The maximum number of triangles in graphs without vertex disjoint friendship graphs

TL;DR

This paper determines the maximum number of triangles in large graphs that do not contain a union of multiple disjoint friendship graphs, extending previous results and characterizing the extremal structures.

Contribution

It explicitly calculates the generalized Turán number for multiple disjoint friendship graphs and characterizes the extremal graphs, generalizing prior work and revealing structural differences.

Findings

Calculated $ ext{ex}(n,K_3,(t+1)F_k)$ for large n.

Characterized the extremal graph structures.

Identified fundamental structural changes in extremal graphs.

Abstract

Given graphs $H$ and $F$, the generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs. The friendship graph $F_k$ consists of $k$ triangles sharing a common vertex. In this paper, we determine the value of $\mathrm{ex}(n,K_3,(t+1)F_k)$, where $K_3$ is a triangle, $t\geq 1$ is an integer, and $(t+1)F_k$ denotes a union of $(t+1)$ pairwise vertex-disjoint copies of $F_k$. Moreover, we characterize the extremal structure. Our result can be viewed as a generalization of the result of Zhu, Chen, Gerbner, Gy\H{o}ri, and Hama Karim, as well as of the remaining case left open by Wang, Ni, Liu, and Kang. In contrast to the extremal graphs of $F_k$, the extremal graphs of $(t+1)F_k$ undergo a fundamental change. This structure is also different from those of previous similar problems.