Biquasiprimitive Oriented Graphs of Valency Four Nemanja Poznanovi´ c and Cheryl E. Praeger Abstract In this short note we describe a recently initiated research programme aiming to use a normal quotient reduction to analyse finite connected, oriented graphs of valency 4, admitting a vertex- and edge-transitive group of automorphisms which preserves the edge orientation. In the first article on this topic (Al-bar et al. Electr J Combin 23, 2016), a subfamily of these graphs was identified as ‘basic’ in the sense that all graphs in this family are normal covers of at least one ‘basic’ member. These basic members can be further divided into three types: quasiprimitive, biquasiprimitive and cycle type. The first and third of these types have been analysed in some detail. Recently, we have begun an analysis of the basic graphs of biquasiprimitive type. We describe our approach and mention some early results. This work is on-going. It began at the Tutte Memorial MATRIX Workshop. A graph Γ is said to be G-oriented for some subgroup G ≤ Aut(Γ ), if G acts transitively on the vertices and edges of Γ , and Γ admits a G-invariant orientation of its edges. Any graph Γ admitting a group of automorphisms which acts transitively on its vertices and edges but which does not act transitively on its arcs can be viewed as a G-oriented graph. These graphs are usually said to be G-half-arc-transitive, and form a well-studied class of vertex-transitive graphs. A G-oriented graph Γ necessarily has even valency, with exactly half of the edges incident to a vertex α being oriented away from α to one of its neighbours. Since such a graph is vertex-transitive, it follows that all of its connected compo- nents are isomorphic, and each connected component is itself a G-oriented graph. We therefore restrict our attention to the study of connected G-oriented graphs. For each even integer m, we let OG (m) denote the family of all graph-group pairs (Γ,G) such that Γ is a finite connected G-oriented graph of valency m. N. Poznanovi´ c() University of Melbourne, Melbourne, VIC, Australia C. E. Praeger University of Western Australia, Perth, Australia e-mail: cheryl.praeger@uwa.edu.au © Springer Nature Switzerland AG 2019 D. R. Wood et al. (eds.), 2017 MATRIX Annals, MATRIX Book Series 2, https://doi.org/10.1007/978-3-030-04161-8_23 337