Hamiltonian cycles in a generalization of bipartite tournaments with a cycle factor

In 2004, Bang-Jensen introduced H i -free digraphs, for i in { 1 , 2 , 3 , 4 } , as a generalization of semicomplete and semicomplete bipartite digraphs. Bang-Jensen conjectured that an H i -free digraph D , for i in { 1 , 2 , 3 , 4 } , is Hamiltonian if and only if D is strong and contains a cycle factor (that is, a collection of vertex disjoint cycles covering all the vertices of D ). S. Wang and R. Wang proved the conjecture for i in { 1 , 2 } in 2009 and Galeana-Sلnchez, Goldfeder and Urrutia proved the conjecture for i = 3 in 2010. In this paper, we prove the conjecture for i = 4 .