On longest non-Hamiltonian cycles in digraphs with the conditions of Bang-Jensen, Gutin and Li

Let D be a strongly connected directed graph of order n≥4. In Bang-Jensen et al. (1996), (J. of Graph Theory 22 (2) (1996) 181-187), J. Bang-Jensen, G. Gutin and H. Li proved the following theorems: If (∗)d(x)+d(y)≥2n−1 and min{d(x),d(y)}≥n−1 for every pair of non-adjacent vertices x,y with a common in-neighbour or (∗∗)min{d+(x)+d−(y),d−(x)+d+(y)}≥n for every pair of non-adjacent vertices x,y with a common in-neighbour or a common out-neighbour, then D is Hamiltonian. In this paper we show that: (i) if D satisfies condition (∗) and the minimum semi-degree of D at least two or (ii) if D is not directed cycle and satisfies condition (∗∗), then either D contains a cycle of length n−1 or n is even and D is isomorphic to the complete bipartite digraph or to the complete bipartite digraph minus one arc.