Highly linked tournaments with large minimum out-degree

TL;DR

This paper proves that strongly connected tournaments with sufficiently large minimum out-degree contain complex linkages, advancing the understanding of tournament structure and connectivity.

Contribution

It establishes a function linking out-degree and connectivity to k-linkage, progressing towards Pokrovskiy's conjecture and showing large out-degree tournaments contain complete directed graph subdivisions.

Findings

Existence of a function f(k) for k-linkage in strongly 4k-connected tournaments.

Large out-degree tournaments contain subdivisions of complete directed graphs.

Progress towards resolving a key conjecture in tournament theory.

Abstract

We prove that there exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integer $k$, if $T$ is a strongly $4k$-connected tournament with minimum out-degree at least $f(k)$, then $T$ is $k$-linked. This makes progress towards resolving a conjecture of Pokrovskiy. Along the way, we show that a tournament with sufficiently large minimum out-degree contains a subdivision of a complete directed graph. This result may be of independent interest.