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Highly-arc-transitive and descendant-homogeneous digraphs with finite out-valency. (English) Zbl 1449.05119

A digraph \(D\) is said to have property \(Z\) if there is a digraph homomorphism from \(D\) onto the digraph \(Z\). The descendant set of a vertex in a digraph \(D\) is the induced subdigraph on the set of vertices reachable from the given vertex by an outward-directed path. A digraph \(D\) is descendant-homogeneous if it is vertex-transitive and any isomorphism between finitely generated subdigraphs of \(D\) extends to an automorphism of \(D\). The present paper investigates highly-arc-transitive digraphs with a homomorphism onto \(Z\) which are, additionally, descendant-homogeneous. It is shown that if \(D\) is a highly-arc-transitive descendant-homogeneous digraph with property \(Z\) and \(F\) is the subdigraph spanned by the descendant sets of a line in \(D\), then \(F\) is a locally finite 2-ended digraph with property \(Z\). In addition if \(D\) has prime out-valency, then there is only one possibility for the subdigraph \(F\), which is then used to classify the highly-arc-transitive descendant-homogeneous digraphs of prime out-valency with property \(Z\).

MSC:

05C20 Directed graphs (digraphs), tournaments
05E18 Group actions on combinatorial structures
20B07 General theory for infinite permutation groups
20B27 Infinite automorphism groups
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