Every $(13k-6)$-strong tournament with minimum out-degree at least $28k-13$ is $k$-linked

Abstract

A digraph $D$ is $k$-linked if it satisfies that for every choice of disjoint sets $\{x_1,\dots ,x_k\}$ and $\{y_1,\dots ,y_k\}$ of vertices of $D$ there are vertex disjoint paths $P_1,\dots ,P_k$ such that $P_i$ is an $(x_i,y_i)$-path. Confirming a conjecture by Kühn et al, Pokrovskiy proved in 2015 that every $452k$-strong tournament is $k$-linked and asked for a better linear bound. Very recently Meng et al proved that every $(40k-31)$-strong tournament is $k$-linked. In this note we use an important lemma from their paper to give a short proof that every $(13k-6)$-strong tournament of minimum out-degree at least $28k-13$ is $k$-linked.