Classification of nonorientable regular embeddings of cartesian products of graphs

Abstract

In this paper, we consider the classification of nonorientable regular embeddings of cartesian products Gd of a graph G. We show that if G is a bipartite graph and d≥3, then there is no nonorientable regular embedding. This is a generalization of the result that there is no nonorientable regular embeddings of Qn for n≥3 shown by R. Nedela and the author in 2007. When G is non-bipartite and d≥3, we show that if there is a nonorientable regular embedding of Gd, then there is a nonorientable regular embedding of G. Furthermore, we show that any nonorientable regular embedding of Gd with d≥3 is isomorphic to a Petrie dual of some orientable regular embedding of Gd.Using these results, we classify the nonorientable regular embeddings of cartesian products Cnd of a cycle Cn; for even n there is no nonorientable regular embedding except when d=1, and for odd n there is a unique nonorientable regular embedding for each d.