Maximum cliques in a graph without disjoint given subgraph

The generalized Turán number \(\ex(n,K_s,F)\) denotes the maximum number of copies of \(K_s\) in an \(n\)-vertex \(F\)-free graph. Let \(kF\) denote \(k\) disjoint copies of \(F\). Gerbner, Methuku and Vizer [DM, 2019, 3130-3141] gave a lower bound for \(\ex(n,K_3,2C_5)\) and obtained the magnitude...

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Bibliographic Details
Published inarXiv.org
Main Authors Zhang, Fangfang, Chen, Yaojun, Gyori, Ervin, Zhu, Xiutao
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 18.09.2023
Subjects
Graph theory
Lower bounds
Mathematics - Combinatorics
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Summary:The generalized Turán number \(\ex(n,K_s,F)\) denotes the maximum number of copies of \(K_s\) in an \(n\)-vertex \(F\)-free graph. Let \(kF\) denote \(k\) disjoint copies of \(F\). Gerbner, Methuku and Vizer [DM, 2019, 3130-3141] gave a lower bound for \(\ex(n,K_3,2C_5)\) and obtained the magnitude of \(\ex(n, K_s, kK_r)\). In this paper, we determine the exact value of \(\ex(n,K_3,2C_5)\) and described the unique extremal graph for large \(n\). Moreover, we also determine the exact value of \(\ex(n,K_r,(k+1)K_r)\) which generalizes some known results.
ISSN:2331-8422
DOI:10.48550/arxiv.2309.09603