Distance–transitive digraphs: Descendant-homogeneity, property [formula omitted] and reachability

Abstract

A digraph is distance–transitive if for every integer s≥0, the automorphism group of the digraph is transitive on pairs (u,v) of vertices for which there is a directed path of length s from u to v, but no directed path of length t<s. This implies vertex and edge transitivity but is weaker than being highly-arc-transitive. A digraph is said to have property Z if it has a homomorphism onto the two-way infinite directed path. In a digraph D, an edge e′ is reachable from an edge e if there exists an alternating walk in D whose initial and terminal edges are e and e′. Reachability is an equivalence relation on the edge set of D and if D is transitive on edges, then this relation is either universal or all of its equivalence classes induce isomorphic bipartite digraphs.We investigate the class of infinite distance–transitive digraphs of finite out-valency. First, we show that earlier results, proved in the context of highly-arc-transitive digraphs, hold for the class of distance–transitive digraphs. Second, we show that if D is a weakly descendant–homogeneous digraph in our 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. Finally, we describe some distance–transitive, weakly descendant–homogeneous digraphs for which the reachability relation is not universal.