{"title":"拉姆齐树的数量与多本图书的数量","authors":"Xiao-bing Guo, Si-nan Hu, Yue-jian Peng","doi":"10.1007/s10255-024-1117-4","DOIUrl":null,"url":null,"abstract":"<div><p>Given two graphs <i>G</i> and <i>H</i>, the Ramsey number <i>R</i>(<i>G,H</i>) is the minimum integer <i>N</i> such that any two-coloring of the edges of <i>K</i><sub><i>N</i></sub> in red or blue yields a red <i>G</i> or a blue <i>H</i>. Let <i>v</i>(<i>G</i>) be the number of vertices of <i>G</i> and <i>χ</i>(<i>G</i>) be the chromatic number of <i>G</i>. Let <i>s</i>(<i>G</i>) denote the chromatic surplus of <i>G</i>, the number of vertices in a minimum color class among all proper <i>χ</i>(<i>G</i>)-colorings of <i>G</i>. Burr showed that <span>\\(R(G,H) \\ge (v(G) - 1)(\\chi (H) - 1) + s(H)\\)</span> if <i>G</i> is connected and <span>\\(v(G) \\ge s(H)\\)</span>. A connected graph <i>G</i> is <i>H</i>-good if <span>\\(R(G,H) = (v(G) - 1)(\\chi (H) - 1) + s(H)\\)</span>. Let <i>tH</i> denote the disjoint union of <i>t</i> copies of graph <i>H</i>, and let <span>\\(G \\vee H\\)</span> denote the join of <i>G</i> and <i>H</i>. Denote a complete graph on <i>n</i> vertices by <i>K</i><sub><i>n</i></sub>, and a tree on <i>n</i> vertices by <i>T</i><sub><i>n</i></sub>. Denote a book with <i>n</i> pages by <i>B</i><sub><i>n</i></sub>, i.e., the join <span>\\({K_2} \\vee \\overline {{K_n}} \\)</span>. Erdős, Faudree, Rousseau and Schelp proved that <i>T</i><sub><i>n</i></sub> is <i>B</i><sub><i>m</i></sub>-good if <span>\\(n \\ge 3m - 3\\)</span>. In this paper, we obtain the exact Ramsey number of <i>T</i><sub><i>n</i></sub> versus 2<i>B</i><sub>2</sub>- Our result implies that <i>T</i><sub><i>n</i></sub> is 2<i>B</i><sub>2</sub>-good if <i>n</i> ≥ 5.</p></div>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"40 3","pages":"600 - 612"},"PeriodicalIF":0.9000,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Ramsey Numbers of Trees Versus Multiple Copies of Books\",\"authors\":\"Xiao-bing Guo, Si-nan Hu, Yue-jian Peng\",\"doi\":\"10.1007/s10255-024-1117-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Given two graphs <i>G</i> and <i>H</i>, the Ramsey number <i>R</i>(<i>G,H</i>) is the minimum integer <i>N</i> such that any two-coloring of the edges of <i>K</i><sub><i>N</i></sub> in red or blue yields a red <i>G</i> or a blue <i>H</i>. Let <i>v</i>(<i>G</i>) be the number of vertices of <i>G</i> and <i>χ</i>(<i>G</i>) be the chromatic number of <i>G</i>. Let <i>s</i>(<i>G</i>) denote the chromatic surplus of <i>G</i>, the number of vertices in a minimum color class among all proper <i>χ</i>(<i>G</i>)-colorings of <i>G</i>. Burr showed that <span>\\\\(R(G,H) \\\\ge (v(G) - 1)(\\\\chi (H) - 1) + s(H)\\\\)</span> if <i>G</i> is connected and <span>\\\\(v(G) \\\\ge s(H)\\\\)</span>. A connected graph <i>G</i> is <i>H</i>-good if <span>\\\\(R(G,H) = (v(G) - 1)(\\\\chi (H) - 1) + s(H)\\\\)</span>. Let <i>tH</i> denote the disjoint union of <i>t</i> copies of graph <i>H</i>, and let <span>\\\\(G \\\\vee H\\\\)</span> denote the join of <i>G</i> and <i>H</i>. Denote a complete graph on <i>n</i> vertices by <i>K</i><sub><i>n</i></sub>, and a tree on <i>n</i> vertices by <i>T</i><sub><i>n</i></sub>. Denote a book with <i>n</i> pages by <i>B</i><sub><i>n</i></sub>, i.e., the join <span>\\\\({K_2} \\\\vee \\\\overline {{K_n}} \\\\)</span>. Erdős, Faudree, Rousseau and Schelp proved that <i>T</i><sub><i>n</i></sub> is <i>B</i><sub><i>m</i></sub>-good if <span>\\\\(n \\\\ge 3m - 3\\\\)</span>. In this paper, we obtain the exact Ramsey number of <i>T</i><sub><i>n</i></sub> versus 2<i>B</i><sub>2</sub>- Our result implies that <i>T</i><sub><i>n</i></sub> is 2<i>B</i><sub>2</sub>-good if <i>n</i> ≥ 5.</p></div>\",\"PeriodicalId\":6951,\"journal\":{\"name\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"volume\":\"40 3\",\"pages\":\"600 - 612\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-12-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematicae Applicatae Sinica, English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10255-024-1117-4\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematicae Applicatae Sinica, English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10255-024-1117-4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
摘要 给定两个图 G 和 H,拉姆齐数 R(G,H)是最小整数 N,使得 KN 边上的任意两个红色或蓝色的着色都能得到一个红色的 G 或蓝色的 H。伯尔(Burr)证明了如果 G 是连通的,并且 \(v(G) \ge s(H)\) ,则 \(R(G,H) \ge (v(G) - 1)(\chi (H) - 1) + s(H)\) 。如果 \(R(G,H) = (v(G) - 1)(\chi (H) - 1) + s(H)\) ,则连通图 G 是 H-good 的。让 tH 表示图 H 的 t 个副本的不相联,让 \(G \vee H\) 表示 G 和 H 的连接。用 Kn 表示 n 个顶点上的完整图,用 Tn 表示 n 个顶点上的树。用 Bn 表示一本有 n 页的书,即 join ({K_2} \vee \overline {{K_n}} \)。Erdős, Faudree, Rousseau 和 Schelp 证明了如果 \(n \ge 3m - 3\) Tn 是 Bm-good 的。在本文中,我们得到了 Tn 相对于 2B2 的精确拉姆齐数。我们的结果意味着,如果 n ≥ 5,Tn 是 2B2-good 的。
Ramsey Numbers of Trees Versus Multiple Copies of Books
Given two graphs G and H, the Ramsey number R(G,H) is the minimum integer N such that any two-coloring of the edges of KN in red or blue yields a red G or a blue H. Let v(G) be the number of vertices of G and χ(G) be the chromatic number of G. Let s(G) denote the chromatic surplus of G, the number of vertices in a minimum color class among all proper χ(G)-colorings of G. Burr showed that \(R(G,H) \ge (v(G) - 1)(\chi (H) - 1) + s(H)\) if G is connected and \(v(G) \ge s(H)\). A connected graph G is H-good if \(R(G,H) = (v(G) - 1)(\chi (H) - 1) + s(H)\). Let tH denote the disjoint union of t copies of graph H, and let \(G \vee H\) denote the join of G and H. Denote a complete graph on n vertices by Kn, and a tree on n vertices by Tn. Denote a book with n pages by Bn, i.e., the join \({K_2} \vee \overline {{K_n}} \). Erdős, Faudree, Rousseau and Schelp proved that Tn is Bm-good if \(n \ge 3m - 3\). In this paper, we obtain the exact Ramsey number of Tn versus 2B2- Our result implies that Tn is 2B2-good if n ≥ 5.
期刊介绍:
Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.