Note on Polychromatic Coloring of Hereditary Hypergraph Families
We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but all its restricted subhypergraphs with edges of size at least 3 are 2-colorable. This disproves a bold conjecture of Keszegh and the author, and can be considered as the first step to understand polychromatic colorings of he...
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| Published in | Graphs and combinatorics Vol. 40; no. 6; p. 131 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Tokyo
Springer Japan
01.12.2024
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0911-0119 1435-5914 1435-5914 |
| DOI | 10.1007/s00373-024-02836-y |
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| Abstract | We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but all its restricted subhypergraphs with edges of size at least 3 are 2-colorable. This disproves a bold conjecture of Keszegh and the author, and can be considered as the first step to understand polychromatic colorings of hereditary hypergraph families better since the seminal work of Berge. We also show that our method cannot give hypergraphs of arbitrary high uniformity, and mention some connections to panchromatic colorings. |
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| AbstractList | We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but all its restricted subhypergraphs with edges of size at least 3 are 2-colorable. This disproves a bold conjecture of Keszegh and the author, and can be considered as the first step to understand polychromatic colorings of hereditary hypergraph families better since the seminal work of Berge. We also show that our method cannot give hypergraphs of arbitrary high uniformity, and mention some connections to panchromatic colorings. We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but all its restricted subhypergraphs with edges of size at least 3 are 2-colorable. This disproves a bold conjecture of Keszegh and the author, and can be considered as the first step to understand polychromatic colorings of hereditary hypergraph families better since the seminal work of Berge. We also show that our method cannot give hypergraphs of arbitrary high uniformity, and mention some connections to panchromatic colorings.We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but all its restricted subhypergraphs with edges of size at least 3 are 2-colorable. This disproves a bold conjecture of Keszegh and the author, and can be considered as the first step to understand polychromatic colorings of hereditary hypergraph families better since the seminal work of Berge. We also show that our method cannot give hypergraphs of arbitrary high uniformity, and mention some connections to panchromatic colorings. |
| ArticleNumber | 131 |
| Author | Pálvölgyi, Dömötör |
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| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/39582740$$D View this record in MEDLINE/PubMed |
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| Cites_doi | 10.1016/j.comgeo.2023.102020 10.1007/s00493-021-4846-5 10.1007/s00454-019-00096-9 10.1137/19M1290231 10.1007/BF01584535 10.1002/jgt.20527 10.37236/1670 10.1007/978-3-031-15914-5_11 |
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| Keywords | Polychromatic coloring Hypergraphs Graph colorings 05C15 (Coloring of graphs and hypergraphs) Panchromatic coloring |
| Language | English |
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| References | B Keszegh (2836_CR9) 2019; 10 C Berge (2836_CR2) 1972; 2 A Kostochka (2836_CR11) 2002; 9 2836_CR3 G Damásdi (2836_CR6) 2022; 42 2836_CR4 B Keszegh (2836_CR10) 2019; 62 2836_CR5 D Shabanov (2836_CR14) 2011; 67 2836_CR7 2836_CR8 2836_CR13 J Pach (2836_CR12) 2014 E Ackerman (2836_CR1) 2020; 34 |
| References_xml | – ident: 2836_CR4 doi: 10.1016/j.comgeo.2023.102020 – volume: 42 start-page: 1027 year: 2022 ident: 2836_CR6 publication-title: Combinatorica doi: 10.1007/s00493-021-4846-5 – volume: 62 start-page: 674 year: 2019 ident: 2836_CR10 publication-title: Discret. Comput. Geom. doi: 10.1007/s00454-019-00096-9 – volume: 34 start-page: 2250 year: 2020 ident: 2836_CR1 publication-title: SIAM J. Discret. Math. doi: 10.1137/19M1290231 – volume: 10 start-page: 1 year: 2019 ident: 2836_CR9 publication-title: J. Comput. Geom. – volume: 2 start-page: 19 year: 1972 ident: 2836_CR2 publication-title: Math. Program. doi: 10.1007/BF01584535 – volume: 67 start-page: 226 year: 2011 ident: 2836_CR14 publication-title: J. Graph. Theory doi: 10.1002/jgt.20527 – ident: 2836_CR7 – ident: 2836_CR8 – start-page: 219 volume-title: Geometry, Intuitive, Discrete, and Convex, Bolyai Society Mathematical Studies year: 2014 ident: 2836_CR12 – ident: 2836_CR13 – ident: 2836_CR3 – volume: 9 start-page: 1 year: 2002 ident: 2836_CR11 publication-title: Electron. J. Combin. doi: 10.37236/1670 – ident: 2836_CR5 doi: 10.1007/978-3-031-15914-5_11 |
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| Snippet | We exhibit a 5-uniform hypergraph that has no polychromatic 3-coloring, but all its restricted subhypergraphs with edges of size at least 3 are 2-colorable.... |
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| SubjectTerms | Coloring Combinatorics Engineering Design Graph theory Graphs Mathematics Mathematics and Statistics Original Paper |
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| Title | Note on Polychromatic Coloring of Hereditary Hypergraph Families |
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