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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 strongly10  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.

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