A classification of prime-valent regular Cayley maps on abelian, dihedral and dicyclic groups
Bull. Korean Math. Soc. 2010 Vol. 47, No. 1, 17-27
Printed January 1, 2010
Dongseok Kim, Young Soo Kwon, and Jaeun Lee
Kyunggi University, Yeungnam University, and Yeungnam University
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
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