Classification of nonorientable regular embeddings of complete bipartite graphs

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 Kn,n into nonorientable surfaces. Such a regular embedding of Kn,n exists only when n is of the form where the pi are primes congruent to ±1 mod 8. In this case, up to isomorphism the number of those regular embeddings of Kn,n is k2.