Martin Škoviera

Lifting Graph Automorphisms by Voltage Assignments

The problem of lifting graph automorphisms along covering projections and the analysis of lifted groups is considered in a purely combinatorial setting. The main tools employed are: (1) a systematic use of the fundamental groupoid; (2) unification of ordinary, relative and permutation voltage constructions into the concept of a voltage space; (3) various kinds of invariance of voltage spaces relative to automorphism groups; and (4) investigation of geometry of the lifted actions by means of transversals over a localization set. Some applications of these results to regular maps on surfaces are given. Because of certain natural applications and greater generality, graphs are allowed to have semiedges. This requires careful re-examination of the whole subject and at the same time leads to simplification and generalization of several known results.

Decompositions and reductions of snarks

According to M. Gardner [“Mathematical Games: Snarks, Boojums, and Other Conjectures Related to the Four-Color-Map Theorem,” Scientific American, vol. 234 (1976), pp. 126–130], a snark is a nontrivial cubic graph whose edges cannot be properly colored by three colors. The problem of what “nontrivial” means is implicitly or explicitly present in most papers on snarks, and is the main motivation of the present paper. Our approach to the discussion is based on the following observation. If G is a snark with a k-edge-cut producing components G1 and G2, then either one of G1 and G2 is not 3-edge-colorable, or by adding a “small” number of vertices to either component one can obtain snarks G1 and G2 whose order does not exceed that of G. The two situations lead to a definition of a k-reduction and k-decomposition of G. Snarks that for m < k do not admit m-reductions, m-decompositions, or both are k-irreducible, k-indecomposable, and k-simple, respectively. The irreducibility, indecomposability, and simplicity provide natural measures of nontriviality of snarks closely related to cyclic connectivity. The present paper is devoted to a detailed investigation of these invariants. The results give a complete characterization of irreducible snarks and characterizations of k-simple snarks for k ≤ 6. A number of problems that have arisen in this research conclude the paper.

Which generalized Petersen graphs are Cayley graphs

The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {uj; i ϵ Zn} ∪ {vj; i ϵ Zn}, and edge-set {uiui, uivi, vivi+k, iϵZn}. In the paper we prove that (i) GP(n, k) is a Cayley graph if and only if k2 1 (mod n); and (ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k2 -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon. The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218].

Regular Embeddings of Canonical Double Coverings of Graphs

This paper addresses the question of determining, for a given graphG, all regular maps havingGas their underlying graph, i.e., all embeddings ofGin closed surfaces exhibiting the highest possible symmetry. We show that ifGsatisfies certain natural conditions, then all orientable regular embeddings of its canonical double covering, isomorphic to the tensor productG?K2, can be described in terms of regular embeddings ofG. This allows us to “lift” the classification of regular embeddings of a given graph to a similar classification for its canonical double covering and to establish various properties of the “derived” maps by employing those of the “base” maps. We apply these results to determining all orientable regular embeddings of the tensor productsKn?K2(known as the cocktail-party graphs) and of then-dipolesDn, the graphs consisting of two vertices and n parallel edges joining them. In the first case we show, in particular, that regular embeddings ofKn?K2exist only ifnis a prime powerpl, and there are 2?(n?1) or?(n?1) isomorphism classes of such maps (where?is Eulers function) according to whetherlis even or odd. Forleven an interesting new infinite family of regular maps is discovered. In the second case, orientable regular embeddings ofDnexist for each positive integern, and their number is a power of 2 depending on the decomposition ofninto primes.

Regular Maps from Voltage Assignments and Exponent Groups

In the paper is developed a common generalization of two methods of construction of regular maps on surfaces. The first one produces graph covering projections that extend to coverings of regular embeddings of the graphs involved. The second method employs a double covering projection of graphs which, in general, need not be extendable to a covering of regular maps. In a more general approach, the latter property remains preserved but the multiplicity of the graph covering may be arbitrary. As an application, some new regular embeddings ofn-cubes and complete bipartite graphs will be constructed. Several open problems are included.

Regular embeddings of complete bipartite graphs

Abstract We prove that for any prime number p the complete bipartite graph K p , p has, up to isomorphism, precisely one regular embedding on an orientable surface—the well-known embedding with faces bounded by hamiltonian cycles.

The maximum genus of graphs of diameter two

Abstract Let G be a (finite) graph of diameter two. We prove that if G is loopless then it is upper embeddable, i.e. the maximum genus γM(G) equals ⌊β(G⧸2⌋, where β(G)=|E(G)|− |V(G)| + 1 is the Betti number of G. For graphs with loops we show that ⌈β(G)⧸2⌉ − 2⩽γM(G)⩽⌊β(G)⧸2⌋ if G is vertex 2-connected, and compute the exact value of γM(G) if the vertex-connectivity of G is 1. We note that by a result of Jungerman [2] and Xuong [10] 4-connected graphs are upper embeddable.

Fano colourings of cubic graphs and the Fulkerson conjecture

A Fano colouring is a colouring of the edges of a cubic graph by points of the Fano plane such that the colours of any three mutually adjacent edges form a line of the Fano plane. It has recently been shown by Holroyd and Skoviera [Colouring of cubic graphs by Steiner triple systems, J. Combin. Theory Ser. B 91 (2004) 57-66] that a cubic graph has a Fano colouring if and only if it is bridgeless. In this paper we prove that six, and conjecture that four, lines of the Fano plane are sufficient to colour any bridgeless cubic graph. We establish connections of our conjecture to other conjectures concerning bridgeless cubic graphs, in particular to the well-known conjecture of Fulkerson about the existence of a double covering by 1-factors in every bridgeless cubic graph.

The Maximum Genus of Graph Bundles

The upper embeddability of Cartesian and strong graph bundles with non-trivial base and fibre is proved. A similar result is obtained for both versions of lexicographic bundles. As a corollary the upper embeddability of Cartesian, strong and lexicographic products is obtained. The results cannot be generalized to the Cartesian and strong bundles with discrete fibres, i.e., to covering graphs. In this case sharp upper and lower bounds for Betti defficiency are obtained.

Colouring of cubic graphs by Steiner triple systems

Let J be a Steiner triple system and G a cubic graph. We say that G is J-colourable if its edges can be coloured so that at each vertex the incident colours form a triple of J. We show that if J is a projective system PG(n, 2), n≥2, then G is J-colourable if and only if it is bridgeless, and that every bridgeless cubic graph has an J-colouring for every Steiner triple system of order greater than 3. We establish a condition on a cubic graph with a bridge which ensures that it fails to have an J-colouring if J is an affine system, and we conjecture that this is the only obstruction to colouring any cubic graph with any non-projective system of order greater than 3.