Abstract
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.
Highlights
We initiate a new approach to studying finite connected oriented graphs of valency four, which admit a vertex-transitive and edge-transitive group of automorphisms preserving an edge orientation
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
Conditions on G-action on vertices quasiprimitive biquasiprimitive at least one quotient action D2r or Zr graphs which either admit quasiprimitive or biquasiprimitive actions on vertices, or are degenerate cycles. This new approach has been used before in other problems dealing with symmetries of graphs, and we believe that it will bear fruit when studying oriented graphs, and in particular, half-arctransitive graphs of valency four
Summary
We initiate a new approach to studying finite connected oriented graphs of valency four, which admit a vertex-transitive and edge-transitive group of automorphisms preserving an edge orientation. Theorem 1.1 and the remarks above suggest a new framework for studying oriented graph-group pairs in OG(4), consisting of the following general steps (see Table 1). A structure theorem is available to study quasiprimitive permutations groups in [38] analogous to the O’Nan–Scott theorem for studying finite primitive permutation groups We apply this theory to determine the possible types of quasiprimitive groups G that can arise for (Γ, G) ∈ OG(4), that is, (Γ, G) is a quasiprimitive basic pair (Step 1 of the framework). (i) As commentary on this result we note that Li, Lu and Marusic [20, Theorem 1.4] showed in 2004 that there are no vertex-primitive graph-group pairs in OG(4) This confirms the suitability of normal quotient reduction to quasiprimitive actions as being the appropriate group theoretic reduction in the non-bipartite case. These types are sometimes called As, Tw, Pa for cases (a)–(c) respectively
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