Local version of Vizing's theorem for multigraphs

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-07-16 DOI:10.1002/jgt.23155

Clinton T. Conley, Jan Grebík, Oleg Pikhurko

{"title":"Local version of Vizing's theorem for multigraphs","authors":"Clinton T. Conley, Jan Grebík, Oleg Pikhurko","doi":"10.1002/jgt.23155","DOIUrl":null,"url":null,"abstract":"Extending a result of Christiansen, we prove that every multigraph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n \n <mo>=</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>V</mi>\n \n <mo>,</mo>\n \n <mi>E</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $G=(V,E)$</annotation>\n </semantics></math> admits a proper edge colouring <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ϕ</mi>\n \n <mo>:</mo>\n \n <mi>E</mi>\n \n <mo>→</mo>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mn>1</mn>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mo>,</mo>\n \n <mo>…</mo>\n <mspace></mspace>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $\\phi :E\\to \\{1,2,\\ldots \\,\\}$</annotation>\n </semantics></math> which is local, that is, <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ϕ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>e</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>⩽</mo>\n \n <mi>max</mi>\n \n <mrow>\n <mo>{</mo>\n \n <mrow>\n <mi>d</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>x</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>+</mo>\n \n <mi>π</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>x</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>,</mo>\n \n <mi>d</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>y</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>+</mo>\n \n <mi>π</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>y</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $\\phi (e)\\leqslant \\max \\{d(x)+\\pi (x),d(y)+\\pi (y)\\}$</annotation>\n </semantics></math> for every edge <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>e</mi>\n </mrow>\n </mrow>\n <annotation> $e$</annotation>\n </semantics></math> with end-points <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>x</mi>\n \n <mo>,</mo>\n \n <mi>y</mi>\n \n <mo>∈</mo>\n \n <mi>V</mi>\n </mrow>\n </mrow>\n <annotation> $x,y\\in V$</annotation>\n </semantics></math>, where <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>d</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>z</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $d(z)$</annotation>\n </semantics></math> (resp. <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>π</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>z</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $\\pi (z)$</annotation>\n </semantics></math>) denotes the degree of a vertex <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>z</mi>\n </mrow>\n </mrow>\n <annotation> $z$</annotation>\n </semantics></math> (resp. the maximum edge multiplicity at <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>z</mi>\n </mrow>\n </mrow>\n <annotation> $z$</annotation>\n </semantics></math>). This is derived from a local version of the Fan Equation.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 4","pages":"693-701"},"PeriodicalIF":0.9000,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23155","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23155","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

Extending a result of Christiansen, we prove that every multigraph $G = (V, E)$ admits a proper edge colouring $ϕ : E \to {1, 2, \dots}$ which is local, that is, $ϕ (e) ⩽ \max {d (x) + π (x), d (y) + π (y)}$ for every edge $e$ with end-points $x, y \in V$ , where $d (z)$ (resp. $π (z)$ ) denotes the degree of a vertex $z$ (resp. the maximum edge multiplicity at $z$ ). This is derived from a local version of the Fan Equation.

查看原文本刊更多论文

多图维京定理的局部版本

通过扩展克里斯蒂安森的一个结果，我们证明了每一个多图都有一个适当的边着色，这个边着色是局部的，也就是说，对于每一条有端点的边，（resp. ）表示顶点的度数（resp. 最大边乘）。这是从范式方程的局部版本中推导出来的。

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

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 .