{"title":"On the chromatic number of powers of subdivisions of graphs","authors":"Michael Anastos , Simona Boyadzhiyska , Silas Rathke , Juanjo Rué","doi":"10.1016/j.dam.2024.10.002","DOIUrl":null,"url":null,"abstract":"<div><div>For a given graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span>, we define its <span><math><mi>n</mi></math></span><em>th subdivision</em> as the graph obtained from <span><math><mi>G</mi></math></span> by replacing every edge by a path of length <span><math><mi>n</mi></math></span>. We also define the <span><math><mi>m</mi></math></span><em>th power</em> of <span><math><mi>G</mi></math></span> as the graph on vertex set <span><math><mi>V</mi></math></span> where we connect every pair of vertices at distance at most <span><math><mi>m</mi></math></span> in <span><math><mi>G</mi></math></span>. In this paper, we study the chromatic number of powers of subdivisions of graphs and resolve the case <span><math><mrow><mi>m</mi><mo>=</mo><mi>n</mi></mrow></math></span> asymptotically. In particular, our result confirms a conjecture of Mozafari-Nia and Iradmusa in the case <span><math><mrow><mi>m</mi><mo>=</mo><mi>n</mi><mo>=</mo><mn>3</mn></mrow></math></span> in a strong sense.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"360 ","pages":"Pages 506-511"},"PeriodicalIF":1.0000,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X24004323","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
For a given graph , we define its th subdivision as the graph obtained from by replacing every edge by a path of length . We also define the th power of as the graph on vertex set where we connect every pair of vertices at distance at most in . In this paper, we study the chromatic number of powers of subdivisions of graphs and resolve the case asymptotically. In particular, our result confirms a conjecture of Mozafari-Nia and Iradmusa in the case in a strong sense.
对于给定的图 G=(V,E)，我们将其第 n 次细分定义为将每条边替换为长度为 n 的路径后从 G 中得到的图。我们还将 G 的第 m 次幂定义为顶点集 V 上的图，在该图中，我们以最多 m 的距离连接 G 中的每对顶点。特别是，我们的结果在强意义上证实了 Mozafari-Nia 和 Iradmusa 在 m=n=3 情况下的猜想。
期刊介绍：
The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal.
Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
