A conjecture on 3-anti-quasi-transitive digraphs

A digraph D is a 3-anti-quasi-transitive digraph, if for any four distinct vertices x0, x1, x2, x3∈V(D) such that x0→x1←x2→x3, x0 and x3 are adjacent. Bang-Jensen conjectured (Bang-Jensen, 2004) that a 3-anti-quasi-transitive digraph is Hamiltonian if and only if it is strong and has a cycle factor. In this paper, we shall prove that this conjecture is true.