A conjecture on 3-anti-quasi-transitive digraphs

A digraph D is a 3-anti-quasi-transitive digraph, if for any four distinct vertices x 0 , x 1 , x 2 , x 3 ∈ V ( D ) such that x 0 → x 1 ← x 2 → x 3 , x 0 and x 3 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.