An upper bound for the Hamiltonicity exponent of finite digraphs

Let D be a strongly k−connected digraph of order n ≥ 2. We prove that for every l ≥ n 2k the power Dl of D is Hamiltonian. Moreover, for any n > 2k ≥ 2 we exhibit strongly k-connected digraphs D of order n such that Dl−1 is non-Hamiltonian for l = d n 2ke. We use standard terminology, unless otherwise stated. A digraph D = (V, A) of order n ≥ 2 is said to be strongly q-arc Hamiltonian if for any system S of mutually vertex-disjoint paths of the complete symmetric digraph with vertex set V of total length not greater than q, the digraph D′ = (V ; A ∪ S) has a Hamiltonian cycle containing S. If D is such a digraph that for every set W of vertices such that | W |≤ p the digraph D−W is strongly q-arc Hamiltonian, then we say that D is strongly (p,q)Hamiltonian. It is clear that a strongly (0, 1)-Hamiltonian digraph is strongly Hamiltonian connected, that is, for every pair of vertices u, v ∈ V there is a Hamiltonian path from u to v. In [1] Bermond proved the following