Cubic one-regular graphs of order twice a square-free integer

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. Let n be a square-free integer. In this paper, we show that a cubic one-regular graph of order 2n exists if and only if n = 3(t)p(1)p(2)...p(s) >= 13, where t <= 1, s >= 1 and p(i)'s are distinct primes such that 3 vertical bar(p(i) - 1). For such an integer n, there are 2(s-1) non-isomorphic cubic one-regular graphs of order 2n, which are all Cayley graphs on the dihedral group of order 2n. As a result, no cubic one-regular graphs of order 4 times an odd square-free integer exist.