On enumeration of a class of maps on Klein bottle

TL;DR

This paper classifies semi-equivelar maps on the Klein bottle, a generalization of equivelar maps, by enumerating eleven specific types based on their face configurations.

Contribution

It provides a comprehensive classification and enumeration of eleven types of semi-equivelar maps on the Klein bottle, expanding understanding of their structure.

Findings

Enumerated eleven types of semi-equivelar maps on the Klein bottle.

Classified these maps based on their face configurations.

Provided a framework for further study of Klein bottle maps.

Abstract

We present enumerations of a class of maps on Klein bottle which give rise to semi-equivelar maps. Semi-equivelar maps are generalizations of equivelar maps. There are eleven types of semi-equivelar maps on the Klein bottle. These are of the types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$, $\{3^{3},$ $4^{2}\}$, $\{3^{2},$ $4,$ $3,$ $4\}$, $\{3,$ $6,$ $3,$ $6\}$, $\{3^{4}, 6\}$, $\{4,$ $8^{2}\}$, $\{3, 12^{2}\}$, $\{4,$ $6,$ $12\}$, $\{3,$ $4,$ $6,$ $4\}$. In this article, we attempt to classify these maps.