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

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