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We initiate a systematic study of decompositions of the complete graph into circulants. Among other things, we completely determine the existence spectrum for Moebius ladders M 4 , M 5 , M 6 and the prism P r 5 . [ABSTRACT FROM AUTHOR]

Abstract: 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 . 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 arc-transitive 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. [Copyright &y& Elsevier]

Abstract: A circulant with connection set is the graph with vertex set , the cyclic group of order , and edge set . The chromatic number of connected circulants of degree at most four has been previously determined completely by Heuberger [C. Heuberger, On planarity and colorability of circulant graphs, Discrete Math. 268 (2003) 153–169]. In this paper, we determine completely the chromatic number of connected circulants of degree 5. The methods used are essentially extensions of Heuberger’s method but the formulae developed are much more complex. [Copyright &y& Elsevier]

Abstract: 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 [W.T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. 43 (1947) 459–474; W.T. Tutte, On the symmetry of cubic graphs, Canad. J. Math. 11 (1959) 621–624] showed that every cubic finite symmetric cubic graph is s-regular for some . We show that a symmetric cubic graph of girth at most 9 is either 1-regular or 2′-regular (following the notation of Djoković), or belongs to a small family of exceptional graphs. On the other hand, we show that there are infinitely many 3-regular cubic graphs of girth 10, so that the statement for girth at most 9 cannot be improved to cubic graphs of larger girth. Also we give a characterisation of the 1- or 2′-regular cubic graphs of girth , proving that with five exceptions these are closely related with quotients of the triangle group in each case, or of the group in the case . All the 3-transitive cubic graphs and exceptional 1- and 2-regular cubic graphs of girth at most 9 appear in the list of cubic symmetric graphs up to 768 vertices produced by Conder and Dobcsányi [M. Conder, P. Dobcsányi, Trivalent symmetric graphs up to 768 vertices, J. Combin. Math. Combin. Comput. 40 (2002) 41–63]; the largest is the 3-regular graph F570 of order 570 (and girth 9). The proofs of the main results are computer-assisted. [Copyright &y& Elsevier]

Abstract: We show that if where is an odd prime and , then the complete bipartite graph has regular embeddings in orientable surfaces. These maps, which are Cayley maps for cyclic and dihedral groups, have type and genus ; one is reflexible, and the rest are chiral. The method involves groups which factorise as a product of two cyclic groups of order . We deduce that if is odd then has at least orientable regular embeddings, and that this lower bound is attained if and only if no two primes and dividing satisfy . [Copyright &y& Elsevier]

Abstract: By a regular embedding of a graph K into a surface we mean a two-cell embedding of K into a compact connected surface with the automorphism group acting regularly on flags. Regular embeddings of the n-dimensional cubes into orientable surfaces exist for any positive integer n. In contrast to this, we prove the nonexistence of nonorientable regular embeddings of for . [Copyright &y& Elsevier]

