有限射影空间的色指数

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs Pub Date : 2023-06-30 DOI:10.1002/jcd.21904

Lei Xu, Tao Feng

{"title":"有限射影空间的色指数","authors":"Lei Xu, Tao Feng","doi":"10.1002/jcd.21904","DOIUrl":null,"url":null,"abstract":"<p>A line coloring of <math>\n <semantics>\n <mrow>\n <mtext>PG</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{PG}(n,q)$</annotation>\n </semantics></math>, the <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math>-dimensional projective space over GF<math>\n <semantics>\n <mrow>\n <mo>(</mo>\n \n <mi>q</mi>\n \n <mo>)</mo>\n </mrow>\n <annotation> $(q)$</annotation>\n </semantics></math>, is an assignment of colors to all lines of <math>\n <semantics>\n <mrow>\n <mtext>PG</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{PG}(n,q)$</annotation>\n </semantics></math> so that any two lines with the same color do not intersect. The chromatic index of <math>\n <semantics>\n <mrow>\n <mtext>PG</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{PG}(n,q)$</annotation>\n </semantics></math>, denoted by <math>\n <semantics>\n <mrow>\n <mi>χ</mi>\n \n <mo>′</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mtext>PG</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\chi ^{\\prime} (\\text{PG}(n,q))$</annotation>\n </semantics></math>, is the least number of colors for which a coloring of <math>\n <semantics>\n <mrow>\n <mtext>PG</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{PG}(n,q)$</annotation>\n </semantics></math> exists. This paper translates the problem of determining the chromatic index of <math>\n <semantics>\n <mrow>\n <mtext>PG</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{PG}(n,q)$</annotation>\n </semantics></math> to the problem of examining the existences of <math>\n <semantics>\n <mrow>\n <mtext>PG</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mn>3</mn>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{PG}(3,q)$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mtext>PG</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mn>4</mn>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{PG}(4,q)$</annotation>\n </semantics></math> with certain properties. In particular, it is shown that for any odd integer <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>q</mi>\n \n <mo>∈</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mn>3</mn>\n \n <mo>,</mo>\n \n <mn>4</mn>\n \n <mo>,</mo>\n \n <mn>8</mn>\n \n <mo>,</mo>\n \n <mn>16</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n \n <mo>,</mo>\n \n <mi>χ</mi>\n \n <mo>′</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mtext>PG</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msup>\n <mi>q</mi>\n \n <mi>n</mi>\n </msup>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∕</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>q</mi>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $q\\in \\{3,4,8,16\\},\\chi ^{\\prime} (\\text{PG}(n,q))=({q}^{n}-1)\\unicode{x02215}(q-1)$</annotation>\n </semantics></math>, which implies the existence of a parallelism of <math>\n <semantics>\n <mrow>\n <mtext>PG</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>q</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{PG}(n,q)$</annotation>\n </semantics></math> for any odd integer <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n <mi>q</mi>\n \n <mo>∈</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mn>3</mn>\n \n <mo>,</mo>\n \n <mn>4</mn>\n \n <mo>,</mo>\n \n <mn>8</mn>\n \n <mo>,</mo>\n \n <mn>16</mn>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n <annotation> $q\\in \\{3,4,8,16\\}$</annotation>\n </semantics></math>.</p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 9","pages":"432-446"},"PeriodicalIF":0.5000,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The chromatic index of finite projective spaces\",\"authors\":\"Lei Xu, Tao Feng\",\"doi\":\"10.1002/jcd.21904\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A line coloring of <math>\\n <semantics>\\n <mrow>\\n <mtext>PG</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>q</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{PG}(n,q)$</annotation>\\n </semantics></math>, the <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math>-dimensional projective space over GF<math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n \\n <mi>q</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n <annotation> $(q)$</annotation>\\n </semantics></math>, is an assignment of colors to all lines of <math>\\n <semantics>\\n <mrow>\\n <mtext>PG</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>q</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{PG}(n,q)$</annotation>\\n </semantics></math> so that any two lines with the same color do not intersect. The chromatic index of <math>\\n <semantics>\\n <mrow>\\n <mtext>PG</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>q</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{PG}(n,q)$</annotation>\\n </semantics></math>, denoted by <math>\\n <semantics>\\n <mrow>\\n <mi>χ</mi>\\n \\n <mo>′</mo>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mtext>PG</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>q</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\chi ^{\\\\prime} (\\\\text{PG}(n,q))$</annotation>\\n </semantics></math>, is the least number of colors for which a coloring of <math>\\n <semantics>\\n <mrow>\\n <mtext>PG</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>q</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{PG}(n,q)$</annotation>\\n </semantics></math> exists. This paper translates the problem of determining the chromatic index of <math>\\n <semantics>\\n <mrow>\\n <mtext>PG</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>q</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{PG}(n,q)$</annotation>\\n </semantics></math> to the problem of examining the existences of <math>\\n <semantics>\\n <mrow>\\n <mtext>PG</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mn>3</mn>\\n \\n <mo>,</mo>\\n \\n <mi>q</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{PG}(3,q)$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <mtext>PG</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mn>4</mn>\\n \\n <mo>,</mo>\\n \\n <mi>q</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{PG}(4,q)$</annotation>\\n </semantics></math> with certain properties. In particular, it is shown that for any odd integer <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <mi>q</mi>\\n \\n <mo>∈</mo>\\n \\n <mrow>\\n <mo>{</mo>\\n \\n <mrow>\\n <mn>3</mn>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n \\n <mo>,</mo>\\n \\n <mn>8</mn>\\n \\n <mo>,</mo>\\n \\n <mn>16</mn>\\n </mrow>\\n \\n <mo>}</mo>\\n </mrow>\\n \\n <mo>,</mo>\\n \\n <mi>χ</mi>\\n \\n <mo>′</mo>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mtext>PG</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>q</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>=</mo>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <msup>\\n <mi>q</mi>\\n \\n <mi>n</mi>\\n </msup>\\n \\n <mo>−</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>∕</mo>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>q</mi>\\n \\n <mo>−</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $q\\\\in \\\\{3,4,8,16\\\\},\\\\chi ^{\\\\prime} (\\\\text{PG}(n,q))=({q}^{n}-1)\\\\unicode{x02215}(q-1)$</annotation>\\n </semantics></math>, which implies the existence of a parallelism of <math>\\n <semantics>\\n <mrow>\\n <mtext>PG</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <mi>n</mi>\\n \\n <mo>,</mo>\\n \\n <mi>q</mi>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{PG}(n,q)$</annotation>\\n </semantics></math> for any odd integer <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <mi>q</mi>\\n \\n <mo>∈</mo>\\n \\n <mrow>\\n <mo>{</mo>\\n \\n <mrow>\\n <mn>3</mn>\\n \\n <mo>,</mo>\\n \\n <mn>4</mn>\\n \\n <mo>,</mo>\\n \\n <mn>8</mn>\\n \\n <mo>,</mo>\\n \\n <mn>16</mn>\\n </mrow>\\n \\n <mo>}</mo>\\n </mrow>\\n </mrow>\\n <annotation> $q\\\\in \\\\{3,4,8,16\\\\}$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":15389,\"journal\":{\"name\":\"Journal of Combinatorial Designs\",\"volume\":\"31 9\",\"pages\":\"432-446\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-06-30\",\"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.21904\",\"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.21904","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

