On the generalised Saxl graphs of permutation groups

A base for a finite permutation group G ⩽ Sym ( Ω ) is a subset of Ω 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 Ω , 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 ) ⩾ 2 : here, the edges are the pairs of elements of Ω 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.