Regular Cayley maps of skew-type 3 for abelian groups

Let M=CM(G,X,p)M=CM(G,X,p) be a regular Cayley map, and let φφ be the corresponding skew-morphism. If the power function of φφ assumes exactly ii values in Z|X|, then MM is called of skew-type ii. Note that regular balanced Cayley maps are of skew-type 1, and regular tt-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.