Tetravalent -transitive graphs of order

Let s be a positive integer. A graph is s -transitive if its automorphism group is transitive on s -arcs but not on ( s + 1 ) -arcs, and 1 2 -arc-transitive if its automorphism group is transitive on vertices, edges but not on arcs. Let p be a prime. Feng et al. [Y.-Q. Feng, K.S. Wang, C.X. Zhou, Tetravalent half-trasnitive graphs of order 4 p , European J. Combin. 28 (2007) 726–733] classified tetravalent 1 2 -arc-transitive graphs of order 4 p . In this article a complete classification of tetravalent s -transitive graphs of order 4 p is given. It follows from this classification that with the exception of two graphs of orders 8 or 28 , all such graphs are 1 -transitive. As a result, all tetravalent vertex- and edge-transitive graphs of order 4 p are known.