Regular t-balanced Cayley maps

The concept of a t-balanced Cayley map is a natural generalization of the previously studied notions of balanced and anti-balanced Cayley maps (the terms coined by [J. Širáň, M. Škoviera, Groups with sign structure and their antiautomorphisms, Discrete Math. 108 (1992) 189–202. [12]]). We develop a general theory of t-balanced Cayley maps based on the use of skew-morphisms of groups [R. Jajcay, J. Širáň, Skew-morphisms of regular Cayley maps, Discrete Math. 244 (1–3) (2002) 167–179], and apply our results to the specific case of regular Cayley maps of abelian groups.