Comenius University and University of Primorska
Cayley maps are orientable embeddings of right Cayley graphs having the property that left-multiplications by the elements of the underlying Cayley group induce orientation-preserving automorphisms of the map giving rise to a group of automorphisms acting regularly on the vertices of the map. A Cayley map is orientably-regular if the entire group of orientation-preserving automorphisms acts regularly on the darts of the map. This property has been shown to be equivalent to the existence of a skew-morphism of the Cayley group of the map fixing the identity of the group and preserving the set of neighbors of the identity in the underlying graph. In our talk, we report on recent developments in general (as well as some subclasses of) skew-morphism classifications for specific classes of groups, as well as on generalizations of regular Cayley maps, and the corresponding generalizations of skew-morphisms.