Abstract. The paper focuses on the classification of vertex-transitive polyhedral maps of genus from 2 to 4. These maps naturally generalise the spherical maps associated with the classical Archimedean solids. Our analysis is based on the fact that each Archimedean map on an orientable surface projects onto a one-or a two-vertex quotient map. For a given genus g ≥ 2 the number of quotients to consider is bounded by a function of g. All Archimedean maps of genus g can be reconstructed from these quotients as regular covers with covering transformation group isomorphic to a group G from a set of g-admissible groups. Since the lists of groups acting on surfaces of genus 2, 3 and 4 are known, the problem can be solved by a computer-aided case-to-case analysis.

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.

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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. © 1996 John Wiley & Sons, Inc.

A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ , and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ . Tutte (1947, 1959) showed that every finite symmetric cubic graph is s-regular for some s 5. Djoković and Miller (1980) proved that there are seven types of arc-transitive group action on finite cubic graphs, characterised by the stabilisers of a vertex and an edge. A given finite symmetric cubic graph, however, may admit more than one type of arctransitive group action. In this paper we determine exactly which combinations of types are possible. Some combinations are easily eliminated by existing theory, and others can be eliminated by elementary extensions of that theory. The remaining combinations give 17 classes of finite symmetric cubic graph, and for each of these, we prove the class is infinite, and determine at least one representative. For at least 14 of these 17 classes the representative we give has the minimum possible number of vertices (and we show that in two of these 14 cases every graph in the class is a cover of the smallest representative), while for the other three classes, we give the smallest examples known to us. In an appendix, we give a table showing the class of every symmetric cubic graph on up to 768 vertices.

Let N g (f ) denote the number of rooted maps of genus g having f edges. An exact formula for N g (f ) is known for g = 0 (Tutte, 1963), g = 1 (Arques, 1987), g = 2, 3 (Bender and Canfield, 1991). In the present paper we derive an enumeration formula for the number Θ γ (e) of unrooted maps on an orientable surface S γ of a given genus γ and with a given number of edges e. It has a form of a linear combination i,j c i,j N g j (f i ) of numbers of rooted maps N g j (f i ) for some g j γ and f i e. The coefficients c i,j are functions of γ and e. We consider the quotient S γ /Z of S γ by a cyclic group of automorphisms Z as a two-dimensional orbifold O. The task of determining c i,j requires solving the following two subproblems:(a) to compute the number Epi o (Γ, Z ) of order-preserving epimorphisms from the fundamental group Γ of the orbifold O = S γ /Z onto Z ; (b) to calculate the number of rooted maps on the orbifold O which lifts along the branched covering S γ → S γ /Z to maps on S γ with the given number e of edges.The number Epi o (Γ, Z ) is expressed in terms of classical number-theoretical functions. The other problem is reduced to the standard enumeration problem of determining the numbers N g (f ) for some g γ and f e. It follows that Θ γ (e) can be calculated whenever the numbers N g (f ) are known for g γ and f e. In the end of the paper the above approach is applied to derive the functions Θ γ (e) explicitly for γ 3. We note that the function Θ γ (e) was known only for γ = 0 (Liskovets, 1981). Tables containing the numbers of isomorphism classes of maps with up to 30 edges for genus γ = 1, 2, 3 are presented.

We discuss a special case of the Hamilton-Waterloo problem in which a 2-factorization of Kn is sought consisting of 2-factors of two kinds: Hamiltonian cycles, and triangle-factors. We determine completely the spectrum of solutions for several inÿnite classes of orders n.

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