A sufficient condition for the hamiltonian property of digraphs with large semi-degrees

Let $D$ be a digraph on $p\ge 5$ vertices with minimum degree at least $p-1$ and with minimum semi-degree at least $p\mathrm{/}2-1$. For $D$ (unless some extremal cases) we present a detailed proof of the following results [12]: (i) $D$ contains cycles of length 3, 4 and $p-1$; (ii) if $p=2n$, then $D$ is hamiltonian.