Highly Arc Transitive Digraphs

Finite digrahs Г with a group G of automorphisms acting transitively on the set of s-arcs, for some s ⩾ 2, are investigated. For each valency v and each s ⩾ 2 an infinite family of finite digraphs of valency v which are s-arc transitive but not (s + 1)-arc transitive are constructed. These examples are the result of a general construction given in the paper. Several classification results are proved and the case of G primitive on vertices is discussed