Stability of Generalized Turán Number for Linear Forests

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 \(ex(n,T,{\mathcal {F}})\). 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.