Classification of regular maps with prime number of faces and the asymptotic behaviour of their reflexible to chiral ratio

Abstract

In this paper we classify the reflexible and chiral regular oriented maps with p faces of valency n, and then we compute the asymptotic behaviour of the reflexible to chiral ratio of the regular oriented maps with p faces. The limit depends on p and for certain primes p we show that the limit can be 1, greater than 1 and less than 1. In contrast, the reflexible to chiral ratio of regular polyhedra (which are regular maps) with Suzuki automorphism groups, computed by Hubard and Leemans (2014), has produced a nill asymptotic ratio.