Almost Covers Of 2-Arc Transitive Graphs

Let Γ be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition B with block size v. Let ΓB be the quotient graph of Γ relative to B and Γ [B,C] the bipartite subgraph of Γ induced by adjacent blocks B,C of B. In this paper we study such graphs for which ΓB is connected, (G,2)-arc transitive and is almost covered by Γ in the sense that Γ [B,C] is a matching of v−1≥2 edges. Such graphs arose as a natural extremal case in a previous study by the author with Li and Praeger. The case ΓB∼=Kv+1 is covered by results of Gardiner and Praeger. We consider here the general case where ΓB ∼=Kv+1, and prove that, for some even integer n≥4, ΓB is a near n-gonal graph with respect to a certain G-orbit on n-cycles of ΓB. Moreover, we prove that every (G,2)-arc transitive near n-gonal graph with respect to a G-orbit on n-cycles arises as a quotient ΓB of a graph with these properties. (A near n-gonal graph is a connected graph Σ of girth at least 4 together with a set E of n-cycles of Σ such that each 2-arc of Σ is contained in a unique member of E .)