The Hoàng–Reed Conjecture for

The Hoàng–Reed Conjecture states that a digraph with minimum out-degree d contains d dicycles C 1 , C 2 , … , C d such that C k intersects ⋃ i < k C i on at most one vertex, for each k . It was made as a more structural approach to the Caccetta–Häggkvist Conjecture. We introduce the concept of a nearest separator and use it to prove a stronger version of the Hoàng–Reed Conjecture for the δ + = 2 case. With these two tools we go on to prove the δ + = 3 case.