A new sufficient condition for a digraph to be Hamiltonian

In Bang-Jensen et al. (Sufficient conditions for a digraph to be Hamiltonian, J. Graph Theory 22 (1996) 181-187) the following extension of Meyniels theorem was conjectured: If D is a strongly connected digraph on it vertices with the property that d(x)+d(y)greater than or equal to 2n-1 for every pair of non-adjacent vertices x,y with a common out-neighbour or a common in-neighbour, then D is Hamiltonian. We verify the conjecture in the special case where we also require that min{d(+)(x)+d(-)(y), d(-)(x)+d(+)(y)}greater than or equal to n(-1) for all pairs of vertices x, y as above. This generalizes one of the results in [2], Furthermore we provide additional support for the conjecture above by showing that such a digraph always has a factor (a spanning collection of disjoint cycles). Finally, we show that if D satisfies that d(x)+d(y)greater than or equal to 5/2n-4 for every pair of non-adjacent vertices x,y with a common out-neighbour or a common in-neighbour, then D is Hamiltonian. (C) 1999 Elsevier Science B.V. All rights reserved.