Qingqiong Cai , Shinya Fujita , Henry Liu , Boram Park
{"title":"Monochromatic k-connection of graphs","authors":"Qingqiong Cai , Shinya Fujita , Henry Liu , 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 -connected graph , the monochromatic-connection number of , denoted by , is the maximum number of colours in an edge-colouring of such that, any two vertices are connected by internally vertex-disjoint monochromatic paths. In this paper, we shall study the parameter . We obtain bounds for , for general graphs . We also compute exactly when is small, and is a graph on vertices, with a spanning -connected subgraph having the minimum possible number of edges, namely . We prove a similar result when is a bipartite graph.
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。
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.