Abstract
The generalized Turán number ex(n,Ks,F) denotes the maximum number of copies of Ks in an n-vertex F-free graph. Let kF denote k disjoint copies of F. Gerbner et al. (2019) [8] gave a lower bound for ex(n,K3,2C5) and obtained the magnitude of ex(n,Ks,kKr). In this paper, we determine the exact value of ex(n,K3,2C5) and described the unique extremal graph for large n. Moreover, we also determine the exact value of ex(n,Kr,(k+1)Kr) which generalizes some known results.
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