An infinite family of simple graphs underlying chiral, orientable reflexible and non-orientable rotary maps

Abstract

In this paper, we provide the first known infinite family of simple graphs, each of which is the skeleton of a chiral map, a skeleton of a reflexible map on an orientable surface, as well as a skeleton of a reflexible map on a non-orientable surface. This family consists of all lexicographic products $C_n\left[m{K}_1\right]$, where $m\ge 3$, $n=sm$, with s an integer not divisible by 4. This answers a question posed by Wilson in 2002.