Classification of nonorientable regular embeddings of complete bipartite graphs

Abstract

A 2-cell embedding of a graph G into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags - mutually incident vertex-edge-face triples. In this paper, we classify the regular embeddings of complete bipartite graphs K-n,K-n into nonorientable surfaces. Such a regular embedding of K-n,K-n exists only when n is of the form n = 2p(1)(a1) p(2)(a2) ... p(k)(ak) where the p(i) are primes congruent to +/- 1 mod 8. In this case, up to isomorphism the number of those regular embeddings of K-n,K-n is 2(k). (C) 2011 Elsevier Inc. All rights reserved.