{"title":"The list r-hued coloring of Halin graph","authors":"Shudan Lu , Fengxia Liu , Hong-Jian Lai","doi":"10.1016/j.dam.2025.06.044","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mi>L</mi></math></span> be a list assignment of colors available for vertices of a graph <span><math><mi>G</mi></math></span>. An <span><math><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></math></span>-coloring of a graph <span><math><mi>G</mi></math></span> is a proper coloring <span><math><mi>c</mi></math></span> such that every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> receives at least <span><math><mrow><mo>min</mo><mrow><mo>{</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>,</mo><mi>r</mi><mo>}</mo></mrow></mrow></math></span> colors in its neighbors and <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>∈</mo><mi>L</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span>. The list <span><math><mi>r</mi></math></span>-hued chromatic number <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a graph <span><math><mi>G</mi></math></span>, is the smallest <span><math><mi>k</mi></math></span> such that for each list assignment <span><math><mi>L</mi></math></span> satisfying <span><math><mrow><mrow><mo>|</mo><mi>L</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>=</mo><mi>k</mi></mrow></math></span>, for any <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mi>G</mi></math></span> has an <span><math><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></math></span>-coloring. An <span><math><mrow><mo>(</mo><mi>L</mi><mo>,</mo><mi>r</mi><mo>)</mo></mrow></math></span>-coloring is called an <span><math><mi>r</mi></math></span>-hued <span><math><mi>k</mi></math></span>-coloring if <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Let <span><math><mi>G</mi></math></span> be a Halin graph. We determine the upper bounds of the list <span><math><mi>r</mi></math></span>-hued chromatic number of Halin graphs when <span><math><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>r</mi><mo>=</mo><mi>Δ</mi></mrow></math></span>. This improves former results on 2-hued coloring of Halin graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"377 ","pages":"Pages 32-42"},"PeriodicalIF":1.0000,"publicationDate":"2025-07-04","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/S0166218X25003543","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
Let be a list assignment of colors available for vertices of a graph . An -coloring of a graph is a proper coloring such that every vertex receives at least colors in its neighbors and . The list -hued chromatic number of a graph , is the smallest such that for each list assignment satisfying , for any , has an -coloring. An -coloring is called an -hued -coloring if for all . Let be a Halin graph. We determine the upper bounds of the list -hued chromatic number of Halin graphs when and . This improves former results on 2-hued coloring of Halin graphs.
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.
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