Highly linked tournaments with large minimum out-degree
Ant\'onio Gir\~ao, Richard Snyder

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 such that for any positive integer , if is a strongly -connected tournament with minimum out-degree at least , then is -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.
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