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Disjoint 3-cycles in tournaments: a proof of the Bermond-Thomassen conjecture for tournaments. (English) Zbl 1292.05119
Summary: 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 \(\epsilon >0 \), when \(k\) is large enough, every tournament with minimum out-degree at least \((1.5 + \epsilon)k\) contains \(k\) disjoint cycles. The linear factor 1.5 is best possible as shown by the regular tournaments.

MSC:
05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles
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