Cubic Vertex-Transitive Non-Cayley Graphs of Order 8p

A graph is vertex-transitive if its automorphism group acts transitively on its vertices. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the cubic vertextransitive non-Cayley graphs of order 8p are classified for each prime p. It follows from this classification that there are two sporadic and two infinite families of such graphs, of which the sporadic ones have order 56, one infinite family exists for every prime p > 3 and the other family exists if and only if p ≡ 1 (mod 4). For each family there is a unique graph for a given order.