{"title":"Infinite Ramsey-Minimal Graphs for Star Forests","authors":"Fawwaz Fakhrurrozi Hadiputra, Valentino Vito","doi":"10.1007/s00373-024-02752-1","DOIUrl":null,"url":null,"abstract":"<p>For graphs <i>F</i>, <i>G</i>, and <i>H</i>, we write <span>\\(F \\rightarrow (G,H)\\)</span> if every red-blue coloring of the edges of <i>F</i> produces a red copy of <i>G</i> or a blue copy of <i>H</i>. The graph <i>F</i> is said to be (<i>G</i>, <i>H</i>)-minimal if it is subgraph-minimal with respect to this property. The characterization problem for Ramsey-minimal graphs is classically done for finite graphs. In 2021, Barrett and the second author generalized this problem to infinite graphs. They asked which pairs (<i>G</i>, <i>H</i>) admit a Ramsey-minimal graph and which ones do not. We show that any pair of star forests such that at least one of them involves an infinite-star component admits no Ramsey-minimal graph. Also, we construct a Ramsey-minimal graph for a finite star forest versus a subdivision graph. This paper builds upon the results of Burr et al. (Discrete Math 33:227–237, 1981) on Ramsey-minimal graphs for finite star forests.</p>","PeriodicalId":12811,"journal":{"name":"Graphs and Combinatorics","volume":"106 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Graphs and Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00373-024-02752-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For graphs F, G, and H, we write \(F \rightarrow (G,H)\) if every red-blue coloring of the edges of F produces a red copy of G or a blue copy of H. The graph F is said to be (G, H)-minimal if it is subgraph-minimal with respect to this property. The characterization problem for Ramsey-minimal graphs is classically done for finite graphs. In 2021, Barrett and the second author generalized this problem to infinite graphs. They asked which pairs (G, H) admit a Ramsey-minimal graph and which ones do not. We show that any pair of star forests such that at least one of them involves an infinite-star component admits no Ramsey-minimal graph. Also, we construct a Ramsey-minimal graph for a finite star forest versus a subdivision graph. This paper builds upon the results of Burr et al. (Discrete Math 33:227–237, 1981) on Ramsey-minimal graphs for finite star forests.
对于图 F、G 和 H，如果 F 边的每一个红蓝着色都会产生 G 的一个红色副本或 H 的一个蓝色副本，我们就将其写为（F /rightarrow (G,H)/ ）。拉姆齐最小图的表征问题是针对有限图的经典问题。2021 年，巴雷特和第二位作者将这一问题推广到了无限图。他们问哪些图对（G, H）允许有拉姆齐最小图，哪些不允许有拉姆齐最小图。我们证明，任何一对星形森林，只要其中至少有一个涉及无限星形成分，就不会有拉姆齐最小图。此外，我们还为有限星形森林与细分图构建了拉姆齐最小图。本文建立在 Burr 等人（Discrete Math 33:227-237, 1981）关于有限星形林的拉姆齐最小图的研究成果之上。
期刊介绍：
Graphs and Combinatorics is an international journal devoted to research concerning all aspects of combinatorial mathematics. In addition to original research papers, the journal also features survey articles from authors invited by the editorial board.
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