Regular Cayley maps for dihedral groups

Abstract An orientably-regular map M is a 2-cell embedding of a finite connected graph in a closed orientable surface such that the group Aut ∘ M of orientation-preserving automorphisms of M acts transitively on the set of arcs. Such a map M is called a Cayley map for the finite group G if Aut ∘ M contains a subgroup, which is isomorphic to G and acts regularly on the set of vertices. Conder and Tucker (2014) classified the regular Cayley maps for finite cyclic groups, and obtain two two-parameter families M ( n , r ) , one for odd n and one for even n, where n is the order of the regular cyclic group and r is a positive integer satisfying certain arithmetical conditions. In this paper, we classify the regular Cayley maps for dihedral groups in the same fashion. Five two-parameter families M i ( n , r ) , 1 ≤ i ≤ 5 , are derived, where 2n is the order of the regular dihedral group and r is an integer satisfying certain arithmetical conditions. For each map M i ( n , r ) , we determine its valence and covalence, and also describe the structure of the group Aut ∘ M i ( n , r ) . Unlike the approach of Conder and Tucker, which is entirely algebraic, we follow the traditional combinatorial representation of Cayley maps, and use a combination of permutation group theoretical techniques, the method of quotient Cayley maps, and computations with skew morphisms.