Vertex-primitive $s$-arc-transitive Cayley digraphs

TL;DR

This paper proves that the maximum value of s for finite vertex-primitive s-arc-transitive Cayley digraphs is 2 and fully characterizes their structure when s=2.

Contribution

It establishes the exact upper bound of 2 for s and provides a complete structural characterization for the case s=2.

Findings

Maximum s is exactly 2 for these digraphs.

Complete classification of structures when s=2.

Progress on a longstanding open problem.

Abstract

Determining an upper bound on $s$ for vertex-primitive $s$-arc-transitive digraphs has been an open problem of considerable interest since a question asked by Praeger in 1990. Although much progress has been made and an upper bound is conjectured to be $2$, a complete classification for $s=2$ remains out of reach. In this paper, we prove that the tight upper bound on $s$ for finite vertex-primitive $s$-arc-transitive Cayley digraphs is exactly $2$. Furthermore, we completely characterize the structure of these digraphs when $s=2$.