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 $K_{n,n}$ into nonorientable surfaces. Such a regular embedding of $K_{n,n}$ exists only when n is of the form $n=2p_1^{a_1}{p}_2^{a_2}\cdots p_k^{a_k}$ where the $p_i$ are primes congruent to ±1 mod 8. In this case, up to isomorphism the number of those regular embeddings of $K_{n,n}$ is $2^k$.