{"title":"Total-rainbow connection and forbidden subgraphs","authors":"Jingshu Zhang , Hui Jiang , Wenjing Li","doi":"10.1016/j.dam.2024.07.051","DOIUrl":null,"url":null,"abstract":"<div><p>A path in a total-colored graph <span><math><mi>G</mi></math></span> is called a <em>total-rainbow path</em> if its edges and internal vertices have distinct colors. The total-colored graph <span><math><mi>G</mi></math></span> is <em>total-rainbow connected</em> if for any two distinct vertices of <span><math><mi>G</mi></math></span>, there is a total-rainbow path connecting them. The <em>total-rainbow connection number</em> of <span><math><mi>G</mi></math></span>, denoted as <span><math><mrow><mi>t</mi><mi>r</mi><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, represents the minimum number of colors that are required to make <span><math><mi>G</mi></math></span> total-rainbow connected. This paper characterizes all the families <span><math><mi>F</mi></math></span> of connected graphs with <span><math><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></mrow></math></span>, for which there exists a constant <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> such that <span><math><mi>G</mi></math></span> being a connected <span><math><mi>F</mi></math></span>-free graph implies <span><math><mrow><mi>t</mi><mi>r</mi><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>d</mi><mi>i</mi><mi>a</mi><mi>m</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>F</mi></mrow></msub></mrow></math></span>, where <span><math><mrow><mi>d</mi><mi>i</mi><mi>a</mi><mi>m</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denotes the diameter of <span><math><mi>G</mi></math></span>.</p></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"359 ","pages":"Pages 364-370"},"PeriodicalIF":1.0000,"publicationDate":"2024-08-29","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/S0166218X24003524","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
A path in a total-colored graph is called a total-rainbow path if its edges and internal vertices have distinct colors. The total-colored graph is total-rainbow connected if for any two distinct vertices of , there is a total-rainbow path connecting them. The total-rainbow connection number of , denoted as , represents the minimum number of colors that are required to make total-rainbow connected. This paper characterizes all the families of connected graphs with , for which there exists a constant such that being a connected -free graph implies , where denotes the diameter of .
如果全彩色图 G 中的边和内部顶点都有不同的颜色，则该图中的路径称为全彩虹路径。如果对于 G 中任意两个不同的顶点，存在一条连接它们的全彩虹路径，那么全彩色图 G 就是全彩虹连接图。G 的全彩虹连接数表示使 G 全彩虹连接所需的最少颜色数，用 trc(G) 表示。本文描述了|F|∈{1,2}的所有连通图族 F，对于这些族 F，存在一个常数 kF，使得 G 作为无 F 连通图意味着 trc(G)≤2diam(G)+kF，其中 diam(G) 表示 G 的直径。
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.
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