Generalized Tur\'an problem for a path and a clique

TL;DR

This paper determines the maximum number of cliques in large graphs avoiding a path and a clique, extending previous results and confirming a conjecture for specific cases.

Contribution

It generalizes and strengthens prior work by explicitly calculating the generalized Turán number for paths and cliques, and characterizes extremal graphs.

Findings

Exact value of ex(n, K_r, {P_k, K_m}) for large n

Characterization of all extremal graphs in the case

A tight upper bound confirming a conjecture

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.