Conflict-free Hypergraph Matchings and Coverings
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918353949425664 |
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| author | Joos, Felix Mubayi, Dhruv Smith, Zak |
| author_facet | Joos, Felix Mubayi, Dhruv Smith, Zak |
| contents | Recent work showing the existence of conflict-free almost-perfect hypergraph matchings has found many applications. We show that, assuming certain simple degree and codegree conditions on the hypergraph $ \mathcal{H} $ and the conflicts to be avoided, a conflict-free almost-perfect matching can be extended to one covering all of the vertices in a particular subset of $ V(\mathcal{H}) $, by using an additional set of edges; in particular, we ensure that our matching avoids all of a further set of conflicts, which may consist of both old and new edges. This setup is useful for various applications, and our main theorem provides a black box which encapsulates many long and tedious calculations, massively simplifying the proofs of results in generalised Ramsey theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18144 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conflict-free Hypergraph Matchings and Coverings Joos, Felix Mubayi, Dhruv Smith, Zak Combinatorics Recent work showing the existence of conflict-free almost-perfect hypergraph matchings has found many applications. We show that, assuming certain simple degree and codegree conditions on the hypergraph $ \mathcal{H} $ and the conflicts to be avoided, a conflict-free almost-perfect matching can be extended to one covering all of the vertices in a particular subset of $ V(\mathcal{H}) $, by using an additional set of edges; in particular, we ensure that our matching avoids all of a further set of conflicts, which may consist of both old and new edges. This setup is useful for various applications, and our main theorem provides a black box which encapsulates many long and tedious calculations, massively simplifying the proofs of results in generalised Ramsey theory. |
| title | Conflict-free Hypergraph Matchings and Coverings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.18144 |