{"title":"树和近线稳定集","authors":"Tung Nguyen, Alex Scott, Paul Seymour","doi":"arxiv-2409.09397","DOIUrl":null,"url":null,"abstract":"When $H$ is a forest, the Gy\\'arf\\'as-Sumner conjecture implies that every\ngraph $G$ with no induced subgraph isomorphic to $H$ and with bounded clique\nnumber has a stable set of linear size. We cannot prove that, but we prove that\nevery such graph $G$ has a stable set of size $|G|^{1-o(1)}$. If $H$ is not a\nforest, there need not be such a stable set. Second, we prove that when $H$ is a ``multibroom'', there {\\em is} a stable\nset of linear size. As a consequence, we deduce that all multibrooms satisfy a\n``fractional colouring'' version of the Gy\\'arf\\'as-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Trees and near-linear stable sets\",\"authors\":\"Tung Nguyen, Alex Scott, Paul Seymour\",\"doi\":\"arxiv-2409.09397\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"When $H$ is a forest, the Gy\\\\'arf\\\\'as-Sumner conjecture implies that every\\ngraph $G$ with no induced subgraph isomorphic to $H$ and with bounded clique\\nnumber has a stable set of linear size. We cannot prove that, but we prove that\\nevery such graph $G$ has a stable set of size $|G|^{1-o(1)}$. If $H$ is not a\\nforest, there need not be such a stable set. Second, we prove that when $H$ is a ``multibroom'', there {\\\\em is} a stable\\nset of linear size. As a consequence, we deduce that all multibrooms satisfy a\\n``fractional colouring'' version of the Gy\\\\'arf\\\\'as-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting.\",\"PeriodicalId\":501407,\"journal\":{\"name\":\"arXiv - MATH - Combinatorics\",\"volume\":\"16 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Combinatorics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.09397\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09397","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
When $H$ is a forest, the Gy\'arf\'as-Sumner conjecture implies that every
graph $G$ with no induced subgraph isomorphic to $H$ and with bounded clique
number has a stable set of linear size. We cannot prove that, but we prove that
every such graph $G$ has a stable set of size $|G|^{1-o(1)}$. If $H$ is not a
forest, there need not be such a stable set. Second, we prove that when $H$ is a ``multibroom'', there {\em is} a stable
set of linear size. As a consequence, we deduce that all multibrooms satisfy a
``fractional colouring'' version of the Gy\'arf\'as-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting.