Counterexamples to the linkage conjecture for tournaments

Abstract

Let $k\ge 2$ be an integer. A digraph D is k-linked if for every set of 2k distinct vertices $x_1,\dots ,x_k,y_1,\dots ,y_k$ in D, there exist k pairwise vertex-disjoint paths $P_1,\dots ,P_k$ such that each path $P_i$ starts at $x_i$ and ends at $y_i$ for $i\in \left[k\right]$. In 2015, Pokrovskiy conjectured that there exists a function $g\left(k\right)$ such that every 2k-connected tournament with minimum in-degree and minimum out-degree at least $g\left(k\right)$ is k-linked in Pokrovskiy (2015) [16]. In this paper, we disprove Pokrovskiy's conjecture by constructing a family of 2k-connected tournaments of order $n\ge 14k^2$ with arbitrarily large minimum semi-degree (depending on n) that are not k-linked. The counterexamples, with sufficiently large order n, also provide a negative answer to the question posed by Girão et al. (2021) [8]: whether or not 2k-connectivity is sufficient for k-linkage in every tournament with minimum out-degree at least some polynomial in k.