Semi-equivelar and vertex-transitive maps on the torus

TL;DR

This paper classifies semi-equivelar maps on the torus and Klein bottle, establishing conditions under which they are vertex-transitive, and provides examples illustrating the diversity of these maps.

Contribution

It identifies eleven types of semi-equivelar maps on the torus, determines vertex-transitivity conditions for some types, and presents examples showing non-vertex-transitive maps.

Findings

Eleven semi-equivelar map types on the torus identified.

Vertex-transitivity proven for specific semi-equivelar map types.

Examples of non-vertex-transitive semi-equivelar maps on the Klein bottle provided.

Abstract

A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${\rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. We show that there are eleven types of semi-equivelar maps on the torus. Three of these are equivelar maps. It is known that two of the three types of equivelar maps on the torus are always vertex-transitive. We show that this is true for the remaining one type of equivelar map and one other type of semi-equivelar maps, namely, if $X$ is a semi-equivelar map of type $[6^3]$ or $[3^3, 4^2]$ then $X$ is vertex-transitive. We also show, by presenting examples, that this result is not true for the remaining seven types of semi-equivelar maps. There are ten types of semi-equivelar maps on the Klein bottle. We present examples in each of the ten types which are not vertex-transitive.