Linear colouring of binomial random graphs

IF 0.7 3区 数学 Q2 MATHEMATICS
{"title":"Linear colouring of binomial random graphs","authors":"","doi":"10.1016/j.disc.2024.114278","DOIUrl":null,"url":null,"abstract":"<div><div>We investigate the linear chromatic number <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>lin</mtext></mrow></msub><mo>(</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo><mo>)</mo></math></span> of the binomial random graph <span><math><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span> on <em>n</em> vertices in which each edge appears independently with probability <span><math><mi>p</mi><mo>=</mo><mi>p</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. For a graph <em>G</em>, <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>lin</mtext></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is defined as the smallest <em>k</em> such that <em>G</em> admits a <em>k</em>-colouring with the property that every path <em>P</em> in <em>G</em> receives a colour which appears on only one vertex of <em>P</em>. For dense random graphs (<span><math><mi>n</mi><mi>p</mi><mo>→</mo><mo>∞</mo></math></span> as <span><math><mi>n</mi><mo>→</mo><mo>∞</mo></math></span>), we show that asymptotically almost surely <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mtext>lin</mtext></mrow></msub><mo>(</mo><mi>G</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo><mo>)</mo><mo>≥</mo><mi>n</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>n</mi><mi>p</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo><mo>)</mo><mo>=</mo><mi>n</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></math></span>. Understanding the order of the linear chromatic number for subcritical random graphs (<span><math><mi>n</mi><mi>p</mi><mo>&lt;</mo><mn>1</mn></math></span>) and critical ones (<span><math><mi>n</mi><mi>p</mi><mo>=</mo><mn>1</mn></math></span>) is relatively easy. However, supercritical sparse random graphs (<span><math><mi>n</mi><mi>p</mi><mo>=</mo><mi>c</mi></math></span> for some constant <span><math><mi>c</mi><mo>&gt;</mo><mn>1</mn></math></span>) remain to be investigated.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004096","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

We investigate the linear chromatic number χlin(G(n,p)) of the binomial random graph G(n,p) on n vertices in which each edge appears independently with probability p=p(n). For a graph G, χlin(G) is defined as the smallest k such that G admits a k-colouring with the property that every path P in G receives a colour which appears on only one vertex of P. For dense random graphs (np as n), we show that asymptotically almost surely χlin(G(n,p))n(1O((np)1/2))=n(1o(1)). Understanding the order of the linear chromatic number for subcritical random graphs (np<1) and critical ones (np=1) is relatively easy. However, supercritical sparse random graphs (np=c for some constant c>1) remain to be investigated.
二项式随机图形的线性着色
我们研究了 n 个顶点上的二项式随机图 G(n,p)的线性色度数 χlin(G(n,p)),其中每条边都以 p=p(n) 的概率独立出现。对于一个图 G,χlin(G) 被定义为最小的 k,使得 G 可以接受 k-着色,其特性是 G 中的每条路径 P 得到的颜色只出现在 P 的一个顶点上。对于密集随机图(np→∞ 为 n→∞),我们证明了渐近几乎肯定 χlin(G(n,p))≥n(1-O((np)-1/2))=n(1-o(1))。理解亚临界随机图(np<1)和临界随机图(np=1)的线性色度数阶相对容易。然而,超临界稀疏随机图(np=c,对于某个常数 c>1)仍有待研究。
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
来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
期刊介绍: Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信