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 et al. (2019) [8] gave a lower bound for ex (n , K 3 , 2 C 5) and obtained the magnitude of ex (n , K s , k K r). In this paper, we determine the exact value of ex (n , K 3 , 2 C 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. [ABSTRACT FROM AUTHOR]