Subdivisions and near-linear stable sets

IF 1 2区数学 Q1 MATHEMATICS

Combinatorica Pub Date : 2025-07-04 DOI:10.1007/s00493-025-00154-2

Tung Nguyen, Alex Scott, Paul Seymour

{"title":"Subdivisions and near-linear stable sets","authors":"Tung Nguyen, Alex Scott, Paul Seymour","doi":"10.1007/s00493-025-00154-2","DOIUrl":null,"url":null,"abstract":"<p>We prove that for every complete graph <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.213ex\" role=\"img\" style=\"vertical-align: -0.505ex;\" viewbox=\"0 -735.2 1205.1 952.8\" width=\"2.799ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-4B\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1201\" xlink:href=\"#MJMATHI-74\" y=\"-213\"></use></g></svg></span><script type=\"math/tex\">K_t</script></span>, all graphs <i>G</i> with no induced subgraph isomorphic to a subdivision of <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.213ex\" role=\"img\" style=\"vertical-align: -0.505ex;\" viewbox=\"0 -735.2 1205.1 952.8\" width=\"2.799ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-4B\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1201\" xlink:href=\"#MJMATHI-74\" y=\"-213\"></use></g></svg></span><script type=\"math/tex\">K_t</script></span> have a stable subset of size at least <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.614ex\" role=\"img\" style=\"vertical-align: -0.706ex;\" viewbox=\"0 -821.4 6664.3 1125.3\" width=\"15.479ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-7C\" y=\"0\"></use><use x=\"278\" xlink:href=\"#MJMATHI-47\" y=\"0\"></use><use x=\"1065\" xlink:href=\"#MJMAIN-7C\" y=\"0\"></use><use x=\"1343\" xlink:href=\"#MJMAIN-2F\" y=\"0\"></use><g transform=\"translate(2010,0)\"><use xlink:href=\"#MJMAIN-70\"></use><use x=\"556\" xlink:href=\"#MJMAIN-6F\" y=\"0\"></use><use x=\"1057\" xlink:href=\"#MJMAIN-6C\" y=\"0\"></use><use x=\"1335\" xlink:href=\"#MJMAIN-79\" y=\"0\"></use><use x=\"1864\" xlink:href=\"#MJMAIN-6C\" y=\"0\"></use><use x=\"2142\" xlink:href=\"#MJMAIN-6F\" y=\"0\"></use><use x=\"2643\" xlink:href=\"#MJMAIN-67\" y=\"0\"></use></g><use x=\"5320\" xlink:href=\"#MJMAIN-7C\" y=\"0\"></use><use x=\"5599\" xlink:href=\"#MJMATHI-47\" y=\"0\"></use><use x=\"6385\" xlink:href=\"#MJMAIN-7C\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">|G|/\\operatorname {polylog}|G|</script></span>. This is close to best possible, because for <span><span style=\"\"></span><span style=\"font-size: 100%; display: inline-block;\" tabindex=\"0\"><svg focusable=\"false\" height=\"2.209ex\" role=\"img\" style=\"vertical-align: -0.505ex;\" viewbox=\"0 -733.9 2196.1 951.2\" width=\"5.101ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-74\" y=\"0\"></use><use x=\"639\" xlink:href=\"#MJMAIN-2265\" y=\"0\"></use><use x=\"1695\" xlink:href=\"#MJMAIN-37\" y=\"0\"></use></g></svg></span><script type=\"math/tex\">t\\ge 7</script></span>, not all such graphs <i>G</i> have a stable set of linear size, even if <i>G</i> is triangle-free.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"62 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00493-025-00154-2","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We prove that for every complete graph $K_t$ , all graphs G with no induced subgraph isomorphic to a subdivision of $K_t$ have a stable subset of size at least $|G|/\operatorname {polylog}|G|$ . This is close to best possible, because for $t\ge 7$ , not all such graphs G have a stable set of linear size, even if G is triangle-free.

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细分与近线性稳定集

我们证明了对于每一个完全图K_t，所有没有诱导子图同构于K_t的一个细分的图G都有一个大小至少为|G|/\operatorname {polylog}|G|的稳定子集。这是接近最好的可能，因为对于t\ \ 7，不是所有这样的图G都有一个稳定的线性大小的集合，即使G是无三角形的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Combinatorica 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.