Characterisation of graphs which underlie regular maps on closed surfaces

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 G(upsilon) of every edge e is dihedral of order 4 and the stabiliser G(upsilon) of each vertex It is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with upsilon. Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup Ii of index 2 such that H-v is the cyclic subgroup of index 2 in G(upsilon). An analogous result is proved for orientably-regular embeddings.