Regular -balanced Cayley maps for abelian groups

The inherent high symmetry of Cayley maps makes them an excellent source of orientably regular maps, and the regularity of a Cayley map has been shown to be equivalent to the existence of a skew-morphism of its underlying group that has a generating orbit closed under inverses. We set to investigate the properties of the so-called t -balanced skew-morphisms of abelian groups with the aim of providing the basis for a complete classification of t -balanced regular Cayley maps of abelian groups. In the case of cyclic groups, we show that the only t -balanced regular Cayley maps for the groups Z 2 r , Z 2 p r and Z 4 p r , p an odd prime, r ≥ 1 , are the well understood balanced and antibalanced Cayley maps.