Regular Cayley maps of skew-type 3 for abelian groups

Let M = C M ( 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.