A subgroup G of automorphisms of a graph X is said to be 1/2-arc-transitive if it is vertex- and edge- but not arc-transitive. The graph X is said to be 1/2-arc-transitive if AutX is 1/2-arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and 1/2-arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and 1/2-arc-transitive group actions on graphs of valency 4 given via the well known concept of medial graphs (European J. Combin. 19 (1998) 345) is generalised. Any orientably regular hypermap H gives rise to a uniquely determined medial map whose underlying graph Y admits a 1/2-arc-transitive group action of the automorphism group G of the original hypermap H. Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G≤AutX acting 1/2-arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is 1/2-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1/2-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1/2-arc-transitive group action. Indeed, in last section we have applied general results to construct 1/2-arc-transitive if it is vertex- and edge- but not arc-transitive. The graph X is said to be 1/2-arc-transitive if AutX is 1/2-arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and 1/2-arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and 1/2-arc-transitive group actions on graphs of valency 4 given via the well known concept of medial graphs (European J. Combin. 19 (1998) 345) is generalised. Any orientably regular hypermap H gives rise to a uniquely determined medial map whose underlying graph Y admits a 1/2-arc-transitive group action of the automorphism group G of the original hypermap H. Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G≤AutX acting 1/2-arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is 1/2-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1/2-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1/2-arc-transitive group action. Indeed, in last section we have applied general results to construct 1/2-arc-transitive if Aut X is 1/2-arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and 1/2-arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and 1/2-arc-transitive group actions on graphs of valency 4 given via the well known concept of medial graphs (European J. Combin. 19 (1998) 345) is generalised. Any orientably regular hypermap H gives rise to a uniquely determined medial map whose underlying graph Y admits a 1/2-arc-transitive group action of the automorphism group G of the original hypermap H. Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G≤AutX acting 1/2-arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is 1/2-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1/2-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1/2-arc-transitive group action. Indeed, in last section we have applied general results to construct 1/2-arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and 1/2-arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and 1/2-arc-transitive group actions on graphs of valency 4 given via the well known concept of medial graphs (European J. Combin. 19 (1998) 345) is generalised. Any orientably regular hypermap H gives rise to a uniquely determined medial map whose underlying graph Y admits a 1/2-arc-transitive group action of the automorphism group G of the original hypermap H. Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G≤AutX acting 1/2-arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is 1/2-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1/2-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1/2-arc-transitive group action. Indeed, in last section we have applied general results to construct 1/2-arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and 1/2-arc-transitive group actions on graphs of valency 4 given via the well known concept of medial graphs (European J. Combin. 19 (1998) 345) is generalised. Any orientably regular hypermap H gives rise to a uniquely determined medial map whose underlying graph Y admits a 1/2-arc-transitive group action of the automorphism group G of the original hypermap H. Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G≤AutX acting 1/2-arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is 1/2-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1/2-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1/2-arc-transitive group action. Indeed, in last section we have applied general results to construct 1/2-arc-transitive group actions on graphs of valency 4 given via the well known concept of medial graphs (European J. Combin. 19 (1998) 345) is generalised. Any orientably regular hypermap H gives rise to a uniquely determined medial map whose underlying graph Y admits a 1/2-arc-transitive group action of the automorphism group G of the original hypermap H. Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G≤AutX acting 1/2-arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is 1/2-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1/2-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1/2-arc-transitive group action. Indeed, in last section we have applied general results to construct 1/2-arc-transitive group action of the automorphism group G of the original hypermap H. Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G≤Aut X acting 1/2-arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is 1/2-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1/2-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1/2-arc-transitive group action. Indeed, in last section we have applied general results to construct 1/2-arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is 1/2-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1/2-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1/2-arc-transitive group action. Indeed, in last section we have applied general results to construct 1/2-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1/2-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1/2-arc-transitive group action. Indeed, in last section we have applied general results to construct 1/2-arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1/2-arc-transitive group action. Indeed, in last section we have applied general results to construct 1/2-arc-transitive group action. Indeed, in last section we have applied general results to construct 1/2-arc-transitive graphs with cycle stabilisers of arbitrarily large orders. [Copyright &y& Elsevier]

A map is a 2-cell embedding of a connected graph into a closed surface. A map is orientable if the supporting surface is orientable. An orientable map is regular if its group of orientation-preserving automorphisms acts transitively on the darts. Using an equivalent algebraic description of regular maps and their coverings, we employ the theory of group extensions to classify the almost totally branched coverings of the platonic maps with non-abelian covering transformation groups, generalising the results of Hu, Nedela and Wang. [ABSTRACT FROM AUTHOR]

Abstract: A snark is a cubic graph with no proper 3-edge-colouring. In 1996, Nedela and Škoviera proved the following theorem: Let be a snark with an -edge-cut, , whose removal leaves two 3-edge-colourable components and . Then both and can be completed to two snarks and of order not exceeding that of by adding at most vertices, where the number only depends on . The known values of the function are , , (Goldberg, 1981) [6], and (Cameron et al. 1987) [4]. The value is not known and is apparently difficult to calculate. In 1979, Jaeger conjectured that there are no 7-cyclically-connected snarks. If this conjecture holds true, then is the last important value to determine. The paper is aimed attacking the problem of determining by investigating the structure and colour properties of potential complements in 6-decompositions of snarks. We find a set of complements that suffice to perform -decompositions of snarks with at most vertices. We show that if this set is not complete to perform 6-decompositions of all snarks, then and there are strong restrictions on the structure of (possibly) missing complements. Part of the proofs are computer assisted. [Copyright &y& Elsevier]

Abstract: By a regular embedding of a graph into a closed surface we mean a 2-cell embedding with the automorphism group acting regularly on flags. Recently, Kwon and Nedela [Non-existence of nonorientable regular embeddings of -dimensional cubes, Discrete Math., to appear] showed that no regular embeddings of the n-dimensional cubes into nonorientable surfaces exist for any positive integer . In 1997, Nedela and Škoviera [Regular maps from voltage assignments and exponent groups, European J. Combin. 18 (1997) 807–823] presented a construction giving for each solution of the congruence a regular embedding of the hypercube into an orientable surface. It was conjectured that all regular embeddings of into orientable surfaces can be constructed in this way. This paper gives a classification of regular embeddings of hypercubes into orientable surfaces for n odd, proving affirmatively the conjecture of Nedela and Škoviera for every odd n. [Copyright &y& Elsevier]

In this paper, we classify all regular embeddings of the complete multipartite graphs Kp,…,p for a prime p into orientable surfaces. Also, the same work is done for the regular embeddings of the lexicographical product of any connected arc-transitive graph of prime order q with the complement of the complete graph of prime order p, where q and p are not necessarily distinct. Lots of regular maps found in this paper are Cayley maps. [Copyright &y& Elsevier]