Construction of k-arc transitive digraphs

A digraph is k-arc transitive if it has a group of automorphisms which acts transitively on the set of k-arcs. Unlike the undirected case, in which the cycles are the only k-arc transitive finite graphs for k⩾8, there are k-arc transitive finite digraphs with arbitrary out-degree for every positive integer k. We show that every regular finite digraph admits a covering digraph which is k-arc transitive. The result provides a technique to construct k-arc transitive digraphs from arbitrary regular digraphs. Some examples are given from complete graphs with and without loops.