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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 ε>0 , when k is large enough, every tournament with minimum out‐degree at least (1.5+ε)k contains k disjoint cycles. The linear factor 1.5 is best possible as shown by the regular tournaments.