On Hamiltonian bypasses in digraphs with the condition of Y. Manoussakis

Let D be a strongly connected directed graph of order n ≥ 4 which satisfies the following condition for every triple x, y, z of vertices such that x and y are nonadjacent: If there is no arc from x to z, then d(x)+d(y)+d⁺(x)+d^-(z) ≥ 3n-2. If there is no arc from z to x, then d(x)+d(y)+d^-(x)+d⁺(z) ≥ 3n-2. In [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 is obtained from a Hamiltonian cycle by reversing exactly one arc) or D is isomorphic to one tournament of order five.