Locally arc-transitive graphs of valence {3,4} with trivial edge kernel

Abstract

In this paper, we consider connected locally G-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel G [1] uv of the action of an edge-stabiliser on the neighbourhood Γ (u) ∪ Γ (v) is trivial. We find 19 finitely presented groups with the property that any such group G is a quotient of one of these groups. As an application, we enumerate all connected locally arc-transitive graphs of valence {3, 4} on at most 350 vertices whose automorphism group contains a locally arc-transitive subgroup G with G

Danzer’s configuration revisited

“…Note that similarly to the proof of the previous lemma, this latter can also be inferred from the labelling of Figure 4, which depicts the Cayley-Salmon configuration. [31] that there is exactly one biregular graph with 20 vertices of valence 3 and 15 vertices of valence 4 that is edge-transitive of girth 6. In his Table 2 it has ID = {35, 2}.…”

“…Remark 4.9. Potočnik proved [31] that there is exactly one biregular graph with 20 vertices of valence 3 and 15 vertices of valence 4 that is edge-transitive of girth 6. In his Table 2 it has ID = {35, 2}.…”

On the stabilisers of locally 2-transitive graphs

“…van Bon [3] proved that if G [1] vw = 1 then Γ is not locally (G, s)-arc-transitive. Potocnik [12] determined the amalgams for valency {3, 4} in the case where G [1] vw = 1. This paper is one of a series of papers which aim to classify the amalgams (G v , G w , G vw ) with trivial edge kernel.…”