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 regular embedding of $K_{n,n}$ exists only when $n=2p_1^{a_1}{p}_2^{a_2}...p_k^{a_k}$ (a prime decomposition of $n$) and all $p_i\equiv \pm 1(\mathrm{mod}8)$. In this case, the number of those regular embeddings of $K_{n,n}$ up to isomorphism is $2^k$.