Highly arc-transitive digraphs with no homomorphism onto \(\mathbb Z\)
The authors' work is based on three previous papers: \textit{P. J. Cameron, C. E. Praeger} and \textit{N. C. Wormald} [Combinatorica 13, 377-396 (1993; Zbl 0793.05065)], \textit{C. E. Praeger} [Australas. J. Comb. 3, 207-210 (1991; Zbl 0758.05055)] and \textit{D. M. Evans} [Eur. J. Comb. 18, 281-286 (1997; Zbl 0873.05050)]. An \(s\)-arc of \(D\) is a walk on \(s+1\) vertices. A walk \(W\) is alternating, if the directions of edges are alternating, while one travels through \(W\). The digraph \(D\) is \(s\)-arc transitive if its automorphism group is transitive on the set of \(s\)-arcs, while \(D\) is said to be highly arc transitive if it is \(s\)-arc transitive for all \(s >0\). A digraph has property \(Z\) if it has a homomorphism onto the two-way infinite path \(Z\). The main result of the paper is the exhibition of highly arc-transitive digraphs for all odd \(n \geq 3\) without property \(Z\) that also satisfy that all in and out degrees are 2, and the reachability equivalance classes induce alternating cycles of length \(2n\).
scientific article; zbMATH DE number 1912036