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

Authors


  • Part of this work was done while the first author was on sabbatical at AlGCo, LIRMM, Université Montpellier 2, France whose hospitality is gratefully acknowledged. Financial support from the Danish National Science research council (FNU) (under grant no. 09-066741) is gratefully acknowledged.Contract grant sponsor: ANR GRATOS; Contract grant number: ANR-09-JCJC-0041-01 (to S.B.).

Abstract

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

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