One-regular graphs of square-free order of prime valency

Abstract

A graph is one-regular if its automorphism group acts regularly on the set of arcs of the graph. Marušič and Pisanski [D. Marušič and T. Pisanski, Symmetries of hexagonal graphs on the torus, Croat. Chemica Acta 73 (2000) 969–981] classified one-regular Cayley graphs on a dihedral group of valency 3, and Kwak et al. [J.H. Kwak, Y.S. Kwon, J.M. Oh, Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency, J. Combin. Theory B 98 (2008) 585–598] classified those of valency 5. In this paper one-regular Cayley graphs on a dihedral group of any prime valency are classified and enumerated. It is shown that for an odd prime q , there exists a q -valent one-regular Cayley graph on the dihedral group of order 2 m if and only if m = q t p 1 e 1 p 2 e 2 ⋯ p s e s ≥ 13 , where t ≤ 1 , s ≥ 1 , e i ≥ 1 and p i ’s are distinct primes such that q ∣ ( p i − 1 ) . There are exactly ( q − 1 ) s − 1 non-isomorphic such graphs for a given order. Consequently, one-regular cyclic Haar graphs of prime valency are classified and enumerated. Furthermore, it is shown that every q -valent one-regular graph of square-free order is a Cayley graph on a dihedral group, and as a result, q -valent one-regular graphs of square-free order are classified and enumerated.