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

• Authors: Yan-Quan Feng;Yan-Tao Li
• Venue: European Journal of Combinatorics
• Year: 2011

A graph is one-regular if its automorphism group acts regularly on the set of arcs of the graph. Marusic and Pisanski [D. Marusic 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 2m if and only if m=q^tp"1^e^"^1p"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.