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arXiv · 1303.4213

Proof of a conjecture of Thomassen on Hamilton cycles in highly connected tournaments

Daniela Kühn·John Lapinskas·Deryk Osthus·Viresh Patel
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Abstract

A conjecture of Thomassen from 1982 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 Camion, 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^2(k)). 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 Ajtai, Komlós and Szemerédi on asymptotically optimal sorting networks.

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Daniela Kühn, John Lapinskas, Deryk Osthus, Viresh Patel. 2013-03-18. Proof of a conjecture of Thomassen on Hamilton cycles in highly connected tournaments. https://doi.org/10.1112/plms%2Fpdu019

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