Half-arc-transitive graphs and chiral hypermaps

A subgroup G of automorphisms of a graph X is said to be 12-arc-transitive if it is vertex- and edge- but not arc-transitive. The graph X is said to be 12-arc-transitive if Aut X is 12-arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and 12-arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and 12-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 12-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 12-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 12-arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 12-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 12-arc-transitive group action. Indeed, in last section we have applied general results to construct 12-arc-transitive graphs with cycle stabilisers of arbitrarily large orders.