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

TL;DR

This paper proves Thomassen's conjecture that highly connected tournaments contain multiple edge-disjoint Hamilton cycles, establishing a quadratic logarithmic bound on the connectivity needed, and introduces improved linking properties using sorting networks.

Contribution

It confirms Thomassen's conjecture by providing an explicit bound on connectivity and demonstrates that strongly connected tournaments are k-linked with improved bounds.

Findings

Proved that f(k)=O(k^2*log^2(k)) for Hamilton cycles in tournaments.

Showed that strongly 10^4*k*log(k)-connected tournaments are k-linked.

Improved previous exponential bounds on linking in tournaments.

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, Koml\'os and Szemer\'edi on asymptotically optimal sorting networks.