Cubic semisymmetric graphs of order 6p3

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 2p or 2p^2 for a prime p and by Malnic, et al. [A. Malnic, D. Marusic, C.Q. Wang, Cubic edge-transitive graphs of order 2p^3, Discrete Math. 274 (2004) 187-198], there exists a unique cubic semisymmetric graph of order 2p^3, the so-called Gray graph of order 54. In this paper it is shown that a connected cubic semisymmetric graph of order 6p^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.

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