A new sufficient condition for a 2-strong digraph to be Hamiltonian

In this paper we prove the following new sufficient condition for a digraph to be Hamiltonian: Let D be a 2 -strong digraph of order n >= 9 . If n - 1 vertices of D have degrees at least n + k and the remaining vertex has degree at least n - k - 4 , where k is a non-negative integer, then D is Hamiltonian. This is an extension of Ghouila-Houri's theorem for 2-strong digraphs and is a generalization of an early result of the author (DAN Arm. SSR (91(2):6-8, 1990). The obtained result is best possible in the sense that for k = 0 there is a digraph of order n = 8 (respectively, n = 9) with the minimum degree n - 4 = 4 (respectively, with the minimum degree n - 5 = 4) whose n - 1 vertices have degrees at least n - 1, but it is not Hamiltonian. We also give a new sufficient condition for a 3-strong digraph to be Hamiltonian-connected.