{"title":"关于图形色度数的最大奇数诱导子图","authors":"Tao Wang, Baoyindureng Wu","doi":"10.1002/jgt.23148","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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.</p>","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":"{\"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\":\"<p>Let <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><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.</p>\",\"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}","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}
Maximum odd induced subgraph of a graph concerning its chromatic number
Let be the maximum order of an odd induced subgraph of . In 1992, Scott proposed a conjecture that for a graph of order without isolated vertices, where is the chromatic number of . 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 for a connected graph of order . Scott's conjecture is open for graphs with chromatic number at least 3.
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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 .
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