Disjoint 3-Cycles in Tournaments: A Proof of The Bermond-Thomassen Conjecture for Tournaments

We prove that every tournament with minimum out-degree at least 2k− 1 contains k disjoint 3-cycles. This provides additional support for the conjecture by Bermond and Thomassen that every digraph D of minimum out-degree 2k − 1 contains k vertex disjoint cycles. We also prove that for every > 0, when k is large enough, every tournament with minimum out-degree at least (1.5+ )k contains k disjoint cycles. The linear factor 1.5 is best possible as shown by the regular tournaments.