On the Saxl graphs of primitive groups with soluble stabilisers

TL;DR

This paper proves a conjecture about the connectivity and diameter of Saxl graphs for certain primitive groups with soluble stabilisers, using probabilistic and computational methods, and explores related graph properties.

Contribution

It establishes the conjecture for all almost simple primitive groups with soluble stabilisers and determines bounds on graph invariants and unique regular suborbits.

Findings

Saxl graphs are connected with diameter at most 2 for the studied groups.

Established optimal bounds on clique and independence numbers.

Identified groups with a unique regular suborbit.

Abstract

Let $G$ be a transitive permutation group on a finite set $\Omega$ and recall that a base for $G$ is a subset of $\Omega$ with trivial pointwise stabiliser. The base size of $G$, denoted $b(G)$, is the minimal size of a base. If $b(G)=2$ then we can study the Saxl graph $\Sigma(G)$ of $G$, which has vertex set $\Omega$ and two vertices are adjacent if they form a base. This is a vertex-transitive graph, which is conjectured to be connected with diameter at most $2$ when $G$ is primitive. In this paper, we combine probabilistic and computational methods to prove a strong form of this conjecture for all almost simple primitive groups with soluble point stabilisers. In this setting, we also establish best possible lower bounds on the clique and independence numbers of $\Sigma(G)$ and we determine the groups with a unique regular suborbit, which can be interpreted in terms of the valency of $\Sigma(G)$.