Characterisation of Graphs which Underlie Regular Maps on Closed Surfaces

Abstract

It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser Ge of every edge e is dihedral of order 4 and the stabiliser Gυ of each vertex υ is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with υ. Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that Hυ is the cyclic subgroup of index 2 in Gυ. An analogous result is proved for orientably-regular embeddings.