Ramsey-type problems on induced covers and induced partitions toward the Gyárfás–Sumner conjecture

IF 0.9 3区 数学 Q2 MATHEMATICS
Shuya Chiba, Michitaka Furuya
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This conjecture gives a solution of a Ramsey-type problem on the chromatic number. For a graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>, the <i>induced SP-cover number</i> <span></span><math>\n <semantics>\n <mrow>\n <mtext>inspc</mtext>\n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{inspc}(G)$</annotation>\n </semantics></math> (resp. the <i>induced SP-partition number</i> <span></span><math>\n <semantics>\n <mrow>\n <mtext>inspp</mtext>\n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\text{inspp}(G)$</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 the minimum cardinality of a family <span></span><math>\n <semantics>\n <mrow>\n <mi>P</mi>\n </mrow>\n <annotation> ${\\mathscr{P}}$</annotation>\n </semantics></math> of induced subgraphs of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> such that each element of <span></span><math>\n <semantics>\n <mrow>\n <mi>P</mi>\n </mrow>\n <annotation> ${\\mathscr{P}}$</annotation>\n </semantics></math> is a star or a path and <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mo>⋃</mo>\n <mrow>\n <mi>P</mi>\n \n <mo>∈</mo>\n \n <mi>P</mi>\n </mrow>\n </msub>\n \n <mi>V</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>P</mi>\n \n <mo>)</mo>\n </mrow>\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> ${\\bigcup }_{P\\in {\\mathscr{P}}}V(P)=V(G)$</annotation>\n </semantics></math> (resp. <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mover>\n <mo>⋃</mo>\n \n <mo>˙</mo>\n </mover>\n <mrow>\n <mi>P</mi>\n \n <mo>∈</mo>\n \n <mi>P</mi>\n </mrow>\n </msub>\n \n <mi>V</mi>\n <mrow>\n <mo>(</mo>\n \n <mi>P</mi>\n \n <mo>)</mo>\n </mrow>\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> ${\\dot{\\bigcup }}_{P\\in {\\mathscr{P}}}V(P)=V(G)$</annotation>\n </semantics></math>). Such two invariants are directly related concepts to the chromatic number. From the viewpoint of this fact, we focus on Ramsey-type problems for two invariants <span></span><math>\n <semantics>\n <mrow>\n <mtext>inspc</mtext>\n </mrow>\n <annotation> $\\text{inspc}$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mtext>inspp</mtext>\n </mrow>\n <annotation> $\\text{inspp}$</annotation>\n </semantics></math>, which are analogies of the Gyárfás-Sumner conjecture, and settle them. As a corollary of our results, we also settle other Ramsey-type problems for widely studied invariants.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"419-441"},"PeriodicalIF":0.9000,"publicationDate":"2024-06-06","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.23124","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

Gyárfás and Sumner independently conjectured that for every tree T $T$ , there exists a function f T : N N ${f}_{T}:{\mathbb{N}}\to {\mathbb{N}}$ such that every T $T$ -free graph G $G$ satisfies χ ( G ) f T ( ω ( G ) ) $\chi (G)\le {f}_{T}(\omega (G))$ , where χ ( G ) $\chi (G)$ and ω ( G ) $\omega (G)$ are the chromatic number and the clique number of G $G$ , respectively. This conjecture gives a solution of a Ramsey-type problem on the chromatic number. For a graph G $G$ , the induced SP-cover number inspc ( G ) $\text{inspc}(G)$ (resp. the induced SP-partition number inspp ( G ) $\text{inspp}(G)$ ) of G $G$ is the minimum cardinality of a family P ${\mathscr{P}}$ of induced subgraphs of G $G$ such that each element of P ${\mathscr{P}}$ is a star or a path and P P V ( P ) = V ( G ) ${\bigcup }_{P\in {\mathscr{P}}}V(P)=V(G)$ (resp. ˙ P P V ( P ) = V ( G ) ${\dot{\bigcup }}_{P\in {\mathscr{P}}}V(P)=V(G)$ ). Such two invariants are directly related concepts to the chromatic number. From the viewpoint of this fact, we focus on Ramsey-type problems for two invariants inspc $\text{inspc}$ and inspp $\text{inspp}$ , which are analogies of the Gyárfás-Sumner conjecture, and settle them. As a corollary of our results, we also settle other Ramsey-type problems for widely studied invariants.

走向 Gyárfás-Sumner 猜想的诱导盖和诱导分区上的拉姆齐型问题
Gyárfás 和 Sumner 独立猜想,对于每一棵树 T $T$ ,存在一个函数 f T : N → N ${f}_{T}:{\mathbb{N}}\to {\mathbb{N}}$ ,使得每一个 T $T$ -free graph G $G$ 满足 χ ( G ) ≤ f T ( ω ( G ) ) $\chi (G)\le {f}_{T}(\omega (G))$ ,其中 χ ( G ) $\chi (G)$ 和 ω ( G ) $\omega (G)$ 分别是 G $G$ 的色度数和小群数。这个猜想给出了关于色度数的拉姆齐式问题的解。对于图 G $G$,G $G$的诱导 SP-cover 数 inspc ( G ) $\text{inspc}(G)$ (或者诱导 SP-partition 数 inspp ( G ) $\text{inspp}(G)$ )是 G $G$ 的诱导子图的族 P ${\mathscr{P}}$ 的最小卡片度,使得 P ${\mathscr{P}}$ 的每个元素都是星或路径,并且⋃ P ∈ P V ( P ) = V ( G ) ${bigcup }_{P\in {\mathscr{P}}}V(P)=V(G)$ (或者诱导 SP-partition 数 inspp ( G ) $\text{inspp}(G)$ )是 G $G$ 的诱导子图的最小卡片度。
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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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