{"title":"On the chromatic number of $$P_5$$ -free graphs with no large intersecting cliques","authors":"Weilun Xu, Xia Zhang","doi":"10.1007/s10878-023-01088-5","DOIUrl":null,"url":null,"abstract":"<p>A graph <i>G</i> is called <span>\\((H_1, H_2)\\)</span>-free if <i>G</i> contains no induced subgraph isomorphic to <span>\\(H_1\\)</span> or <span>\\(H_2\\)</span>. Let <span>\\(P_k\\)</span> be a path with <i>k</i> vertices and <span>\\(C_{s,t,k}\\)</span> (<span>\\(s\\le t\\)</span>) be a graph consisting of two intersecting complete graphs <span>\\(K_{s+k}\\)</span> and <span>\\(K_{t+k}\\)</span> with exactly <i>k</i> common vertices. In this paper, using an iterative method, we prove that the class of <span>\\((P_5,C_{s,t,k})\\)</span>-free graphs with clique number <span>\\(\\omega \\)</span> has a polynomial <span>\\(\\chi \\)</span>-binding function <span>\\(f(\\omega )=c(s,t,k)\\omega ^{\\max \\{s,k\\}}\\)</span>. In particular, we give two improved chromatic bounds: every <span>\\((P_5, butterfly)\\)</span>-free graph <i>G</i> has <span>\\(\\chi (G)\\le \\frac{3}{2}\\omega (G)(\\omega (G)-1)\\)</span>; every <span>\\((P_5, C_{1,3})\\)</span>-free graph <i>G</i> has <span>\\(\\chi (G)\\le 9\\omega (G)\\)</span>.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-023-01088-5","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
A graph G is called \((H_1, H_2)\)-free if G contains no induced subgraph isomorphic to \(H_1\) or \(H_2\). Let \(P_k\) be a path with k vertices and \(C_{s,t,k}\) (\(s\le t\)) be a graph consisting of two intersecting complete graphs \(K_{s+k}\) and \(K_{t+k}\) with exactly k common vertices. In this paper, using an iterative method, we prove that the class of \((P_5,C_{s,t,k})\)-free graphs with clique number \(\omega \) has a polynomial \(\chi \)-binding function \(f(\omega )=c(s,t,k)\omega ^{\max \{s,k\}}\). In particular, we give two improved chromatic bounds: every \((P_5, butterfly)\)-free graph G has \(\chi (G)\le \frac{3}{2}\omega (G)(\omega (G)-1)\); every \((P_5, C_{1,3})\)-free graph G has \(\chi (G)\le 9\omega (G)\).
期刊介绍:
The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering.
The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.