalized in 1974 in the work of Gross [11] 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 [4]. Much of the theory of combinatorial graph coverings in its own right was subsequently developed by Gross and Tucker in the seventies. We refer the reader to [14, 41] and the references therein. The theory was extended to combinatorial graph bundles introduced by Pisanski and Vrabec [32] in the eighties. 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 [4] and in a paper of Djoković[8]. 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 [16] and Kwak and Lee [18, 19]; constructions of regular maps on surfaces based on covering space techniques due to Archdeacon, Gvozdjak, Nedela, Richter, Siráˇn,ˇ Skovieraˇ and Surowski [1, 2, 15, 29, 30, 38]; and construction of transitive graphs with prescribed degree of symmetry, for instance by Du, Malniˇc,Nedela, Maruˇsiˇc,Scapellato,

24 Seifter, Trofimov and Waller [9, 23, 24, 26, 27, 35]. Lifting and/or projecting techniques play a prominent role also in the study of imprimitive graphs, cf. Gardiner and Praeger [10] among others. The lifting problem in terms of voltages in the context of general topological spaces was considered by Malniˇc[21]. Very recently, Venkatesh [40] obtained structural results which refine the work of Biggs [4, 5] and Djoković[8] on lifting groups. 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 [3, 6, 20, 42] and many others.

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