Some Semi - Equivelar Maps

TL;DR

This paper classifies semi-equivelar maps on surfaces with various Euler characteristics, demonstrating their existence and properties, including non-vertex transitivity and constructions on different surfaces.

Contribution

It provides the first classification of semi-equivelar maps on surfaces with Euler characteristic -1 and constructs new examples on double torus and other surfaces.

Findings

Classified semi-equivelar maps on surface with Euler characteristic -1.

Proved none of these maps are vertex transitive.

Constructed semi-equivelar maps on double torus and other surfaces.

Abstract

Semi-Equivelar maps are generalizations of Archimedean Solids (as are equivelar maps of the Platonic solids) to the surfaces other than $2-$Sphere. We classify some semi equivelar maps on surface of Euler characteristic -1 and show that none of these are vertex transitive. We establish existence of 12-covered triangulations for this surface. We further construct double cover of these maps to show existence of semi-equivelar maps on the surface of double torus. We also construct several semi-equivelar maps on the surfaces of Euler characteristics -8 and -10 and on non-orientable surface of Euler characteristics -2.