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

摘要

A graph G is one-regular if its automorphism group Aut(G) acts transitively and semi-regularly on the arc set. A Cayley graph Cay(F, S) is normal if F is a normal subgroup of the fullautomorphism group of Cay(F, 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. Marui, D.,Pisanski, T. (Croat. Chemica Acta, 73, 969-981 (2000)) classified cubic one-regular Cayley graphs ona dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4-valentone-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.