Bounding $s$ for vertex-primitive $s$-arc-transitive digraphs of alternating and symmetric groups

TL;DR

This paper establishes that for finite vertex-primitive s-arc-transitive digraphs with alternating or symmetric groups, the maximum s is 2, advancing understanding of symmetry bounds in graph theory.

Contribution

It proves that the upper bound s for such digraphs is at most 2 when the group is an alternating or symmetric group, refining previous bounds and confirming conjectures.

Findings

s ≤ 2 for alternating and symmetric groups

Advances bounds on symmetry in vertex-primitive digraphs

Builds on prior work of Pan, Wu, and Yin

Abstract

Determining an upper bound on $s$ for finite vertex-primitive $s$-arc-transitive digraphs has received considerable attention dating back to a question of Praeger in 1990. It was shown by Giudici and Xia that the smallest upper bound on $s$ is attained for some digraph admitting an almost simple $s$-arc-transitive group. In this paper, based on the work of Pan, Wu and Yin, we prove that $s\leqslant 2$ in the case where the group is an alternating or symmetric group.