Fractional hypergraph coloring
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909582140375040 |
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| author | Akhmejanova, Margarita Longbrake, Sean |
| author_facet | Akhmejanova, Margarita Longbrake, Sean |
| contents | We investigate proper $(a:b)$-fractional colorings of $n$-uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned $b$ distinct colors from a set of $a$ colors, and an edge is properly colored if no single color is shared by all vertices of the edge. A hypergraph is $(a:b)$-colorable if every edge is properly colored. We prove that for any $2\leq b\leq a-2\leq n/\ln n$, every $n$-uniform hypergraph $H$ with $ |E(H)| \leq (ab^3)^{-1/2}\left(\frac{n}{\log n}\right)^{1/2} \left(\frac{a}{b}\right)^{n-1} $ is proper $(a:b)$-colorable. We also address specific cases, including $(a:a-1)$-colorability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12430 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional hypergraph coloring Akhmejanova, Margarita Longbrake, Sean Combinatorics 05C15, 05D40 We investigate proper $(a:b)$-fractional colorings of $n$-uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned $b$ distinct colors from a set of $a$ colors, and an edge is properly colored if no single color is shared by all vertices of the edge. A hypergraph is $(a:b)$-colorable if every edge is properly colored. We prove that for any $2\leq b\leq a-2\leq n/\ln n$, every $n$-uniform hypergraph $H$ with $ |E(H)| \leq (ab^3)^{-1/2}\left(\frac{n}{\log n}\right)^{1/2} \left(\frac{a}{b}\right)^{n-1} $ is proper $(a:b)$-colorable. We also address specific cases, including $(a:a-1)$-colorability. |
| title | Fractional hypergraph coloring |
| topic | Combinatorics 05C15, 05D40 |
| url | https://arxiv.org/abs/2504.12430 |