On Hamiltonian bypasses in orgraphs with large semi-degrees

A Hamiltonian path in a digraph $D$ in which the initial vertex dominates the terminal vertex is called a Hamiltonian bypass. A digraph with at most one arc between any pair of vertices is called an oriented graph (or orgraph, for short). Let $D$ be an orgraph of order $p$ with minimum semi-degree at least $\lfloor p/2\rfloor -1$. In this paper, we show that if $p\ge 10$ and $p\ne 11$, then $D$ contains a Hamiltonian bypass. We present examples of orgraphs which shows that this result is sharp in the following sense: both lower bounds $\lfloor p/2\rfloor -1$ and 10 are tight.