Spanning spheres in Dirac hypergraphs
Fuente:
arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866913894357794816 |
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| author | Illingworth, Freddie Lang, Richard Müyesser, Alp Parczyk, Olaf Sgueglia, Amedeo |
| author_facet | Illingworth, Freddie Lang, Richard Müyesser, Alp Parczyk, Olaf Sgueglia, Amedeo |
| contents | We show that a $k$-uniform hypergraph on $n$ vertices has a spanning subgraph homeomorphic to the $(k - 1)$-dimensional sphere provided that $H$ has no isolated vertices and each set of $k - 1$ vertices supported by an edge is contained in at least $n/2 + o(n)$ edges. This gives a topological extension of Dirac's theorem and asymptotically confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan.
Unlike typical results in the area, our proof does not rely on the Absorption Method, the Regularity Lemma or the Blow-up Lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host graph with a family of complete blow-ups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06275 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spanning spheres in Dirac hypergraphs Illingworth, Freddie Lang, Richard Müyesser, Alp Parczyk, Olaf Sgueglia, Amedeo Combinatorics 05C35, 05C65, 05C70 We show that a $k$-uniform hypergraph on $n$ vertices has a spanning subgraph homeomorphic to the $(k - 1)$-dimensional sphere provided that $H$ has no isolated vertices and each set of $k - 1$ vertices supported by an edge is contained in at least $n/2 + o(n)$ edges. This gives a topological extension of Dirac's theorem and asymptotically confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan. Unlike typical results in the area, our proof does not rely on the Absorption Method, the Regularity Lemma or the Blow-up Lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host graph with a family of complete blow-ups. |
| title | Spanning spheres in Dirac hypergraphs |
| topic | Combinatorics 05C35, 05C65, 05C70 |
| url | https://arxiv.org/abs/2407.06275 |