Almost covers of 2-arc transitive graphs

Abstract

Let Γ be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition ℬ with block size v. Let Γ ℬ 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 Γ ℬ 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 Γ ℬ ≅K v+1 is covered by results of Gardiner and Praeger. We consider here the general case where Γ ℬ ≇K v+1, and prove that, for some even integer n ≥ 4, Γ ℬ is a near n-gonal graph with respect to a certain G-orbit on n-cycles of Γ ℬ. 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 Γ ℬ 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.)