Half-Regular Cayley Maps

We use the term half-regular map to describe an orientable map with an orientation preserving automorphism group that is transitive on vertices and half-transitive on darts. We present a full classification of half-regular Cayley maps using the concept of skew-morphisms. We argue that half-regular Cayley maps come in two types: those that arise from two skew-morphism orbits of equal size that are both closed under inverses and those that arise from two equal-sized orbits that do not contain involutions or inverses but one contains the inverses of the other. In addition, half-regular Cayley maps of the first type are shown to be half-edge-transitive, while half-regular Cayley maps of the second type are shown to be necessarily edge-transitive. A connection between half-regular Cayley maps and regular hypermaps is also investigated.