细分与近线性稳定集

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

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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":"{\"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>. 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引用次数: 0

摘要

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。

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Subdivisions and near-linear stable sets

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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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.