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Highly linked tournaments with large minimum out-degree. (English) Zbl 1428.05123
Summary: We prove that there exists a function \(f : \mathbb{N} \to \mathbb{N}\) such that for any positive integer \(k\), if \(T\) is a strongly \(4k\)-connected tournament with minimum out-degree at least \(f(k)\), then \(T\) is \(k\)-linked. This resolves a conjecture of A. Pokrovskiy [ibid. 115, 339–347 (2015; Zbl 1319.05063)] up to a factor of 2 of the required connectivity. Along the way, we show that a tournament with sufficiently large minimum out-degree contains a subdivision of a complete directed graph. This result may be of independent interest.

MSC:
05C20 Directed graphs (digraphs), tournaments
05C40 Connectivity
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