Highly arc-transitive digraphs -- counterexamples and structure

TL;DR

This paper constructs new examples of highly arc-transitive digraphs with specific properties, resolving longstanding open problems and expanding understanding of their structure.

Contribution

It provides the first known locally finite highly arc-transitive digraph with a universal reachability relation and constructs 2-ended examples with non-trivial building blocks, challenging previous conjectures.

Findings

Constructed a locally finite highly arc-transitive digraph with universal reachability.

Developed 2-ended highly arc transitive digraphs with non-complete bipartite building blocks.

Described the structure of 2-ended highly arc transitive digraphs more generally.

Abstract

We resolve two problems of [Cameron, Praeger, and Wormald -- Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 1993]. First, we construct a locally finite highly arc-transitive digraph with universal reachability relation. Second, we provide constructions of 2-ended highly arc transitive digraphs where each `building block' is a finite bipartite graph that is not a disjoint union of complete bipartite graphs. This was conjectured impossible in the above paper. We also describe the structure of 2-ended highly arc transitive digraphs in more generality, although complete characterization remains elusive.