Finite Edge-Transitive Oriented Graphs of Valency Four: a Global Approach

We develop a new framework for analysing finite connected, oriented graphs of valency four, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of ‘basic’ graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restrictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.