{"title":"The optimal proper connection number of a graph with given independence number","authors":"Shinya Fujita , Boram Park","doi":"10.1016/j.disopt.2021.100660","DOIUrl":null,"url":null,"abstract":"<div><p><span>An edge-colored connected graph </span><span><math><mi>G</mi></math></span> is <em>properly connected</em> if between every pair of distinct vertices, there exists a path such that no two adjacent edges have the same color. Fujita (2019) introduced the <em>optimal proper connection number</em> pc<sub>opt</sub>(G) for a monochromatic connected graph <span><math><mi>G</mi></math></span>, to make a connected graph properly connected efficiently. More precisely, pc<sub>opt</sub>\n(<span><math><mi>G</mi></math></span>) is the smallest integer <span><math><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></math></span> when one converts a given monochromatic graph <span><math><mi>G</mi></math></span> into a properly connected graph by recoloring <span><math><mi>p</mi></math></span> edges with <span><math><mi>q</mi></math></span> colors. In this paper, we show that pc<sub>opt</sub>\n(<span><math><mi>G</mi></math></span>) has an upper bound in terms of the independence number <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Namely, we prove that for a connected graph <span><math><mi>G</mi></math></span>, pc<sub>opt</sub>\n(<span><math><mi>G</mi></math></span>)<span><math><mrow><mo>≤</mo><mfrac><mrow><mn>5</mn><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span>. Moreover, for the case <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>3</mn></mrow></math></span>, we improve the upper bound to 4, which is tight.</p></div>","PeriodicalId":50571,"journal":{"name":"Discrete Optimization","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.disopt.2021.100660","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Optimization","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1572528621000396","RegionNum":4,"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-colored connected graph is properly connected if between every pair of distinct vertices, there exists a path such that no two adjacent edges have the same color. Fujita (2019) introduced the optimal proper connection number pcopt(G) for a monochromatic connected graph , to make a connected graph properly connected efficiently. More precisely, pcopt
() is the smallest integer when one converts a given monochromatic graph into a properly connected graph by recoloring edges with colors. In this paper, we show that pcopt
() has an upper bound in terms of the independence number . Namely, we prove that for a connected graph , pcopt
(). Moreover, for the case , we improve the upper bound to 4, which is tight.
期刊介绍:
Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.