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.
Categories: math.CO