One–regular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer

A graph G is one–regular if its automorphism group Aut(G) acts transitively and semiregularly on the arc set. A Cayley graph Cay(Γ, S) is normal if Γ is a normal subgroup of the full automorphism group of Cay(Γ, S). Xu, M. Y., Xu, J. (Southeast Asian Bulletin of Math., 25, 355–363 (2001)) classified one–regular Cayley graphs of valency at most 4 on finite abelian groups. Marušič, D., Pisanski, T. (Croat. Chemica Acta, 73, 969–981 (2000)) classified cubic one–regular Cayley graphs on a dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4–valent one–regular normal Cayley graphs G on a dihedral group whose vertex stabilizers in Aut(G) are cyclic. A classification of the same kind of graphs of valency 6 is also discussed.