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

Abstract

Given graphs $H$ and $F$, the generalized Turán 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, $t\ge 1$ 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, Győri, 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.

24 pages