A new sufficient condition for a digraph to be Hamiltonian Original Research Article

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 n vertices with the property that d(x)+d(y)⩾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)}⩾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)⩾52n−4 for every pair of non-adjacent vertices x,y with a common out-neighbour or a common in-neighbour, then D is Hamiltonian.