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(Gamma, S) is normal if Gamma is a normal subgroup of the full automorphism group of Cay(Gamma, 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. Marusic, 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.