Fangfang Wu, Hajo Broersma, Shenggui Zhang, Binlong Li
{"title":"Turán Numbers for Vertex-disjoint Triangles and Pentagons","authors":"Fangfang Wu, Hajo Broersma, Shenggui Zhang, Binlong Li","doi":"10.1007/s10114-025-3272-9","DOIUrl":null,"url":null,"abstract":"<div><p>The Turán number, denoted by ex (<i>n</i>, <i>H</i>), is the maximum number of edges of a graph on <i>n</i> vertices containing no graph <i>H</i> as a subgraph. Denote by <i>kC</i><sub><i>ℓ</i></sub> the union of <i>k</i> vertex-disjoint copies of <i>C</i><sub><i>ℓ</i></sub>. In this paper, we present new results for the Turán numbers of vertex-disjoint cycles. Our first results deal with the Turán number of vertex-disjoint triangles ex (<i>n</i>, <i>kC</i><sub>3</sub>). We determine the Turán number ex(<i>n</i>, <i>kC</i><sub>3</sub>) for <span>\\(n \\geq {k^{2}+5k \\over 2}\\)</span> when <i>k</i> ≤ 4, and <i>n</i> ≥ <i>k</i><sup>2</sup> + 2 when <i>k</i> ≥ 4. Moreover, we give lower and upper bounds for ex (<i>n</i>, <i>kC</i><sub>3</sub>) with <span>\\(3k \\leq n \\leq {k^{2}+5k \\over 2}\\)</span> when <i>k</i> ≤ 4, and 3<i>k</i> ≤ <i>n</i> ≤ <i>k</i><sup>2</sup> + 2 when <i>k</i> ≥ 4. Next, we give a lower bound for the Turán number of vertex-disjoint pentagons ex (<i>n</i>, <i>kC</i><sub>5</sub>). Finally, we determine the Turán number ex (<i>n</i>, <i>kC</i><sub>5</sub>) for <i>n</i> = 5<i>k</i>, and propose two conjectures for ex (<i>n</i>, <i>kC</i><sub>5</sub>) for the other values of <i>n</i>.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 4","pages":"1181 - 1195"},"PeriodicalIF":0.8000,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-025-3272-9","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
The Turán number, denoted by ex (n, H), is the maximum number of edges of a graph on n vertices containing no graph H as a subgraph. Denote by kCℓ the union of k vertex-disjoint copies of Cℓ. In this paper, we present new results for the Turán numbers of vertex-disjoint cycles. Our first results deal with the Turán number of vertex-disjoint triangles ex (n, kC3). We determine the Turán number ex(n, kC3) for \(n \geq {k^{2}+5k \over 2}\) when k ≤ 4, and n ≥ k2 + 2 when k ≥ 4. Moreover, we give lower and upper bounds for ex (n, kC3) with \(3k \leq n \leq {k^{2}+5k \over 2}\) when k ≤ 4, and 3k ≤ n ≤ k2 + 2 when k ≥ 4. Next, we give a lower bound for the Turán number of vertex-disjoint pentagons ex (n, kC5). Finally, we determine the Turán number ex (n, kC5) for n = 5k, and propose two conjectures for ex (n, kC5) for the other values of n.
期刊介绍:
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.