According to the supposition by Bang-Jensen et al.

for a strong digraph D with n vertices

D is Hamiltonian if the sum of their degrees is no less than 2n-1 for every pair of non-adjacent vertices x

y with a common out-neighbor or a common in-neighbor.In this paper

the authors have proved that

if the sum of their degrees is no less than the maximum between 2n-1 and (5/2)n-9/2 for every pair of vertices with above property

D is Hamiltonian.