In Bang-Jensen et al. (Sufficient conditions for a digraph to be Hamiltonia
n, J. Graph Theory 22 (1996) 181-187) the following extension of Meyniels t
heorem 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 pa
ir 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 c
ase 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 gener
alizes one of the results in [2], Furthermore we provide additional support
for the conjecture above by showing that such a digraph always has a facto
r (a spanning collection of disjoint cycles). Finally, we show that if D sa
tisfies 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
.