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 ℓ,k⩾2 except for (ℓ,k)∈{(2,3),(2,4)}, the Cayley graph Cay(Dn,S) on dihedral groups Dn=〈a,b|an=b2=2(ab)=1〉 with S={a1+ℓ+⋯+ℓtb|0⩽t⩽k−1} and is one-regular. 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 one-regular 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.