A Study On The Generalized Turán Problem

Let F and T are simple graphs.A graph G is called F-free if G does not contain F as a subgraph.The Turán number ex(n,F)is the maximum possible number of edges in an n-vertex graph G that is F-free.Turán number has become the basis of extremal graph theory.Many scholars have found the Turán number of a given graph,such as a circle,a path,a bipartite graph,etc.,but there are still many Turán numbers of graphs that are difficult to find.On the basis of Turán number,the generalized Turán number is proposed by Erdos.The Generalized Turán number ex(n,T,F)is the maximum possible number of copies of T in an n-vertex graph which is F-free.Now,Turán number and the generalized Turán number have attracted the attention of many scholars.This paper mainly studies the following three aspects:1.By using the closure technique,we determined the generalized Turán number of linear forest,that is,calculating the maximum number of edges of G and the maximum number of triangles in G if the linear forest of k edges does not contained in graph G.2.We extend the matching theorem of Erdos-Gallai and prove two stable versions of the matching theorem of Erdos-Gallai.3.By using the probability method-the technique of dependent random choice,we deter-mined asymptotic upper and lower bounds of the generalized Turán number of W_2l^s and P)l^s.