PG（n，q）的一个线性着色$\text{PG}（n，q）$上的n$n$维投影空间，是PG（n，q）的所有行的颜色分配$\text｛PG｝（n，q）$，使得任何两条具有相同颜色的线都不相交。PG（n，q）的色指数$\text｛PG｝（n，q）$，用χ′（PG（n，q）$\chi^｛\prime｝（\text｛PG｝（n，q））$，是PG（n，q）的着色所针对的颜色的最少数目$\text｛PG｝（n，q）$存在。本文讨论了PG色指数的确定问题（n，q）$\text｛PG｝（n，q）$到检验PG（3，q）$\text｛PG｝（3，q）$和PG（4，q）$\text｛PG｝（4，q）$属性。特别地，证明了对于任意奇整数n$n$和q∈｛3、4、8、16}，χ′（PG（n，q）=（q n−1）/（q−1）$q\in\｛3,4,8,16\｝，\chi^｛\prime｝（\text｛PG｝（n，q））=（｛q｝^{n}-1)\unicode{x02215}（q-1）$，这意味着PG（n$\text｛PG｝（n，q）$对于任何奇整数n$n$和q∈｛3、4、8、16$q\in\{3,4,8,16\}$。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

The chromatic index of finite projective spaces

A line coloring of $PG (n, q)$ , the $n$ -dimensional projective space over GF $(q)$ , is an assignment of colors to all lines of $PG (n, q)$ so that any two lines with the same color do not intersect. The chromatic index of $PG (n, q)$ , denoted by $χ' (PG (n, q))$ , is the least number of colors for which a coloring of $PG (n, q)$ exists. This paper translates the problem of determining the chromatic index of $PG (n, q)$ to the problem of examining the existences of $PG (3, q)$ and $PG (4, q)$ with certain properties. In particular, it is shown that for any odd integer $n$ and $q \in {3, 4, 8, 16}, χ' (PG (n, q)) = (q^{n} - 1) ∕ (q - 1)$ , which implies the existence of a parallelism of $PG (n, q)$ for any odd integer $n$ and $q \in {3, 4, 8, 16}$ .

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Journal of Combinatorial Designs 数学-数学

CiteScore

1.60

自引率

14.30%

发文量

审稿时长

>12 weeks

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.