Maximum odd induced subgraph of a graph concerning its chromatic number

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-07-09 DOI:10.1002/jgt.23148

Tao Wang, Baoyindureng Wu

{"title":"Maximum odd induced subgraph of a graph concerning its chromatic number","authors":"Tao Wang, Baoyindureng Wu","doi":"10.1002/jgt.23148","DOIUrl":null,"url":null,"abstract":"Let <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>f</mi>\n \n <mi>o</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${f}_{o}(G)$</annotation>\n </semantics></math> be the maximum order of an odd induced subgraph of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>. In 1992, Scott proposed a conjecture that <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>f</mi>\n \n <mi>o</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mfrac>\n <mi>n</mi>\n \n <mrow>\n <mi>χ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mfrac>\n </mrow>\n </mrow>\n <annotation> ${f}_{o}(G)\\ge \\frac{n}{\\chi (G)}$</annotation>\n </semantics></math> for a graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> of order <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> without isolated vertices, where <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>χ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $\\chi (G)$</annotation>\n </semantics></math> is the chromatic number of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>. In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang, and Wargo in 1997, which states that <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>f</mi>\n \n <mi>o</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mn>2</mn>\n \n <mfenced>\n <mfrac>\n <mi>n</mi>\n \n <mn>4</mn>\n </mfrac>\n </mfenced>\n </mrow>\n </mrow>\n <annotation> ${f}_{o}(G)\\ge 2\\unicode{x0230A}\\frac{n}{4}\\unicode{x0230B}$</annotation>\n </semantics></math> for a connected graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> of order <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math>. Scott's conjecture is open for graphs with chromatic number at least 3.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"578-596"},"PeriodicalIF":0.9000,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23148","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let $f_{o} (G)$ be the maximum order of an odd induced subgraph of $G$ . In 1992, Scott proposed a conjecture that $f_{o} (G) \geq \frac{n}{χ (G)}$ for a graph $G$ of order $n$ without isolated vertices, where $χ (G)$ is the chromatic number of $G$ . In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang, and Wargo in 1997, which states that $f_{o} (G) \geq 2 (\frac{n}{4})$ for a connected graph $G$ of order $n$ . Scott's conjecture is open for graphs with chromatic number at least 3.

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关于图形色度数的最大奇数诱导子图

让是 .的奇数诱导子图的最大阶数。 1992 年，斯科特（Scott）提出了一个猜想：对于无孤立顶点的阶数图，让是 .的色度数。在本文中，我们证明了这一猜想对于二叉图并不成立，但对于所有线图都成立。此外，我们还推翻了 Berman、Wang 和 Wargo 于 1997 年提出的猜想，即对于秩为 .斯科特的猜想对于色度数至少为 3 的图是开放的。

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .