A conjecture on -transitive digraphs

A digraph is k -transitive, if for any a path x 0 x 1 … x k of length k , x 0 dominates x k . A digraph is a strong k -transitive digraph, if it is k -transitive and it is strongly connected. César Hernández-Cruz proposed the following conjecture: Let k − 1 be a prime and D be a strong k -transitive digraph. If ∣ V ( D ) ∣ ≥ k + 1 , D contains an n -cycle, with n ≥ k and ( n , k − 1 ) = 1 , and D is not a symmetrical ( k + 1 ) -cycle, then D is a complete digraph. In this paper, we shall prove that the conjecture is true.