7 problems

Pokrovskiy's conjecture. For every $k\in\mathbb{N}$, there exists an integer $h(k)$ such that every $2k$-connected tournament with $\delta^0(D)\geq h(k)$ is $k$-linked.

Let $k$ be a positive integer. A tournament is a directed graph in which exactly one of $xy$ and $yx$ is an edge for every pair of distinct vertices $x,y$. A tournament is $k$-link…

Gir{a}o–Popielarz–Snyder conjecture. There exists a constant $C>0$ such that every $(2k+1)$-connected tournament with $\delta^+(T)\geq Ck$ is $k$-linked.

Linear minimum out-degree conjecture. There exists a constant $C>0$ such that every $(2k+1)$-connected tournament with minimum out-degree at least $Ck$ is $k$-linked.

Let $G$ be a biconnected graph. For positive integers $a,b$, say that $G$ is $(a,b)$-linked if, for every pair of disjoint vertex sets $M_1,M_2$ with $|M_1|=a$ and $|M_2|=b$, there…

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

Linear connectivity conjecture for linked tournaments. There exists a constant $C>0$ such that, for every $k\in\mathbb{N}$, every strongly $Ck$-connected tournament is $k$-linked.