Nonorientable regular embeddings of graphs of order p(2)

A map is called regular if its automorphism group acts regularly on the set of all flags (incident vertex-edge-face triples). An orientable map is called orientably regular if the group of all orientation-preserving automorphisms is regular on the set of all arcs (incident vertex-edge pairs). If an orientably regular map admits also orientation-reversing automorphisms, then it is regular, and is called reflexible. A regular embedding and orientably regular embedding of a graph g, are, respectively, 2-cell embeddings of g. as a regular map and orientably regular map on some closed surface. In Du et al. (2004) [7], the orientably regular embeddings of graphs of order pq for two primes p and q (p may be equal to q) have been classified, where all the reflexible maps can be easily read from the classification theorem. In [11], Du and Wang (2007) classified the nonorientable regular embeddings of these graphs for p not equal q. In this paper, we shall classify the nonorientable regular embeddings of graphs of order p(2) where p is a prime so that a complete classification of regular embeddings of graphs of order pq for two primes p and q is obtained. All graphs in this paper are connected and simple. (C) 2009 Elsevier B.V. All rights reserved.X1110sciescopu