  C. Thomassen (Proc. London Math. Soc. (3) 42 (1981), 231-251) gave a
characterization of strongly connected non-Hamiltonian digraphs of order $p\geq
3$ with minimum degree $p-1$. In this paper we give an analogous
characterization of strongly connected non-Hamiltonian digraphs with
Meyniel-type condition (the sum of degrees of every pair of non-adjacent
vertices $x$ and $y$ at least $2p-2$). Moreover, we prove that such digraphs
$D$ contain cycles of all lengths $k$, for $2\leq k\leq m$, where $m$ is the
length of a longest cycle in $D$.
