Cubic semisymmetric graphs of order

A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive. By Folkman [J. Folkman, Regular line-symmetric graphs, J. Combin Theory 3 (1967) 215–232], there is no semisymmetric graph of order 2 p or 2 p 2 for a prime p and by Malnič, et al. [A. Malnič, D. Marušič, C.Q. Wang, Cubic edge-transitive graphs of order 2 p 3 , Discrete Math. 274 (2004) 187–198], there exists a unique cubic semisymmetric graph of order 2 p 3 , the so-called Gray graph of order 54. In this paper it is shown that a connected cubic semisymmetric graph of order 6 p 3 exists if and only if p − 1 is divisible by 3. There are exactly two such graphs for a given order, which are constructed explicitly.