{"title":"Complexity results for two kinds of conflict-free edge-coloring of graphs","authors":"Ping Li","doi":"10.1016/j.dam.2025.02.016","DOIUrl":null,"url":null,"abstract":"<div><div>An edge-coloring of graph <span><math><mi>G</mi></math></span> is called <em>closed-neighborhood</em> (resp. <em>open-neighborhood</em>) <em>conflict-free edge-coloring</em> if for every edge <span><math><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, there is a color assigned to exactly one edge among <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>∪</mo><mi>E</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>∪</mo><mi>E</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><mrow><mo>{</mo><mi>u</mi><mi>v</mi><mo>}</mo></mrow></mrow></math></span>). The smallest number of colors needed in any possible closed-neighborhood (resp. open-neighborhood) conflict-free edge-coloring of <span><math><mi>G</mi></math></span>, denoted <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>C</mi><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>[</mo><mi>G</mi><mo>]</mo></mrow></mrow></math></span> (resp. <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>C</mi><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>), is called the <em>closed-neighborhood</em> (resp. <em>open-neighborhood</em>) <em>conflict-free index</em> of <span><math><mi>G</mi></math></span>. In this paper, we prove that decide whether <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>C</mi><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>[</mo><mi>G</mi><mo>]</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span> or <span><math><mrow><msubsup><mrow><mi>χ</mi></mrow><mrow><mi>C</mi><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn></mrow></math></span> is NP-complete, even if <span><math><mi>G</mi></math></span> is a bipartite graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 218-225"},"PeriodicalIF":1.0000,"publicationDate":"2025-02-17","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/S0166218X25000800","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
An edge-coloring of graph is called closed-neighborhood (resp. open-neighborhood) conflict-free edge-coloring if for every edge , there is a color assigned to exactly one edge among (resp. ). The smallest number of colors needed in any possible closed-neighborhood (resp. open-neighborhood) conflict-free edge-coloring of , denoted (resp. ), is called the closed-neighborhood (resp. open-neighborhood) conflict-free index of . In this paper, we prove that decide whether or is NP-complete, even if is a bipartite graph.
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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