On pre-Hamiltonian Cycles in Hamiltonian Digraphs - Darbinyan, Samvel Kh. (Mathematical Problems of Computer Science, 2021)
Let D be a strongly connected directed graph of order n ≥ 4. In [14] (J. of Graph Theory, Vol.16, No. 5, 51-59, 1992) Y. Manoussakis proved the following theorem: Suppose that D 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. Then D is Hamiltonian. In this paper we show that: If D satisfies the condition of Manoussakis' theorem, then 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 n=2 and n=2.