{"title":"On \n \n \n λ\n \n $\\lambda $\n -backbone coloring of cliques with tree backbones in linear time","authors":"Krzysztof Michalik, Krzysztof Turowski","doi":"10.1002/jgt.23108","DOIUrl":null,"url":null,"abstract":"<p>A <span></span><math>\n <semantics>\n <mrow>\n <mi>λ</mi>\n </mrow>\n <annotation> $\\lambda $</annotation>\n </semantics></math>-backbone coloring of a graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> with its subgraph (also called a <i>backbone</i>) <span></span><math>\n <semantics>\n <mrow>\n <mi>H</mi>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> is a function <span></span><math>\n <semantics>\n <mrow>\n <mi>c</mi>\n \n <mo>:</mo>\n \n <mi>V</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>→</mo>\n <mrow>\n <mo>{</mo>\n <mrow>\n <mn>1</mn>\n \n <mo>,</mo>\n \n <mi>…</mi>\n \n <mo>,</mo>\n \n <mi>k</mi>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n </mrow>\n <annotation> $c:V(G)\\to \\{1,\\ldots ,k\\}$</annotation>\n </semantics></math> ensuring that <span></span><math>\n <semantics>\n <mrow>\n <mi>c</mi>\n </mrow>\n <annotation> $c$</annotation>\n </semantics></math> is a proper coloring of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> and for each <span></span><math>\n <semantics>\n <mrow>\n <mrow>\n <mo>{</mo>\n <mrow>\n <mi>u</mi>\n \n <mo>,</mo>\n \n <mi>v</mi>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n \n <mo>∈</mo>\n \n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>H</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\{u,v\\}\\in E(H)$</annotation>\n </semantics></math> it holds that <span></span><math>\n <semantics>\n <mrow>\n <mo>|</mo>\n \n <mi>c</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>u</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>−</mo>\n \n <mi>c</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>v</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>|</mo>\n \n <mo>≥</mo>\n \n <mi>λ</mi>\n </mrow>\n <annotation> $|c(u)-c(v)|\\ge \\lambda $</annotation>\n </semantics></math>. In this paper we propose a way to color cliques with tree and forest backbones in linear time that the largest color does not exceed <span></span><math>\n <semantics>\n <mrow>\n <mi>max</mi>\n <mrow>\n <mo>{</mo>\n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mi>λ</mi>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n \n <mo>+</mo>\n \n <mi>Δ</mi>\n \n <msup>\n <mrow>\n <mo>(</mo>\n \n <mi>H</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mn>2</mn>\n </msup>\n <mrow>\n <mo>⌈</mo>\n <mrow>\n <mi>log</mi>\n \n <mi>n</mi>\n </mrow>\n \n <mo>⌉</mo>\n </mrow>\n </mrow>\n <annotation> $\\max \\{n,2\\lambda \\}+{\\rm{\\Delta }}{(H)}^{2}\\lceil \\mathrm{log}n\\rceil $</annotation>\n </semantics></math>. This result improves on the previously existing approximation algorithms as it is <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>Δ</mi>\n \n <msup>\n <mrow>\n <mo>(</mo>\n \n <mi>H</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mn>2</mn>\n </msup>\n <mrow>\n <mo>⌈</mo>\n <mrow>\n <mi>log</mi>\n \n <mi>n</mi>\n </mrow>\n \n <mo>⌉</mo>\n </mrow>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n <annotation> $({\\rm{\\Delta }}{(H)}^{2}\\lceil \\mathrm{log}n\\rceil )$</annotation>\n </semantics></math>-absolutely approximate, that is, with an additive error over the optimum. We also present an infinite family of trees <span></span><math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math> with <span></span><math>\n <semantics>\n <mrow>\n <mi>Δ</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>T</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <mn>3</mn>\n </mrow>\n <annotation> ${\\rm{\\Delta }}(T)=3$</annotation>\n </semantics></math> for which the coloring of cliques with backbones <span></span><math>\n <semantics>\n <mrow>\n <mi>T</mi>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math> requires at least <span></span><math>\n <semantics>\n <mrow>\n <mi>max</mi>\n <mrow>\n <mo>{</mo>\n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mi>λ</mi>\n </mrow>\n \n <mo>}</mo>\n </mrow>\n \n <mo>+</mo>\n \n <mi>Ω</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>log</mi>\n \n <mi>n</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\max \\{n,2\\lambda \\}+{\\rm{\\Omega }}(\\mathrm{log}n)$</annotation>\n </semantics></math> colors for <span></span><math>\n <semantics>\n <mrow>\n <mi>λ</mi>\n </mrow>\n <annotation> $\\lambda $</annotation>\n </semantics></math> close to <span></span><math>\n <semantics>\n <mrow>\n <mfrac>\n <mi>n</mi>\n \n <mn>2</mn>\n </mfrac>\n </mrow>\n <annotation> $\\frac{n}{2}$</annotation>\n </semantics></math>. The construction draws on the theory of Fibonacci numbers, particularly on Zeckendorf representations.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 1","pages":"5-28"},"PeriodicalIF":0.9000,"publicationDate":"2024-04-29","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.23108","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
A -backbone coloring of a graph with its subgraph (also called a backbone) is a function ensuring that is a proper coloring of and for each it holds that . In this paper we propose a way to color cliques with tree and forest backbones in linear time that the largest color does not exceed . This result improves on the previously existing approximation algorithms as it is -absolutely approximate, that is, with an additive error over the optimum. We also present an infinite family of trees with for which the coloring of cliques with backbones requires at least colors for close to . The construction draws on the theory of Fibonacci numbers, particularly on Zeckendorf representations.
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.
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