Flexible list colorings: Maximizing the number of requests satisfied

IF 0.9 3区 数学 Q2 MATHEMATICS
Hemanshu Kaul, Rogers Mathew, Jeffrey A. Mudrock, Michael J. Pelsmajer
{"title":"Flexible list colorings: Maximizing the number of requests satisfied","authors":"Hemanshu Kaul,&nbsp;Rogers Mathew,&nbsp;Jeffrey A. Mudrock,&nbsp;Michael J. Pelsmajer","doi":"10.1002/jgt.23103","DOIUrl":null,"url":null,"abstract":"<p>Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose <span></span><math>\n \n <mrow>\n <mn>0</mn>\n \n <mo>≤</mo>\n \n <mi>ϵ</mi>\n \n <mo>≤</mo>\n \n <mn>1</mn>\n </mrow></math>, <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> is a graph, <span></span><math>\n \n <mrow>\n <mi>L</mi>\n </mrow></math> is a list assignment for <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math>, and <span></span><math>\n \n <mrow>\n <mi>r</mi>\n </mrow></math> is a function with nonempty domain <span></span><math>\n \n <mrow>\n <mi>D</mi>\n \n <mo>⊆</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> such that <span></span><math>\n \n <mrow>\n <mi>r</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>v</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∈</mo>\n \n <mi>L</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>v</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> for each <span></span><math>\n \n <mrow>\n <mi>v</mi>\n \n <mo>∈</mo>\n \n <mi>D</mi>\n </mrow></math> (<span></span><math>\n \n <mrow>\n <mi>r</mi>\n </mrow></math> is called a request of <span></span><math>\n \n <mrow>\n <mi>L</mi>\n </mrow></math>). The triple <span></span><math>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>G</mi>\n \n <mo>,</mo>\n \n <mi>L</mi>\n \n <mo>,</mo>\n \n <mi>r</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> is <span></span><math>\n \n <mrow>\n <mi>ϵ</mi>\n </mrow></math>-satisfiable if there exists a proper <span></span><math>\n \n <mrow>\n <mi>L</mi>\n </mrow></math>-coloring <span></span><math>\n \n <mrow>\n <mi>f</mi>\n </mrow></math> of <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> such that <span></span><math>\n \n <mrow>\n <mi>f</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>v</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <mi>r</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>v</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> for at least <span></span><math>\n \n <mrow>\n <mi>ϵ</mi>\n \n <mo>∣</mo>\n \n <mi>D</mi>\n \n <mo>∣</mo>\n </mrow></math> vertices in <span></span><math>\n \n <mrow>\n <mi>D</mi>\n </mrow></math>. We say <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> is <span></span><math>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>ϵ</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow></math>-flexible if <span></span><math>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>G</mi>\n \n <mo>,</mo>\n \n <msup>\n <mi>L</mi>\n \n <mo>′</mo>\n </msup>\n \n <mo>,</mo>\n \n <msup>\n <mi>r</mi>\n \n <mo>′</mo>\n </msup>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> is <span></span><math>\n \n <mrow>\n <mi>ϵ</mi>\n </mrow></math>-satisfiable whenever <span></span><math>\n \n <mrow>\n <msup>\n <mi>L</mi>\n \n <mo>′</mo>\n </msup>\n </mrow></math> is a <span></span><math>\n \n <mrow>\n <mi>k</mi>\n </mrow></math>-assignment for <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> and <span></span><math>\n \n <mrow>\n <msup>\n <mi>r</mi>\n \n <mo>′</mo>\n </msup>\n </mrow></math> is a request of <span></span><math>\n \n <mrow>\n <msup>\n <mi>L</mi>\n \n <mo>′</mo>\n </msup>\n </mrow></math>. It was shown by Dvořák et al. that if <span></span><math>\n \n <mrow>\n <mi>d</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow></math> is prime, <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> is a <span></span><math>\n \n <mrow>\n <mi>d</mi>\n </mrow></math>-degenerate graph, and <span></span><math>\n \n <mrow>\n <mi>r</mi>\n </mrow></math> is a request for <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> with domain of size 1, then <span></span><math>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>G</mi>\n \n <mo>,</mo>\n \n <mi>L</mi>\n \n <mo>,</mo>\n \n <mi>r</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> is 1-satisfiable whenever <span></span><math>\n \n <mrow>\n <mi>L</mi>\n </mrow></math> is a <span></span><math>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>d</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow></math>-assignment. In this paper, we extend this result to all <span></span><math>\n \n <mrow>\n <mi>d</mi>\n </mrow></math> for bipartite <span></span><math>\n \n <mrow>\n <mi>d</mi>\n </mrow></math>-degenerate graphs.</p><p>The literature on flexible list coloring tends to focus on showing that for a fixed graph <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> and <span></span><math>\n \n <mrow>\n <mi>k</mi>\n \n <mo>∈</mo>\n \n <mi>N</mi>\n </mrow></math> there exists an <span></span><math>\n \n <mrow>\n <mi>ϵ</mi>\n \n <mo>&gt;</mo>\n \n <mn>0</mn>\n </mrow></math> such that <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> is <span></span><math>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>ϵ</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow></math>-flexible, but it is natural to try to find the largest possible <span></span><math>\n \n <mrow>\n <mi>ϵ</mi>\n </mrow></math> for which <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> is <span></span><math>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>ϵ</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow></math>-flexible. In this vein, we improve a result of Dvořák et al., by showing <span></span><math>\n \n <mrow>\n <mi>d</mi>\n </mrow></math>-degenerate graphs are <span></span><math>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>d</mi>\n \n <mo>+</mo>\n \n <mn>2</mn>\n \n <mo>,</mo>\n \n <mn>1</mn>\n \n <mo>∕</mo>\n \n <msup>\n <mn>2</mn>\n \n <mrow>\n <mi>d</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msup>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow></math>-flexible. In pursuit of the largest <span></span><math>\n \n <mrow>\n <mi>ϵ</mi>\n </mrow></math> for which a graph is <span></span><math>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>ϵ</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow></math>-flexible, we observe that a graph <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> is not <span></span><math>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>k</mi>\n \n <mo>,</mo>\n \n <mi>ϵ</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow></math>-flexible for any <span></span><math>\n \n <mrow>\n <mi>k</mi>\n </mrow></math> if and only if <span></span><math>\n \n <mrow>\n <mi>ϵ</mi>\n \n <mo>&gt;</mo>\n \n <mn>1</mn>\n \n <mo>∕</mo>\n \n <mi>ρ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math>, where <span></span><math>\n \n <mrow>\n <mi>ρ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow></math> is the Hall ratio of <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math>, and we initiate the study of the <i>list flexibility number of a graph</i> <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math>, which is the smallest <span></span><math>\n \n <mrow>\n <mi>k</mi>\n </mrow></math> such that <span></span><math>\n \n <mrow>\n <mi>G</mi>\n </mrow></math> is <span></span><math>\n \n <mrow>\n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>k</mi>\n \n <mo>,</mo>\n \n <mn>1</mn>\n \n <mo>∕</mo>\n \n <mi>ρ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow></math>-flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 4","pages":"887-906"},"PeriodicalIF":0.9000,"publicationDate":"2024-04-18","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.23103","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

Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose 0 ϵ 1 , G is a graph, L is a list assignment for G , and r is a function with nonempty domain D V ( G ) such that r ( v ) L ( v ) for each v D ( r is called a request of L ). The triple ( G , L , r ) is ϵ -satisfiable if there exists a proper L -coloring f of G such that f ( v ) = r ( v ) for at least ϵ D vertices in D . We say G is ( k , ϵ ) -flexible if ( G , L , r ) is ϵ -satisfiable whenever L is a k -assignment for G and r is a request of L . It was shown by Dvořák et al. that if d + 1 is prime, G is a d -degenerate graph, and r is a request for G with domain of size 1, then ( G , L , r ) is 1-satisfiable whenever L is a ( d + 1 ) -assignment. In this paper, we extend this result to all d for bipartite d -degenerate graphs.

The literature on flexible list coloring tends to focus on showing that for a fixed graph G and k N there exists an ϵ > 0 such that G is ( k , ϵ ) -flexible, but it is natural to try to find the largest possible ϵ for which G is ( k , ϵ ) -flexible. In this vein, we improve a result of Dvořák et al., by showing d -degenerate graphs are ( d + 2 , 1 2 d + 1 ) -flexible. In pursuit of the largest ϵ for which a graph is ( k , ϵ ) -flexible, we observe that a graph G is not ( k , ϵ ) -flexible for any k if and only if ϵ > 1 ρ ( G ) , where ρ ( G ) is the Hall ratio of G , and we initiate the study of the list flexibility number of a graph G , which is the smallest k such that G is ( k , 1 ρ ( G ) ) -flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.

灵活的列表着色:最大限度地满足请求数量
灵活列表着色由 Dvořák、Norin 和 Postle 于 2019 年提出。假设 ,是一个图,是对 ,的列表赋值,并且是一个具有非空域的函数,这样对于每个 ( 称为 )的请求。如果存在一个适当的 ,且至少对...中的顶点而言,...是可满足的,那么这个三元组就是...可满足的。德沃夏克(Dvořák)等人曾证明,如果是质数,是退化图,并且是域大小为 1 的请求,那么只要是分配,就是可满足的。在本文中,我们将这一结果扩展到所有双向-退化图。关于灵活列表着色的文献往往侧重于证明对于一个固定的图,存在一个这样的-灵活,但很自然的是,我们试图找到最大可能的-灵活。为此,我们改进了德沃夏克等人的一项成果,证明了-退化图是-灵活的。在追求图的最大-柔性时,我们观察到,当且仅当 ,为 ,的霍尔比时,图对于任何都不是-柔性的,因此我们开始研究图的列表柔性数,它是-柔性的最小值。我们研究了图的列表柔性数、列表色度数、列表包装数和退化性之间的关系和联系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Graph Theory
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 .
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