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 close d 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(up silon) 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 wh ich acts regularly on the edges incident with upsilon. Such a regular embed ding can be realised on an orientable surface if and only if the group G ha s 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 embeddi ngs.