{"title":"Generalized Turán problem for a path and a clique","authors":"Xiaona Fang, Xiutao Zhu, Yaojun Chen","doi":"arxiv-2409.10129","DOIUrl":null,"url":null,"abstract":"Let $\\mathcal{H}$ be a family of graphs. The generalized Tur\\'an number\n$ex(n, K_r, \\mathcal{H})$ is the maximum number of copies of the clique $K_r$\nin any $n$-vertex $\\mathcal{H}$-free graph. In this paper, we determine the\nvalue of $ex(n, K_r, \\{P_k, K_m \\} )$ for sufficiently large $n$ with an\nexceptional case, and characterize all corresponding extremal graphs, which\ngeneralizes and strengthens the results of Katona and Xiao [EJC, 2024] on\n$ex(n, K_2, \\{P_k, K_m \\} )$. For the exceptional case, we obtain a tight upper\nbound for $ex(n, K_r, \\{P_k, K_m \\} )$ that confirms a conjecture on $ex(n,\nK_2, \\{P_k, K_m \\} )$ posed by Katona and Xiao.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"117 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10129","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let $\mathcal{H}$ be a family of graphs. The generalized Tur\'an number
$ex(n, K_r, \mathcal{H})$ is the maximum number of copies of the clique $K_r$
in any $n$-vertex $\mathcal{H}$-free graph. In this paper, we determine the
value of $ex(n, K_r, \{P_k, K_m \} )$ for sufficiently large $n$ with an
exceptional case, and characterize all corresponding extremal graphs, which
generalizes and strengthens the results of Katona and Xiao [EJC, 2024] on
$ex(n, K_2, \{P_k, K_m \} )$. For the exceptional case, we obtain a tight upper
bound for $ex(n, K_r, \{P_k, K_m \} )$ that confirms a conjecture on $ex(n,
K_2, \{P_k, K_m \} )$ posed by Katona and Xiao.