Crown-free highly arc-transitive digraphs

Abstract

We construct a family of infinite, non-locally finite highly arc-transitive digraphs which do not have universal reachability relation and which omit special digraphs called 'crowns'. Moreover, there is no homomorphism from any of our digraphs onto Z. The methods are adapted from [6] and [7].

Descendant-homogeneous digraphs

“…We remark that it is shown in a sequel to the present paper [4] that in fact the digraphs whose construction we have just described are the only descendant-homogeneous digraphs in which the descendant digraphs are rooted q-valent trees where 1 < q < ∞.…”

“…Our class of digraphs is a subclass of Evans' class, the digraphs A in our class satisfy all the conditions (C1, C2 and C3) in Evans' class, which guarantee that it has only countably many members up to isomorphism, plus an extra condition, C4. As we shall see below, the addition of this condition will imply that we must allow identification of vertices when defining our notion of amalgamation, in contrast to analogous constructions given in [3].…”

“…In a sequel [4] to this paper we show that the digraphs constructed in Section 4.1 are up to isomorphism the only descendant-homogeneous digraphs whose descendant sets are q-valent rooted trees. We remark on a number of other recent papers [11][12][13][14][15][16][17][18] which treat related material about descendant sets and highly arc-transitive digraphs, though not descendant-homogeneity.…”

“…We give a negative answer, and in fact in Section 4 we construct infinitely many pairwise non-isomorphic descendant-homogeneous digraphs whose descendant sets are trees of any given finite valency greater than 1. These are obtained by omitting 4-crowns (in the sense of [3]) and more general bipartite digraphs. By weakening the definition of descendant-homogeneity to weak descendant-homogeneity, we show that we can actually construct uncountably many pairwise non-isomorphic such digraphs whose descendant sets are trees.…”