On Hamiltonian Bypasses in Digraphs with the Condition of Y. Manoussakis

Let $D$ be a strongly connected directed graph of order $n\geq 4$ vertices which satisfies the following condition for every triple $x,y,z$ of vertices such that $x$ and $y$ are non-adjacent: If there is no arc from $x$ to $z$, then $d(x)+d(y)+d^+(x)+d^-(z)\geq 3n-2$. If there is no arc from $z$ to $x$, then $d(x)+d(y)+d^-(x)+d^+(z)\geq 3n-2$. In \cite{[15]} (J. of Graph Theory, Vol.16, No. 5, 51-59, 1992) Y. Manoussakis proved that $D$ is Hamiltonian. In [9] it was shown that $D$ contains a pre-Hamiltonian cycle (i.e., a cycle of length $n-1$) or $n$ is even and $D$ is isomorphic to the complete bipartite digraph with partite sets of cardinalities of $n/2$ and $n/2$. In this paper we show that $D$ contains also a Hamiltonian bypass, (i.e., a subdigraph obtained from a Hamiltonian cycle by reversing exactly one arc) or $D$ is isomorphic to one tournament of order 5.