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 ( D n , S ) on dihedral groups D n = 〈 a , b | a n = b 2 = ( a b ) 2 = 1 〉 with S = { a 1 + ℓ + ⋯ + ℓ t b | 0 ⩽ t ⩽ k − 1 } and n = ∑ j = 0 k − 1 ℓ j is one-regular. All of these graphs have cyclic vertex stabilizers and girth 6. As a continuation of Marušič and Pisanskiʹs 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.