Proof of the Caccetta–Häggkvist conjecture for oriented graphs with positive minimum out-degree and of independence number two

In his paper, “On the Caccetta–Häggkvist conjecture with forbidden subgraphs” (see Razborov (in press) [5]), A. Razborov points out that Chudnovsky and Seymour proved that an out-regular oriented graph of out-degree d ≥ 2 , of independence number 2 and of order at most 3 d contains a directed triangle. He says also that to the best of his knowledge, the question is still open without the restriction of out-regularity. In this paper, we give a complete answer, by proving that for d ≥ 2 , any oriented graph of minimum out-degree d ≥ 2 , of independence number 2, and of order at most 3 d contains a directed triangle. Additionally, we prove that any oriented graph of minimum out-degree d ≥ 1 , of independence number 2 and of order at most 4 d contains a directed cycle of length at most 4. A simple observation on the girth of a non-acyclic oriented graph of independence number 2, allows to state that the Caccetta–Häggkvist conjecture is true for oriented graphs of minimum out-degree at least 1, and of independence number 2.