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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2408.09391 |
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| _version_ | 1866912661273313280 |
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| author | Lee, Seunghun Smorodinsky, Shakhar |
| author_facet | Lee, Seunghun Smorodinsky, Shakhar |
| contents | We study the conflict-free chromatic number of hypergraphs derived from the family of facets of $d$-dimensional cyclic polytopes with $n$ vertices. While in odd dimensions $d$ the problem is easy, for even dimensions the problem becomes very difficult and exhibits interesting connections to extremal graph theory. We provide sharp asymptotic bounds for the conflict-free chromatic number in several small even dimensions and non-trivial upper and lower bounds for general even dimensions. The main purpose of this paper is revealing a surprising relation between conflict-free colorings and the celebrated Erdős girth conjecture, opening new avenues for future research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09391 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On conflict-free colorings of cyclic polytopes and the girth conjecture for graphs Lee, Seunghun Smorodinsky, Shakhar Combinatorics We study the conflict-free chromatic number of hypergraphs derived from the family of facets of $d$-dimensional cyclic polytopes with $n$ vertices. While in odd dimensions $d$ the problem is easy, for even dimensions the problem becomes very difficult and exhibits interesting connections to extremal graph theory. We provide sharp asymptotic bounds for the conflict-free chromatic number in several small even dimensions and non-trivial upper and lower bounds for general even dimensions. The main purpose of this paper is revealing a surprising relation between conflict-free colorings and the celebrated Erdős girth conjecture, opening new avenues for future research. |
| title | On conflict-free colorings of cyclic polytopes and the girth conjecture for graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.09391 |