Edge disjoint Hamiltonian cycles in highly connected tournaments

TL;DR

This paper proves a conjecture that highly connected tournaments contain a quadratic number of edge-disjoint Hamiltonian cycles, confirming a long-standing hypothesis in graph theory.

Contribution

The authors prove that the function $f(k)$ can be bounded by a constant times $k^2$, confirming the conjecture that highly connected tournaments contain $k$ edge-disjoint Hamiltonian cycles.

Findings

Confirmed that $f(k) ext{ is } O(k^2)$

Established existence of $k$ edge-disjoint Hamiltonian cycles in highly connected tournaments

Resolved a conjecture in graph theory regarding connectivity and Hamiltonian cycles

Abstract

Thomassen conjectured that there is a function $f(k)$ such that every strongly $f(k)$-connected tournament contains $k$ edge-disjoint Hamiltonian cycles. This conjecture was recently proved by K\"uhn, Lapinskas, Osthus, and Patel who showed that $f(k)\leq O(k^2(\log k)^2)$ and conjectured that there is a constant $C$ such that $f(k)\leq Ck^2$. We prove this conjecture.