A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915814344491008 |
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| author | Kathapurkar, Amarja Morris, Patrick Perarnau, Guillem |
| author_facet | Kathapurkar, Amarja Morris, Patrick Perarnau, Guillem |
| contents | A meta-conjecture of Coulson, Keevash, Perarnau and Yepremyan states that above the extremal threshold for a given spanning structure in a (hyper-)graph, one can find a rainbow version of that spanning structure in any suitably bounded colouring of the host (hyper-)graph. We solve one of the most pertinent outstanding cases of this conjecture, by showing that for any $1\leq j\leq k-1$, if $G$ is a $k$-uniform hypergraph above the $j$-degree threshold for a loose Hamilton cycle, then any globally bounded colouring of $G$ contains a rainbow loose Hamilton cycle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07644 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs Kathapurkar, Amarja Morris, Patrick Perarnau, Guillem Combinatorics A meta-conjecture of Coulson, Keevash, Perarnau and Yepremyan states that above the extremal threshold for a given spanning structure in a (hyper-)graph, one can find a rainbow version of that spanning structure in any suitably bounded colouring of the host (hyper-)graph. We solve one of the most pertinent outstanding cases of this conjecture, by showing that for any $1\leq j\leq k-1$, if $G$ is a $k$-uniform hypergraph above the $j$-degree threshold for a loose Hamilton cycle, then any globally bounded colouring of $G$ contains a rainbow loose Hamilton cycle. |
| title | A rainbow Dirac theorem for loose Hamilton cycles in hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.07644 |