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≤AutΓ if X acts regularly on its arc set, while Γ is called arc-regular if X=AutΓ. 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 AutΓ.