DOIAbstract
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(2p)r and Z(4p)r, p an odd prime, r >= 1, are the well understood balanced and antibalanced Cayley maps. (C) 2011 Elsevier B.V. All rights reserv...
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