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Proof of a conjecture of Thomassen on Hamilton cycles in highly connected tournaments. (English) Zbl 1302.05069
Summary: A conjecture of C. Thomassen [Proc. Lond. Math. Soc., III. Ser. 45, 151–168 (1982; Zbl 0486.05049)] states that, for every $$k$$, there is an $$f(k)$$ so that every strongly $$f(k)$$-connected tournament contains $$k$$ edge-disjoint Hamilton cycles. A classical theorem of P. Camion [C. R. Acad. Sci., Paris 249, 2151–2152 (1959; Zbl 0092.15801)], that every strongly connected tournament contains a Hamilton cycle, implies that $$f(1)=1$$. So far, even the existence of $$f(2)$$ was open. In this paper, we prove Thomassen’s conjecture by showing that $$f(k)=O(k^2 \log ^2k)$$. This is best possible up to the logarithmic factor. As a tool, we show that every strongly $$10^4 k \log k$$-connected tournament is $$k$$-linked (which improves a previous exponential bound). The proof of the latter is based on a fundamental result of M. Ajtai et al. [Combinatorica 3, 1–19 (1983; Zbl 0523.68048)] on asymptotically optimal sorting networks.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C35 Extremal problems in graph theory 05C40 Connectivity 05C45 Eulerian and Hamiltonian graphs
##### Keywords:
strongly connected tournament; Hamilton cycle
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##### References:
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