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~tp1p2…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 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.

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| (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. This work was supported by the National Natural Science Foundation of China (Grant No. 10571013), the Key Project of the Chinese Ministry of Education (Grant No. 106029), and the Specialized Research Fund for the Doctoral Program of High Education in China (Grant No. 20060004026)