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 ź 4 . In Bang-Jensen etźal. (1996), (J. of Graph Theory 22 (2) (1996) 181-187), J. Bang-Jensen, G. Gutin and H. Li proved the following theorems: If ( ź ) d ( x ) + d ( y ) ź 2 n - 1 and m i n { d ( x ) , d ( y ) } ź n - 1 for every pair of non-adjacent vertices x , y with a common in-neighbour or ( ź ź ) m i n { d + ( x ) + d - ( y ) , d - ( x ) + d + ( y ) } ź 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 ( ź ) and the minimum semi-degree of D at least two or (ii) if D is not directed cycle and satisfies condition ( ź ź ) , 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.