{"title":"边色图中彩虹小群的更多内容","authors":"Xiao-Chuan Liu , Danni Peng , Xu Yang","doi":"10.1016/j.ejc.2024.104088","DOIUrl":null,"url":null,"abstract":"<div><div>In an edge-colored graph <span><math><mi>G</mi></math></span>, a rainbow clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a complete subgraph on <span><math><mi>k</mi></math></span> vertices in which all the edges have distinct colors. Let <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the number of edges and colors in <span><math><mi>G</mi></math></span>, respectively. In this paper, we show that for any <span><math><mrow><mi>ɛ</mi><mo>></mo><mn>0</mn></mrow></math></span>, if <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>3</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mn>2</mn><mi>ɛ</mi><mo>)</mo></mrow><mfenced><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mfenced></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, then for sufficiently large <span><math><mi>n</mi></math></span>, the number of rainbow cliques <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> in <span><math><mi>G</mi></math></span> is <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>.</div><div>We also characterize the extremal graphs <span><math><mi>G</mi></math></span> without a rainbow clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, for <span><math><mrow><mi>k</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></math></span>, when <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is maximum.</div><div>Our results not only address existing questions but also complete the findings of Ehard and Mohr (2020).</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104088"},"PeriodicalIF":1.0000,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"More on rainbow cliques in edge-colored graphs\",\"authors\":\"Xiao-Chuan Liu , Danni Peng , Xu Yang\",\"doi\":\"10.1016/j.ejc.2024.104088\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In an edge-colored graph <span><math><mi>G</mi></math></span>, a rainbow clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a complete subgraph on <span><math><mi>k</mi></math></span> vertices in which all the edges have distinct colors. Let <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the number of edges and colors in <span><math><mi>G</mi></math></span>, respectively. In this paper, we show that for any <span><math><mrow><mi>ɛ</mi><mo>></mo><mn>0</mn></mrow></math></span>, if <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>3</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mn>2</mn><mi>ɛ</mi><mo>)</mo></mrow><mfenced><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mfenced></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, then for sufficiently large <span><math><mi>n</mi></math></span>, the number of rainbow cliques <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> in <span><math><mi>G</mi></math></span> is <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>.</div><div>We also characterize the extremal graphs <span><math><mi>G</mi></math></span> without a rainbow clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, for <span><math><mrow><mi>k</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></math></span>, when <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is maximum.</div><div>Our results not only address existing questions but also complete the findings of Ehard and Mohr (2020).</div></div>\",\"PeriodicalId\":50490,\"journal\":{\"name\":\"European Journal of Combinatorics\",\"volume\":\"124 \",\"pages\":\"Article 104088\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-11-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"European Journal of Combinatorics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0195669824001732\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669824001732","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
在边色图 G 中，彩虹簇 Kk 是 k 个顶点上的一个完整子图，其中所有的边都有不同的颜色。设 e(G) 和 c(G) 分别为 G 中的边数和颜色数。本文将证明，对于任意ɛ>0，如果 e(G)+c(G)≥(1+k-3k-2+2ɛ)n2 且 k≥3 ，那么对于足够大的 n，G 中彩虹小群 Kk 的数目为 Ω(nk)。我们还描述了在 k=4,5 时，e(G)+c(G) 最大时没有彩虹簇 Kk 的极值图 G 的特征。我们的结果不仅解决了现有问题，还完善了 Ehard 和 Mohr (2020) 的发现。
In an edge-colored graph , a rainbow clique is a complete subgraph on vertices in which all the edges have distinct colors. Let and be the number of edges and colors in , respectively. In this paper, we show that for any , if and , then for sufficiently large , the number of rainbow cliques in is .
We also characterize the extremal graphs without a rainbow clique , for , when is maximum.
Our results not only address existing questions but also complete the findings of Ehard and Mohr (2020).
期刊介绍：
The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.
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