Monochromatic k-connection of graphs

IF 1 3区 数学 Q3 MATHEMATICS, APPLIED
Qingqiong Cai , Shinya Fujita , Henry Liu , Boram Park
{"title":"Monochromatic k-connection of graphs","authors":"Qingqiong Cai ,&nbsp;Shinya Fujita ,&nbsp;Henry Liu ,&nbsp;Boram Park","doi":"10.1016/j.dam.2024.09.025","DOIUrl":null,"url":null,"abstract":"<div><div>An edge-coloured path is <em>monochromatic</em> if all of its edges have the same colour. For a <span><math><mi>k</mi></math></span>-connected graph <span><math><mi>G</mi></math></span>, the <em>monochromatic</em> <span><math><mi>k</mi></math></span><em>-connection number</em> of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>m</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the maximum number of colours in an edge-colouring of <span><math><mi>G</mi></math></span> such that, any two vertices are connected by <span><math><mi>k</mi></math></span> internally vertex-disjoint monochromatic paths. In this paper, we shall study the parameter <span><math><mrow><mi>m</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We obtain bounds for <span><math><mrow><mi>m</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, for general graphs <span><math><mi>G</mi></math></span>. We also compute <span><math><mrow><mi>m</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> exactly when <span><math><mi>k</mi></math></span> is small, and <span><math><mi>G</mi></math></span> is a graph on <span><math><mi>n</mi></math></span> vertices, with a spanning <span><math><mi>k</mi></math></span>-connected subgraph having the minimum possible number of edges, namely <span><math><mrow><mo>⌈</mo><mfrac><mrow><mi>k</mi><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></math></span>. We prove a similar result when <span><math><mi>G</mi></math></span> is a bipartite graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"360 ","pages":"Pages 328-341"},"PeriodicalIF":1.0000,"publicationDate":"2024-10-01","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/S0166218X24004153","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

An edge-coloured path is monochromatic if all of its edges have the same colour. For a k-connected graph G, the monochromatic k-connection number of G, denoted by mck(G), is the maximum number of colours in an edge-colouring of G such that, any two vertices are connected by k internally vertex-disjoint monochromatic paths. In this paper, we shall study the parameter mck(G). We obtain bounds for mck(G), for general graphs G. We also compute mck(G) exactly when k is small, and G is a graph on n vertices, with a spanning k-connected subgraph having the minimum possible number of edges, namely kn2. We prove a similar result when G is a bipartite graph.
图形的单色 k 连接
如果一条边缘着色的路径的所有边缘颜色相同,那么这条路径就是单色的。对于 k 个连接图 G,G 的单色 k 连接数(用 mck(G) 表示)是指在 G 的边缘颜色中,任意两个顶点通过 k 个内部顶点相交的单色路径连接的最大颜色数。本文将研究 mck(G) 参数。对于一般图 G,我们得到了 mck(G) 的边界。当 k 很小时,我们也能精确计算 mck(G)。当 G 是 n 个顶点上的图时,有一个跨 k 连接的子图具有尽可能少的边数,即 ⌈kn2⌉。当 G 是双向图时,我们会证明类似的结果。
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 Applied Mathematics
Discrete Applied Mathematics 数学-应用数学
CiteScore
2.30
自引率
9.10%
发文量
422
审稿时长
4.5 months
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.
×
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