Distance-transitive digraphs: descendant-homogeneity, property $Z$ and reachability

TL;DR

This paper classifies infinite distance-transitive digraphs with finite out-valency based on their homogeneity, property Z, and reachability relations, extending previous results to a broader class.

Contribution

It provides a classification of infinite distance-transitive digraphs with finite out-valency, linking properties Z, reachability, and homogeneity, and extends earlier results to this broader context.

Findings

If weakly descendant-homogeneous, D either has property Z with non-universal reachability or lacks Z with universal reachability and infinite in-valency.

Earlier results on highly-arc-transitive digraphs also hold under weaker distance-transitivity conditions.

A subclass of such digraphs with non-universal reachability is explicitly described.

Abstract

We investigate the class of infinite distance-transitive digraphs $D$ of finite out-valency. We show that if $D$ is a weakly descendant-homogeneous in such a class then either (1) $D$ has property $Z$ and the reachability relation is not universal; or (2) $D$ does not have property $Z$, the reachability relation is universal and $D$ has infinite in-valency. Also, we show that earlier results, proved in the context of highly-arc-transitive digraphs, hold under the weaker condition of distance-transitivity. Finally, we give a description of a subclass of the class distance-transitive weakly descendant-homogeneous digraphs for which the reachability relation is not universal.