{"title":"通过凸优化限定平面图中奇数路径的数量","authors":"Asaf Cohen Antonir, Asaf Shapira","doi":"10.1002/jgt.23120","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>N</mi>\n \n <mi>P</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>H</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${N}_{{\\mathscr{P}}}(n,H)$</annotation>\n </semantics></math> denote the maximum number of copies of <span></span><math>\n <semantics>\n <mrow>\n <mi>H</mi>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> in an <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> vertex planar graph. The problem of bounding this function for various graphs <span></span><math>\n <semantics>\n <mrow>\n <mi>H</mi>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> has been extensively studied since the 70's. A special case that received a lot of attention recently is when <span></span><math>\n <semantics>\n <mrow>\n <mi>H</mi>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> is the path on <span></span><math>\n <semantics>\n <mrow>\n <mn>2</mn>\n \n <mi>m</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> $2m+1$</annotation>\n </semantics></math> vertices, denoted <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>P</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mi>m</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n <annotation> ${P}_{2m+1}$</annotation>\n </semantics></math>. Our main result in this paper is that\n\n </p><p>This improves upon the previously best known bound by a factor <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>e</mi>\n \n <mi>m</mi>\n </msup>\n </mrow>\n <annotation> ${e}^{m}$</annotation>\n </semantics></math>, which is best possible up to the hidden constant, and makes a significant step toward resolving conjectures of Ghosh et al. and of Cox and Martin. The proof uses graph theoretic arguments together with (simple) arguments from the theory of convex optimization.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"330-343"},"PeriodicalIF":0.9000,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23120","citationCount":"0","resultStr":"{\"title\":\"Bounding the number of odd paths in planar graphs via convex optimization\",\"authors\":\"Asaf Cohen Antonir, Asaf Shapira\",\"doi\":\"10.1002/jgt.23120\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>N</mi>\\n \\n <mi>P</mi>\\n </msub>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>H</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> ${N}_{{\\\\mathscr{P}}}(n,H)$</annotation>\\n </semantics></math> denote the maximum number of copies of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n <annotation> $H$</annotation>\\n </semantics></math> in an <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> vertex planar graph. The problem of bounding this function for various graphs <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n <annotation> $H$</annotation>\\n </semantics></math> has been extensively studied since the 70's. A special case that received a lot of attention recently is when <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n <annotation> $H$</annotation>\\n </semantics></math> is the path on <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>2</mn>\\n \\n <mi>m</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n <annotation> $2m+1$</annotation>\\n </semantics></math> vertices, denoted <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>P</mi>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>m</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation> ${P}_{2m+1}$</annotation>\\n </semantics></math>. Our main result in this paper is that\\n\\n </p><p>This improves upon the previously best known bound by a factor <span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>e</mi>\\n \\n <mi>m</mi>\\n </msup>\\n </mrow>\\n <annotation> ${e}^{m}$</annotation>\\n </semantics></math>, which is best possible up to the hidden constant, and makes a significant step toward resolving conjectures of Ghosh et al. and of Cox and Martin. The proof uses graph theoretic arguments together with (simple) arguments from the theory of convex optimization.</p>\",\"PeriodicalId\":16014,\"journal\":{\"name\":\"Journal of Graph Theory\",\"volume\":\"107 2\",\"pages\":\"330-343\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-05-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23120\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23120\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23120","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Bounding the number of odd paths in planar graphs via convex optimization
Let denote the maximum number of copies of in an vertex planar graph. The problem of bounding this function for various graphs has been extensively studied since the 70's. A special case that received a lot of attention recently is when is the path on vertices, denoted . Our main result in this paper is that
This improves upon the previously best known bound by a factor , which is best possible up to the hidden constant, and makes a significant step toward resolving conjectures of Ghosh et al. and of Cox and Martin. The proof uses graph theoretic arguments together with (simple) arguments from the theory of convex optimization.
期刊介绍：
The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.
A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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