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 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.