Regular Cayley maps of skew-type 3 for abelian groups

Abstract

Let M=CM(G,X,p) be a regular Cayley map, and let φ be the corresponding skew-morphism. If the power function of φ assumes exactly i values in Z|X|, then M is called of skew-type i. Note that regular balanced Cayley maps are of skew-type 1, and regular t-balanced (including anti-balanced) Cayley maps are of skew-type 2. Those two classes of maps have been studied by many researchers. In this paper, some results on regular Cayley maps of skew-type 3 for abelian groups are given. As applications, many new regular Cayley maps for abelian groups are constructed.