On the generalised Saxl graphs of permutation groups

A base for a finite permutation group $G\le \mathrm{S}\mathrm{y}\mathrm{m}(\mathrm{\Omega })$ is a subset of $\mathrm{\Omega }$ with trivial pointwise stabiliser in $G$, and the base size of $G$ is the smallest size of a base for $G$. Motivated by the interest in groups of base size two, Burness and Giudici introduced the notion of the Saxl graph. This graph has vertex set $\mathrm{\Omega }$, with edges between elements if they form a base for $G$. We define a generalisation of this graph that encodes useful information about $G$ whenever $b(G)\ge 2$: here, the edges are the pairs of elements of $\mathrm{\Omega }$ that can be extended to bases of size $b(G)$. In particular, for primitive groups, we investigate the completeness and arc-transitivity of the generalised graph, and the generalisation of Burness and Giudici’s Common Neighbour Conjecture on the original Saxl graph.