The optimal proper connection number of a graph with given independence number

IF 0.9 4区数学 Q3 MATHEMATICS, APPLIED

Discrete Optimization Pub Date : 2021-08-01 DOI:10.1016/j.disopt.2021.100660

Shinya Fujita , Boram Park

{"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>An edge-colored connected graph <math><mi>G</mi></math> 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 <math><mi>G</mi></math>, to make a connected graph properly connected efficiently. More precisely, pcopt\n(<math><mi>G</mi></math>) is the smallest integer <math><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></math> when one converts a given monochromatic graph <math><mi>G</mi></math> into a properly connected graph by recoloring <math><mi>p</mi></math> edges with <math><mi>q</mi></math> colors. In this paper, we show that pcopt\n(<math><mi>G</mi></math>) has an upper bound in terms of the independence number <math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>. Namely, we prove that for a connected graph <math><mi>G</mi></math>, pcopt\n(<math><mi>G</mi></math>)<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>. Moreover, for the case <math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>3</mn></mrow></math>, we improve the upper bound to 4, which is tight.</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 $G$ 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 pc_opt(G) for a monochromatic connected graph $G$ , to make a connected graph properly connected efficiently. More precisely, pc_opt ( $G$ ) is the smallest integer $p + q$ when one converts a given monochromatic graph $G$ into a properly connected graph by recoloring $p$ edges with $q$ colors. In this paper, we show that pc_opt ( $G$ ) has an upper bound in terms of the independence number $α (G)$ . Namely, we prove that for a connected graph $G$ , pc_opt ( $G$ ) $\leq \frac{5 α (G) - 1}{2}$ . Moreover, for the case $α (G) \leq 3$ , we improve the upper bound to 4, which is tight.

查看原文本刊更多论文

给定独立数的图的最优固有连接数

如果在每一对不同的顶点之间存在一条路径，使得相邻的两条边没有相同的颜色，则边色连通图G是正确连通的。Fujita(2019)引入单色连通图G的最优适当连接数pcopt(G)，使连通图有效地适当连接。更准确地说，pcopt(G)是将给定的单色图G通过用q种颜色对p条边重新上色而转换为适当连通图时的最小整数p+q。本文用独立数α(G)证明了pcopt(G)有一个上界。也就是说，我们证明了对于连通图G, pcopt(G)≤5α(G)−12。此外，对于α(G)≤3的情况，我们将上界改进为4，它是紧的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Discrete Optimization 管理科学-应用数学

CiteScore

2.10

自引率

9.10%

发文量

审稿时长

>12 weeks

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.