{"title":"关于一些广义Kneer图的色数","authors":"Jozefien D'haeseleer, Klaus Metsch, Daniel Werner","doi":"10.1002/jcd.21875","DOIUrl":null,"url":null,"abstract":"<p>We determine the chromatic number of the Kneser graph <math>\n <semantics>\n <mrow>\n <mi>q</mi>\n \n <msub>\n <mi>Γ</mi>\n <mrow>\n <mn>7</mn>\n \n <mo>,</mo>\n <mrow>\n <mo>{</mo>\n <mrow>\n <mn>3</mn>\n \n <mo>,</mo>\n \n <mn>4</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n </msub>\n </mrow>\n <annotation> $q{{\\rm{\\Gamma }}}_{7,\\{3,4\\}}$</annotation>\n </semantics></math> of flags of vectorial type <math>\n <semantics>\n <mrow>\n <mrow>\n <mo>{</mo>\n <mrow>\n <mn>3</mn>\n \n <mo>,</mo>\n \n <mn>4</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n <annotation> $\\{3,4\\}$</annotation>\n </semantics></math> of a rank 7 vector space over the finite field <math>\n <semantics>\n <mrow>\n <mi>GF</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>q</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\mathrm{GF}(q)$</annotation>\n </semantics></math> for large <math>\n <semantics>\n <mrow>\n <mi>q</mi>\n </mrow>\n <annotation> $q$</annotation>\n </semantics></math> and describe the colorings that attain the bound. This result relies heavily, not only on the independence number, but also on the structure of all <i>large</i> independent sets. Furthermore, our proof is more general in the following sense: it provides the chromatic number of the Kneser graphs <math>\n <semantics>\n <mrow>\n <mi>q</mi>\n \n <msub>\n <mi>Γ</mi>\n <mrow>\n <mn>2</mn>\n \n <mi>d</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n <mrow>\n <mo>{</mo>\n <mrow>\n <mi>d</mi>\n \n <mo>,</mo>\n \n <mi>d</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n </msub>\n </mrow>\n <annotation> $q{{\\rm{\\Gamma }}}_{2d+1,\\{d,d+1\\}}$</annotation>\n </semantics></math> of flags of vectorial type <math>\n <semantics>\n <mrow>\n <mrow>\n <mo>{</mo>\n <mrow>\n <mi>d</mi>\n \n <mo>,</mo>\n \n <mi>d</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n <annotation> $\\{d,d+1\\}$</annotation>\n </semantics></math> of a rank <math>\n <semantics>\n <mrow>\n <mn>2</mn>\n \n <mi>d</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n <annotation> $2d+1$</annotation>\n </semantics></math> vector space over <math>\n <semantics>\n <mrow>\n <mi>GF</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>q</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\mathrm{GF}(q)$</annotation>\n </semantics></math> for large <math>\n <semantics>\n <mrow>\n <mi>q</mi>\n </mrow>\n <annotation> $q$</annotation>\n </semantics></math> as long as the <i>large</i> independent sets of the graphs are only the ones that are known.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 4","pages":"179-204"},"PeriodicalIF":0.5000,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the chromatic number of some generalized Kneser graphs\",\"authors\":\"Jozefien D'haeseleer, Klaus Metsch, Daniel Werner\",\"doi\":\"10.1002/jcd.21875\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We determine the chromatic number of the Kneser graph <math>\\n <semantics>\\n <mrow>\\n <mi>q</mi>\\n \\n <msub>\\n <mi>Γ</mi>\\n <mrow>\\n <mn>7</mn>\\n \\n <mo>,</mo>\\n <mrow>\\n <mo>{</mo>\\n <mrow>\\n <mn>3</mn>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n </mrow>\\n \\n <mo>}</mo>\\n </mrow>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation> $q{{\\\\rm{\\\\Gamma }}}_{7,\\\\{3,4\\\\}}$</annotation>\\n </semantics></math> of flags of vectorial type <math>\\n <semantics>\\n <mrow>\\n <mrow>\\n <mo>{</mo>\\n <mrow>\\n <mn>3</mn>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n </mrow>\\n \\n <mo>}</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\{3,4\\\\}$</annotation>\\n </semantics></math> of a rank 7 vector space over the finite field <math>\\n <semantics>\\n <mrow>\\n <mi>GF</mi>\\n <mrow>\\n <mo>(</mo>\\n \\n <mi>q</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\mathrm{GF}(q)$</annotation>\\n </semantics></math> for large <math>\\n <semantics>\\n <mrow>\\n <mi>q</mi>\\n </mrow>\\n <annotation> $q$</annotation>\\n </semantics></math> and describe the colorings that attain the bound. This result relies heavily, not only on the independence number, but also on the structure of all <i>large</i> independent sets. Furthermore, our proof is more general in the following sense: it provides the chromatic number of the Kneser graphs <math>\\n <semantics>\\n <mrow>\\n <mi>q</mi>\\n \\n <msub>\\n <mi>Γ</mi>\\n <mrow>\\n <mn>2</mn>\\n \\n <mi>d</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n \\n <mo>,</mo>\\n <mrow>\\n <mo>{</mo>\\n <mrow>\\n <mi>d</mi>\\n \\n <mo>,</mo>\\n \\n <mi>d</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n \\n <mo>}</mo>\\n </mrow>\\n </mrow>\\n </msub>\\n </mrow>\\n <annotation> $q{{\\\\rm{\\\\Gamma }}}_{2d+1,\\\\{d,d+1\\\\}}$</annotation>\\n </semantics></math> of flags of vectorial type <math>\\n <semantics>\\n <mrow>\\n <mrow>\\n <mo>{</mo>\\n <mrow>\\n <mi>d</mi>\\n \\n <mo>,</mo>\\n \\n <mi>d</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n \\n <mo>}</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\{d,d+1\\\\}$</annotation>\\n </semantics></math> of a rank <math>\\n <semantics>\\n <mrow>\\n <mn>2</mn>\\n \\n <mi>d</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n <annotation> $2d+1$</annotation>\\n </semantics></math> vector space over <math>\\n <semantics>\\n <mrow>\\n <mi>GF</mi>\\n <mrow>\\n <mo>(</mo>\\n \\n <mi>q</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\mathrm{GF}(q)$</annotation>\\n </semantics></math> for large <math>\\n <semantics>\\n <mrow>\\n <mi>q</mi>\\n </mrow>\\n <annotation> $q$</annotation>\\n </semantics></math> as long as the <i>large</i> independent sets of the graphs are only the ones that are known.</p>\",\"PeriodicalId\":15389,\"journal\":{\"name\":\"Journal of Combinatorial Designs\",\"volume\":\"31 4\",\"pages\":\"179-204\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-02-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Designs\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21875\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Designs","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21875","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the chromatic number of some generalized Kneser graphs
We determine the chromatic number of the Kneser graph of flags of vectorial type of a rank 7 vector space over the finite field for large and describe the colorings that attain the bound. This result relies heavily, not only on the independence number, but also on the structure of all large independent sets. Furthermore, our proof is more general in the following sense: it provides the chromatic number of the Kneser graphs of flags of vectorial type of a rank vector space over for large as long as the large independent sets of the graphs are only the ones that are known.
期刊介绍：
The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including:
block designs, t-designs, pairwise balanced designs and group divisible designs
Latin squares, quasigroups, and related algebras
computational methods in design theory
construction methods
applications in computer science, experimental design theory, and coding theory
graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics
finite geometry and its relation with design theory.
algebraic aspects of design theory.
Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.
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