The quadratic connectivity conjecture for edge-disjoint Hamilton cycles

A tournament is an orientation of a complete graph, a Hamilton cycle is a consistently oriented cycle containing all vertices, and a tournament is strongly $r$-connected when deleting fewer than $r$ vertices leaves a strongly connected digraph.

Quadratic connectivity conjecture. There exists $C”>0$ such that, for every $k\in\mathbb{N}$, every strongly $C”k^2$-connected tournament contains $k$ edge-disjoint Hamilton cycles.

This is the logarithm-free strengthening of the solved general conjecture of Thomassen. The paper establishes an $O(k^2\log^2 k)$ connectivity bound, while the quadratic bound remains open in the supplied text.