A New Sufficient Condition for a Digraph to Be Hamiltonian

In 2] the following extension of Meyniels theorem was conjectured: If D is a 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 minfd + (x)+d ? (y); d ? (x)+d + (y)g 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 satisses that d(x) + d(y) 5 2 n ? 4 for every pair of non-adjacent vertices x; y with a common out-neighbour or a common in-neighbour, then D is Hamiltonian.