Highly Arc-Transitive Digraphs With No Homomorphism Onto Z

In an infinite 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 and if D is 1-arc-transitive, then this relation is either universal or all of its equivalence classes induce isomorphic bipartite digraphs. In Combinatorica, 13 (1993), Cameron, Praeger and Wormald asked if there exist highly arc-transitive digraphs (apart from directed cycles) for which the reachability relation is not universal and which do not have a homomorphism onto the two-way infinite directed path (a Cayley digraph of Z with respect to one generator). In view of an earlier result of Praeger in Australas. J. Combin., 3 (1991), such digraphs are either locally infinite or have equal in- and out-degree. In European J. Combin., 18 (1997), Evans gave an affirmative answer by constructing a locally infinite example.

For each odd integer n >= 3, a construction of a highly arc-transitive digraph without property Z satisfying the additional properties that its in- and out-degree are equal to 2 and that the reachability equivalence classes induce alternating cycles of length 2n, is given. Furthermore, using the line digraph operator, digraphs having the above properties but with alternating cycles of length 4 are obtained.