{"title":"带有针状分支顶点的诱导细分区","authors":"","doi":"10.1016/j.ejc.2024.104072","DOIUrl":null,"url":null,"abstract":"<div><div>We prove that for all <span><math><mrow><mi>r</mi><mo>∈</mo><mi>N</mi><mo>∪</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span> and <span><math><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, there exists <span><math><mrow><mi>Ω</mi><mo>=</mo><mi>Ω</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>N</mi></mrow></math></span> with the following property. Let <span><math><mi>G</mi></math></span> be a graph and let <span><math><mi>H</mi></math></span> be a subgraph of <span><math><mi>G</mi></math></span> isomorphic to a <span><math><mrow><mo>(</mo><mo>≤</mo><mi>r</mi><mo>)</mo></mrow></math></span>-subdivision of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>Ω</mi></mrow></msub></math></span>. Then either <span><math><mi>G</mi></math></span> contains <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> as an induced subgraph, or there is an induced subgraph <span><math><mi>J</mi></math></span> of <span><math><mi>G</mi></math></span> isomorphic to a proper <span><math><mrow><mo>(</mo><mo>≤</mo><mi>r</mi><mo>)</mo></mrow></math></span>-subdivision of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> such that every branch vertex of <span><math><mi>J</mi></math></span> is a branch vertex of <span><math><mi>H</mi></math></span>. This answers in the affirmative a question of Lozin and Razgon. In fact, we show that both the branch vertices and the paths corresponding to the subdivided edges between them can be preserved.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001574/pdfft?md5=ed4f41801de33ce909fbbc25a22d7d22&pid=1-s2.0-S0195669824001574-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Induced subdivisions with pinned branch vertices\",\"authors\":\"\",\"doi\":\"10.1016/j.ejc.2024.104072\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We prove that for all <span><math><mrow><mi>r</mi><mo>∈</mo><mi>N</mi><mo>∪</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span> and <span><math><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, there exists <span><math><mrow><mi>Ω</mi><mo>=</mo><mi>Ω</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>N</mi></mrow></math></span> with the following property. Let <span><math><mi>G</mi></math></span> be a graph and let <span><math><mi>H</mi></math></span> be a subgraph of <span><math><mi>G</mi></math></span> isomorphic to a <span><math><mrow><mo>(</mo><mo>≤</mo><mi>r</mi><mo>)</mo></mrow></math></span>-subdivision of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>Ω</mi></mrow></msub></math></span>. Then either <span><math><mi>G</mi></math></span> contains <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> as an induced subgraph, or there is an induced subgraph <span><math><mi>J</mi></math></span> of <span><math><mi>G</mi></math></span> isomorphic to a proper <span><math><mrow><mo>(</mo><mo>≤</mo><mi>r</mi><mo>)</mo></mrow></math></span>-subdivision of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> such that every branch vertex of <span><math><mi>J</mi></math></span> is a branch vertex of <span><math><mi>H</mi></math></span>. This answers in the affirmative a question of Lozin and Razgon. In fact, we show that both the branch vertices and the paths corresponding to the subdivided edges between them can be preserved.</div></div>\",\"PeriodicalId\":50490,\"journal\":{\"name\":\"European Journal of Combinatorics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-09-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0195669824001574/pdfft?md5=ed4f41801de33ce909fbbc25a22d7d22&pid=1-s2.0-S0195669824001574-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"European Journal of Combinatorics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0195669824001574\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669824001574","RegionNum":3,"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
摘要
我们证明，对于所有 r∈N∪{0} 和 s,t∈N，存在具有以下性质的 Ω=Ω(r,s,t)∈N。设 G 是图，设 H 是 G 的子图，与 KΩ 的（≤r）细分同构。那么要么 G 包含作为诱导子图的 Kt 或 Kt,t，要么 G 的诱导子图 J 与 Ks 的适当 (≤r)- 细分同构，使得 J 的每个分支顶点都是 H 的分支顶点。事实上，我们证明了分支顶点和它们之间对应于细分边的路径都可以保留。
We prove that for all and , there exists with the following property. Let be a graph and let be a subgraph of isomorphic to a -subdivision of . Then either contains or as an induced subgraph, or there is an induced subgraph of isomorphic to a proper -subdivision of such that every branch vertex of is a branch vertex of . This answers in the affirmative a question of Lozin and Razgon. In fact, we show that both the branch vertices and the paths corresponding to the subdivided edges between them can be preserved.
期刊介绍：
The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.
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