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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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##### References:
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