Stability of Generalized Turán Number for Linear Forests

Given a graph T and a family of graphs F , the generalized Turán number of F is the maximum number of copies of T in an F -free graph on n vertices, denoted by e x ( n , T , F ) . A linear forest is a forest whose connected components are all paths and isolated vertices. Let 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 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.