Common Neighbour Conjectures for Saxl Graphs

The paper disproves both the Burness–Giudici common neighbour conjecture for primitive groups of base size $2$ and its later generalisation to arbitrary base size. For every integer $B\ge2$, it constructs infinitely many primitive permutation groups of base size $B$ whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size $2$, it gives infinite counterexample families of affine, product and twisted wreath type, so the conjecture fails in three of the five O'Nan–Scott types.

The paper disproves both the Burness–Giudici common neighbour conjecture for primitive groups of base size $2$ and its later generalisation to arbitrary base size. For every integer $B\ge2$, it constructs infinitely many primitive permutation groups of base size $B$ whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size $2$, it gives infinite counterexample families of affine, product and twisted wreath type, so the conjecture fails in three of the five O'Nan–Scott types.