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Journal Articles Journal of the London Mathematical Society Year : 2022

Conflict‐free hypergraph matchings

Abstract

A celebrated theorem of Pippenger, and Frankl and Rödl states that every almost‐regular, uniform hypergraph with small maximum codegree has an almost‐perfect matching. We extend this result by obtaining a conflict‐free matching, where conflicts are encoded via a collection of subsets . We say that a matching is conflict‐free if does not contain an element of as a subset. Under natural assumptions on , we prove that has a conflict‐free, almost‐perfect matching. This has many applications, one of which yields new asymptotic results for so‐called ‘high‐girth’ Steiner systems. Our main tool is a random greedy algorithm which we call the ‘conflict‐free matching process’.
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Dates and versions

hal-04619462 , version 1 (21-06-2024)

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Stefan Glock, Felix Joos, Jaehoon Kim, Marcus Kühn, Lyuben Lichev. Conflict‐free hypergraph matchings. Journal of the London Mathematical Society, 2022, 109 (5), ⟨10.1112/jlms.12899⟩. ⟨hal-04619462⟩
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