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 D1 of D is Hamiltonian. Moreover, for any k ⩾ 1 and n > 2k we exhibit a strongly k-connected digraph D of order n such that D⌈n2k⌉−1 is non-Hamiltonian.