Every (13k − 6)-strong tournament with minimum out-degree at least 28k − 13 is k-linked
Jørgen Bang‐Jensen, Kasper Skov Johansen · Discrete Mathematics · 2022
A digraph D is k-linked if it satisfies that for every choice of disjoint sets {x1,…,xk} and {y1,…,yk} of vertices of D there are vertex disjoint paths P1,…,Pk such that Pi is an (xi,yi)-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.