Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency

A graph is one-regular if its automorphism group acts regularly on the arc set. In this paper, we construct a new infinite family of one-regular Cayley graphs of any prescribed valency. In fact, for any two positive integers , k 2 except for ( , k) ∈ {(2,3), (2,4)}, the Cayley graph Cay(Dn,S) on dihedral groups Dn = 〈a, b | an = b2 = (ab)2 = 1〉 with S = {a1+ +···+ t b | 0 t k − 1} and n = ∑k−1 j=0 j is oneregular. All of these graphs have cyclic vertex stabilizers and girth 6. As a continuation of Marušič and Pisanski’s classification of cubic one-regular Cayley graphs on dihedral groups in [D. Marušič, T. Pisanski, Symmetries of hexagonal graphs on the torus, Croat. Chemica Acta 73 (2000) 969–981], the 5-valent oneregular Cayley graphs on dihedral groups are classified. Also, with only finitely many possible exceptions, all of one-regular Cayley graphs on dihedral groups of any prescribed prime valency are constructed. © 2007 Elsevier Inc. All rights reserved.