There exist no arc-regular prime-valent graphs of order four times an odd square-free integer

A graph Γ is called X -arc-regular with X ≤ Aut Γ if X acts regularly on its arc set, while Γ is called arc-regular if X = Aut Γ . J.X. Zhou and Y.Q. Feng [Cubic one-regular graphs of order twice a square-free integer, Sci. China Ser. A 51 (2008) 1093–1100] proved that there is no cubic arc-regular graph of order four times an odd square-free integer. In this paper, we shall generalize this result by showing that there is no arc-regular p -valent graph of order four times an odd square-free integer for each odd prime p . Moreover, we prove that there are exactly two specific infinite families of X -arc-regular graphs Γ with X a proper subgroup of Aut Γ .