$(2k+1)$-connected tournaments with large minimum out-degree are $k$-linked

TL;DR

This paper proves that $(2k+1)$-connected tournaments with sufficiently large minimum out-degree are $k$-linked, confirming a conjecture up to an additive constant in connectivity, and demonstrates the necessity of high out-degree with counterexamples.

Contribution

It establishes that $(2k+1)$-connectivity plus polynomial out-degree guarantees $k$-linkedness, resolving a conjecture with an additive connectivity factor and without requiring large in-degree.

Findings

$(2k+1)$-connected tournaments with large out-degree are $k$-linked

Constructs tournaments that are $(2.5k-1)$-connected but not $k$-linked

Polynomial out-degree suffices for $k$-linkedness in highly connected tournaments

Abstract

Pokrovskiy conjectured that there is a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that any $2k$-strongly-connected tournament with minimum out and in-degree at least $f(k)$ is $k$-linked. In this paper, we show that any $(2k+1)$-strongly-connected tournament with minimum out-degree at least some polynomial in $k$ is $k$-linked, thus resolving the conjecture up to the additive factor of $1$ in the connectivity bound, but without the extra assumption that the minimum in-degree is large. Moreover, we show the condition on high minimum out-degree is necessary by constructing arbitrarily large tournaments that are $(2.5k-1)$-strongly-connected but are not $k$-linked.