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

摘要

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=3tp1p2…p_s≥13,where t≤1,s≥1 and p_i’s are distinct primes such that 3|（P_i—1）. For such an integer n,there are 2s-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.