Nonorientable regular embeddings of graphs of order $p^2$

Abstract

A map is called regular if its automorphism group acts regularly on the set of all flags (incident vertex–edge–face triples). An orientable map is called orientably regular if the group of all orientation-preserving automorphisms is regular on the set of all arcs (incident vertex–edge pairs). If an orientably regular map admits also orientation-reversing automorphisms, then it is regular, and is called reflexible. A regular embedding and orientably regular embedding of a graph $\mathcal{G}$ are, respectively, 2-cell embeddings of $\mathcal{G}$ as a regular map and orientably regular map on some closed surface. In Du et al. (2004) [7], the orientably regular embeddings of graphs of order $pq$ for two primes $p$ and $q$ ($p$ may be equal to $q$) have been classified, where all the reflexible maps can be easily read from the classification theorem. In [11], Du and Wang (2007) classified the nonorientable regular embeddings of these graphs for $p\ne q$. In this paper, we shall classify the nonorientable regular embeddings of graphs of order $p^2$ where $p$ is a prime so that a complete classification of regular embeddings of graphs of order $pq$ for two primes $p$ and $q$ is obtained. All graphs in this paper are connected and simple.