{"title":"Induced subdivisions with pinned branch vertices","authors":"Sepehr Hajebi","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}
引用次数: 0
Abstract
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.
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。
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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