Stability of generalized Turán number for linear forests

Abstract

Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized Turán number of $\mathcal{F}$ is the maximum number of copies of $T$ in an $\mathcal{F}$-free graph on $n$ vertices, denoted by $ex(n,T,\mathcal{F})$. When $T=K_r$, $ex(n,K_r,\mathcal{F})$ is a function specifying the maximum possible number of $r$-cliques in an $\mathcal{F}$-free graph on $n$ vertices. A linear forest is a forest whose connected components are all paths and isolated vertices. Let ${\mathcal{L}}_k$ be the family of all linear forests of size $k$ without isolated vertices. In this paper, we obtained the maximum possible number of $r$-cliques in $G$, where $G$ is ${\mathcal{L}}_k$-free with minimum degree at least $d$. Furthermore, we give a stability version of the result. As an application of the stability version of the result, we obtain a clique version of the stability of the Erdős-Gallai Theorem on matchings.

17 pages,1 figure, 14 conferences