{"title":"路径与团的广义Turán问题","authors":"Xiaona Fang , Xiutao Zhu , Yaojun Chen","doi":"10.1016/j.ejc.2025.104137","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mi>H</mi></math></span> be a family of graphs. The generalized Turán number <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is the maximum number of copies of the clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> in any <span><math><mi>n</mi></math></span>-vertex <span><math><mi>H</mi></math></span>-free graph. In this paper, we determine the value of <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> for sufficiently large <span><math><mi>n</mi></math></span> with an exceptional case, and characterize all corresponding extremal graphs. This generalizes and strengthens the results of Katona (2024) on <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>. For the exceptional case, we obtain a tight upper bound for <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> that confirms a conjecture on <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> posed by Katona and Xiao.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104137"},"PeriodicalIF":1.0000,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Generalized Turán problem for a path and a clique\",\"authors\":\"Xiaona Fang , Xiutao Zhu , Yaojun Chen\",\"doi\":\"10.1016/j.ejc.2025.104137\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Let <span><math><mi>H</mi></math></span> be a family of graphs. The generalized Turán number <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is the maximum number of copies of the clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> in any <span><math><mi>n</mi></math></span>-vertex <span><math><mi>H</mi></math></span>-free graph. In this paper, we determine the value of <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> for sufficiently large <span><math><mi>n</mi></math></span> with an exceptional case, and characterize all corresponding extremal graphs. This generalizes and strengthens the results of Katona (2024) on <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>. For the exceptional case, we obtain a tight upper bound for <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> that confirms a conjecture on <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span> posed by Katona and Xiao.</div></div>\",\"PeriodicalId\":50490,\"journal\":{\"name\":\"European Journal of Combinatorics\",\"volume\":\"127 \",\"pages\":\"Article 104137\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2025-02-27\",\"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/S0195669825000198\",\"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/S0195669825000198","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let be a family of graphs. The generalized Turán number is the maximum number of copies of the clique in any -vertex -free graph. In this paper, we determine the value of for sufficiently large with an exceptional case, and characterize all corresponding extremal graphs. This generalizes and strengthens the results of Katona (2024) on . For the exceptional case, we obtain a tight upper bound for that confirms a conjecture on posed by Katona and Xiao.
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.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
