In this paper we prove the following new sufficient condition for a digraph to be Hamiltonian:
{\it Let $D$ be a 2-strong digraph of order $n\geq 9$.
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.

Comment: 20 pages, 2 figures

• 05C20 - Directed graphs (digraphs), tournaments
• 05C45 - Eulerian and Hamiltonian graphs