Kempe equivalent list colorings revisited

IF 0.9 3区 数学 Q2 MATHEMATICS
Dibyayan Chakraborty, Carl Feghali, Reem Mahmoud
{"title":"Kempe equivalent list colorings revisited","authors":"Dibyayan Chakraborty,&nbsp;Carl Feghali,&nbsp;Reem Mahmoud","doi":"10.1002/jgt.23142","DOIUrl":null,"url":null,"abstract":"<p>A <i>Kempe chain</i> on colors <span></span><math>\n <semantics>\n <mrow>\n <mi>a</mi>\n </mrow>\n <annotation> $a$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>b</mi>\n </mrow>\n <annotation> $b$</annotation>\n </semantics></math> is a component of the subgraph induced by colors <span></span><math>\n <semantics>\n <mrow>\n <mi>a</mi>\n </mrow>\n <annotation> $a$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>b</mi>\n </mrow>\n <annotation> $b$</annotation>\n </semantics></math>. A <i>Kempe change</i> is the operation of interchanging the colors of some Kempe chains. For a list-assignment <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math> and an <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math>-coloring <span></span><math>\n <semantics>\n <mrow>\n <mi>φ</mi>\n </mrow>\n <annotation> $\\varphi $</annotation>\n </semantics></math>, a Kempe change is <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math>-<i>valid</i> for <span></span><math>\n <semantics>\n <mrow>\n <mi>φ</mi>\n </mrow>\n <annotation> $\\varphi $</annotation>\n </semantics></math> if performing the Kempe change yields another <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math>-coloring. Two <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math>-colorings are <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math>-<i>equivalent</i> if we can form one from the other by a sequence of <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math>-valid Kempe changes. A <i>degree-assignment</i> is a list-assignment <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math> such that <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>v</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≥</mo>\n \n <mi>d</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>v</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $L(v)\\ge d(v)$</annotation>\n </semantics></math> for every <span></span><math>\n <semantics>\n <mrow>\n <mi>v</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 </mrow>\n <annotation> $v\\in V(G)$</annotation>\n </semantics></math>. Cranston and Mahmoud asked: For which graphs <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> and degree-assignment <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is it true that all the <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math>-colorings of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> are <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math>-equivalent? We prove that for every 4-connected graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> which is not complete and every degree-assignment <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>, all <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math>-colorings of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> are <span></span><math>\n <semantics>\n <mrow>\n <mi>L</mi>\n </mrow>\n <annotation> $L$</annotation>\n </semantics></math>-equivalent.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"410-418"},"PeriodicalIF":0.9000,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23142","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23142","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 Kempe chain on colors a $a$ and b $b$ is a component of the subgraph induced by colors a $a$ and b $b$ . A Kempe change is the operation of interchanging the colors of some Kempe chains. For a list-assignment L $L$ and an L $L$ -coloring φ $\varphi $ , a Kempe change is L $L$ -valid for φ $\varphi $ if performing the Kempe change yields another L $L$ -coloring. Two L $L$ -colorings are L $L$ -equivalent if we can form one from the other by a sequence of L $L$ -valid Kempe changes. A degree-assignment is a list-assignment L $L$ such that L ( v ) d ( v ) $L(v)\ge d(v)$ for every v V ( G ) $v\in V(G)$ . Cranston and Mahmoud asked: For which graphs G $G$ and degree-assignment L $L$ of G $G$ is it true that all the L $L$ -colorings of G $G$ are L $L$ -equivalent? We prove that for every 4-connected graph G $G$ which is not complete and every degree-assignment L $L$ of G $G$ , all L $L$ -colorings of G $G$ are L $L$ -equivalent.

Abstract Image

Kempe 等价表着色再探讨
颜色 a $a$ 和 b $b$ 上的 Kempe 链是颜色 a $a$ 和 b $b$ 诱导的子图的一个组成部分。Kempe 变化是交换某些 Kempe 链颜色的操作。对于一个列表分配 L $L$ 和一个 L $L$ 颜色 φ $\varphi $,如果进行 Kempe 更改能得到另一个 L $L$ 颜色,则 Kempe 更改对 φ $\varphi $ 是 L $L$ 有效的。如果我们可以通过一连串 L $L$ 有效的 Kempe 变换从另一个 L $L$ 着色中得到一个 L $L$ 着色,那么这两个 L $L$ 着色就是 L $L$ 等价的。度赋值是一个列表赋值 L $L$,对于每个 v∈ V ( G ) $v\in V(G)$ 来说,L ( v ) ≥ d ( v ) $L(v)\ge d(v)$ 。克兰斯顿和马哈茂德问对于哪些图 G $G$ 和 G $G$ 的度数赋值 L $L$ 来说,G $G$ 的所有 L $L$ -着色都是 L $L$ -等价的?我们证明,对于每一个不完整的四连图 G $G$ 和 G $G$ 的每一个度数分配 L $L$, G $G$ 的所有 L $L$ -着色都是 L $L$ -等价的。
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
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 .
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信