Petrie lengths of regular maps

Abstract

A regular map $${\mathcal{M}}$$ is an embedding of a graph into a compact surface S such that its automorphism group $${{\rm Aut}^+(\mathcal{M}) \subseteq {\rm Aut}^+(S)}$$ acts transitively on the directed edges. A Petrie polygon of $${\mathcal{M}}$$ is a zig-zag circuit in which every two consecutive edges but no three belong to the same face. It is known that the number of sides of every Petrie polygon of $${\mathcal{M}}$$ is the same and this number is called the Petrie length of $${\mathcal{M}}$$ . In this paper our main concern is the Petrie lengths of toroidal regular maps. There are mainly two types of such regular maps up to duality, and for each type we prove a theorem that enables us to determine the Petrie length of the corresponding map. We also find the Petrie lengths of Accola–Maclachlan, Wiman and Fermat maps, which are well-known families of regular maps. Finally, we consider regular maps with Petrie length 2, and prove that only Accola–Maclachlan and Wiman surfaces underlie such regular maps. In this way, we obtain new characterizations of Accola–Maclachlan and Wiman surfaces.