{"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00373-024-02752-1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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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。
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