A classification of prime-valent regular Cayley maps on some groups

TL;DR

This paper classifies prime-valent regular Cayley maps on abelian, dihedral, and dicyclic groups, revealing their balanced properties and proving the non-existence on dicyclic groups.

Contribution

It provides a complete classification of prime-valent regular Cayley maps on certain groups and establishes new properties and non-existence results.

Findings

All prime-valent regular Cayley maps on dihedral groups are balanced.

All prime-valent regular Cayley maps on abelian groups are either balanced or anti-balanced.

No prime-valent regular Cayley map exists on dicyclic groups.

Abstract

A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are balanced and all prime-valent regular Cayley maps on abelian groups are either balanced or anti-balanced. Furthermore, we prove that there is no prime-valent regular Cayley map on any dicyclic group.