A Note on Hamiltonian Cycles in Digraphs with Large Degrees

Abstract: In this note we prove: {\it Let $D$ be a 2-strong digraph of order $n$ such that its $n-1$ vertices have degrees at least $n+k$ and the remaining vertex $z$ has degree at least $n-k-4$, where $k$ is a positive integer. If $D$ contains a cycle of length at least $n-k-2$ passing through $z$, then $D$ is Hamiltonian}.