Kempe 等价表着色再探讨

IF 0.9 3区 数学 Q2 MATHEMATICS Journal of Graph Theory Pub Date : 2024-06-04 DOI:10.1002/jgt.23142
Dibyayan Chakraborty, Carl Feghali, Reem Mahmoud
{"title":"Kempe 等价表着色再探讨","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":"{\"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? 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引用次数: 0

摘要

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。
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Kempe equivalent list colorings revisited

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.

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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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