A classification of prime-valent regular Cayley maps on abelian, dihedral and dicyclic groups

Dongseok Kim; Young Soo Kwon; Jaeun Lee

doi:10.4134/BKMS.2010.47.1.17

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2010; 47(1): 17-27

Printed January 1, 2010

https://doi.org/10.4134/BKMS.2010.47.1.17

A classification of prime-valent regular Cayley maps on abelian, dihedral and dicyclic groups

Dongseok Kim, Young Soo Kwon, and Jaeun Lee

Kyunggi University, Yeungnam University, and Yeungnam University

Abstract

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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.

Keywords: Cayley map, regular embedding

MSC numbers: 05C10, 05C30