Maximum cliques in a graph without disjoint given subgraph

TL;DR

This paper determines the exact maximum number of triangles in large graphs that avoid two disjoint 5-cycles and extends results to larger cliques avoiding multiple disjoint copies, providing precise extremal configurations.

Contribution

It precisely computes extremal numbers for specific forbidden subgraphs and characterizes the unique extremal graphs for large n, advancing extremal graph theory.

Findings

Exact value of x(n,K_3,2C_5) determined

Unique extremal graph for large n identified

Generalization of extremal results for x(n,K_r,(k+1)K_r)

Abstract

The generalized Tur\'an 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.