On two problems of defective choosability of graphs

IF 0.9 3区 数学 Q2 MATHEMATICS
Jie Ma, Rongxing Xu, Xuding Zhu
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In particular, <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>k</mi>\n <mo>,</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mi>k</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation> $(k,0,k)$</annotation>\n </semantics></math>-choosable means <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-colorable, <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>k</mi>\n <mo>,</mo>\n <mn>0</mn>\n <mo>,</mo>\n <mo>+</mo>\n <mi>∞</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation> $(k,0,+\\infty )$</annotation>\n </semantics></math>-choosable means <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-choosable and <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>k</mi>\n <mo>,</mo>\n <mi>d</mi>\n <mo>,</mo>\n <mo>+</mo>\n <mi>∞</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation> $(k,d,+\\infty )$</annotation>\n </semantics></math>-choosable means <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n </mrow>\n <annotation> $d$</annotation>\n </semantics></math>-defective <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-choosable. This paper proves that there are 1-defective 3-choosable planar graphs that are not 4-choosable, and for any positive integers <span></span><math>\n <semantics>\n <mrow>\n <mi>ℓ</mi>\n <mo>≥</mo>\n <mi>k</mi>\n <mo>≥</mo>\n <mn>3</mn>\n </mrow>\n <annotation> $\\ell \\ge k\\ge 3$</annotation>\n </semantics></math>, and nonnegative integer <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n </mrow>\n <annotation> $d$</annotation>\n </semantics></math>, there are <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>k</mi>\n <mo>,</mo>\n <mi>d</mi>\n <mo>,</mo>\n <mi>ℓ</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation> $(k,d,\\ell )$</annotation>\n </semantics></math>-choosable graphs that are not <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>k</mi>\n <mo>,</mo>\n <mi>d</mi>\n <mo>,</mo>\n <mi>ℓ</mi>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation> $(k,d,\\ell +1)$</annotation>\n </semantics></math>-choosable. These results answer questions asked by Wang and Xu, and Kang, respectively. Our construction of <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>k</mi>\n <mo>,</mo>\n <mi>d</mi>\n <mo>,</mo>\n <mi>ℓ</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation> $(k,d,\\ell )$</annotation>\n </semantics></math>-choosable but not <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>k</mi>\n <mo>,</mo>\n <mi>d</mi>\n <mo>,</mo>\n <mi>ℓ</mi>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n <mo>)</mo>\n </mrow>\n <annotation> $(k,d,\\ell +1)$</annotation>\n </semantics></math>-choosable graphs generalizes the construction of Král' and Sgall for the case <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n <annotation> $d=0$</annotation>\n </semantics></math>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 2","pages":"313-324"},"PeriodicalIF":0.9000,"publicationDate":"2024-09-23","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.23182","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

Given positive integers p k $p\ge k$ , and a nonnegative integer d $d$ , we say a graph G $G$ is ( k , d , p ) $(k,d,p)$ -choosable if for every list assignment L $L$ with L ( v ) k $| L(v)| \ge k$ for each v V ( G ) $v\in V(G)$ and v V ( G ) L ( v ) p $| {\bigcup }_{v\in V(G)}L(v)| \le p$ , there exists an L $L$ -coloring of G $G$ such that each monochromatic subgraph has maximum degree at most d $d$ . In particular, ( k , 0 , k ) $(k,0,k)$ -choosable means k $k$ -colorable, ( k , 0 , + ) $(k,0,+\infty )$ -choosable means k $k$ -choosable and ( k , d , + ) $(k,d,+\infty )$ -choosable means d $d$ -defective k $k$ -choosable. This paper proves that there are 1-defective 3-choosable planar graphs that are not 4-choosable, and for any positive integers k 3 $\ell \ge k\ge 3$ , and nonnegative integer d $d$ , there are ( k , d , ) $(k,d,\ell )$ -choosable graphs that are not ( k , d , + 1 ) $(k,d,\ell +1)$ -choosable. These results answer questions asked by Wang and Xu, and Kang, respectively. Our construction of ( k , d , ) $(k,d,\ell )$ -choosable but not ( k , d , + 1 ) $(k,d,\ell +1)$ -choosable graphs generalizes the construction of Král' and Sgall for the case d = 0 $d=0$ .

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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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