Regular Cayley maps for dihedral groups

A combinatorial map is a pair M=(X,r), where X is a finite simple connected graph and r is a permutation of its arcs (directed edges) such that the orbit of the arc (u,v) under r is the set of arcs emanating from u. Aut(M) consists of the automorphisms of X for which the induced permutations on the arcs commute with r. The order |Aut(M)| is less than equal to 2|E(G)|, and when equality holds M is called regular. M is a Cayley map for a finite group G if X=Cay(G,S), and r is defined by (h,g) -> (h,p(h^{-1}g)), where h^{-1} \in S and p is a fixed cyclic permutation of S.  Regular Cayley maps for cyclic groups were classified by Conder and Tucker (Trans. Amer. Math. Soc., 2014). In this talk, I will show the classification of regular Cayley maps for dihedral groups.