Local degree conditions for K 9 ${K}_{9}$ -minors in graphs

Pub Date : 2024-05-08 DOI:10.1002/jgt.23110

Takashige Akiyama

{"title":"Local degree conditions for \n \n \n \n K\n 9\n \n \n ${K}_{9}$\n -minors in graphs","authors":"Takashige Akiyama","doi":"10.1002/jgt.23110","DOIUrl":null,"url":null,"abstract":"We prove that if each edge of a graph <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> belongs to at least seven triangles, then <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> contains a <math>\n <semantics>\n <mrow>\n <msub>\n <mi>K</mi>\n \n <mn>9</mn>\n </msub>\n </mrow>\n <annotation> ${K}_{9}$</annotation>\n </semantics></math>-minor or contains <math>\n <semantics>\n <mrow>\n <msub>\n <mi>K</mi>\n <mrow>\n <mn>1</mn>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mo>,</mo>\n \n <mn>2</mn>\n </mrow>\n </msub>\n </mrow>\n <annotation> ${K}_{1,2,2,2,2,2}$</annotation>\n </semantics></math> as an induced subgraph. This result was conjectured by Albar and Gonçalves in 2018. Moreover, we apply this result to study the stress freeness of graphs.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23110","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

We prove that if each edge of a graph $G$ belongs to at least seven triangles, then $G$ contains a $K_{9}$ -minor or contains $K_{1, 2, 2, 2, 2, 2}$ as an induced subgraph. This result was conjectured by Albar and Gonçalves in 2018. Moreover, we apply this result to study the stress freeness of graphs.

查看原文

图中 K9 ${K}_{9}$ 未成数的局部度条件

我们证明，如果一个图的每条边至少属于七个三角形，那么就包含一个-小图或包含一个诱导子图。这一结果是阿尔巴和贡萨尔维斯在 2018 年猜想出来的。此外，我们还将这一结果应用于研究图的应力自由性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文