# ALGEBRAIC COMBINATORICS

On the Saxl graphs of primitive groups with soluble stabilisers
Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 1053-1087.

Let $G$ be a transitive permutation group on a finite set $Ø$ and recall that a base for $G$ is a subset of $Ø$ with trivial pointwise stabiliser. The base size of $G$, denoted $b\left(G\right)$, is the minimal size of a base. If $b\left(G\right)=2$ then we can study the Saxl graph $\Sigma \left(G\right)$ of $G$, which has vertex set $Ø$ and two vertices are adjacent if and only 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 $\Sigma \left(G\right)$ and we determine the groups with a unique regular suborbit, which can be interpreted in terms of the valency of $\Sigma \left(G\right)$.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.238
Classification: 20B15,  20E32,  20E28,  05C40
Keywords: Saxl graph, primitive group, base, soluble stabiliser
Burness, Timothy C. 1; Huang, Hong Yi 1

1 School of Mathematics University of Bristol Bristol BS8 1UG (UK)
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Burness, Timothy C.; Huang, Hong Yi. On the Saxl graphs of primitive groups with soluble stabilisers. Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 1053-1087. doi : 10.5802/alco.238. https://alco.centre-mersenne.org/articles/10.5802/alco.238/

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