Local version of Vizing's theorem for multigraphs

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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"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":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23155","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

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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.

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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。

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