Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths

Abstract

In 1982 Thomassen asked whether there exists an integer f(k,t) such that every strongly f(k,t)-connected tournament T admits a partition of its vertex set into t vertex classes V 1,…V t such that for all i the subtournament T[V i] induced on T by V i is strongly k-connected. Our main result implies an affirmative answer to this question. In particular we show that f(k, t)=O(k 7 t 4) suffices. As another application of our main result we give an affirmative answer to a question of Song as to whether, for any integer t, there exists aninteger h(t) such that every strongly h(t)-connected tournament has a 1-factor consisting of t vertex-disjoint cycles of prescribed lengths. We show that h(t)=O(t 5) suffices.

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Correspondence to Timothy Townsend.

Additional information

The research leading to these results was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ ERC Grant Agreements no. 258345 (D. Kühn) and 306349 (D. Osthus).

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Kühn, D., Osthus, D. & Townsend, T. Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle lengths. Combinatorica 36, 451–469 (2016). https://doi.org/10.1007/s00493-015-3186-8

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Mathematics Subject Classification (2000)

  • 05C20
  • 05C35
  • 05C40