Abstarct: A Turan number of a given graph H with n vertices, ex(n,H) is the maximum number of edges in a spanning subgraph of Kn which has no subgraph isomorphic to H. In this thesis, we study some generalization of Turan ’s number, in particular, we set H to be some spanning subgraphs of Kn or a cycle with fixed length. For a graph G, and a subgraph H of it, we define a graph with the vertices set consisting of all subgraphs of G isomorphic to H. Two vertices are adjacent in this graph whenever these two vertices have no edges in common. We will study the chromatic number of this graph for some fixed subgraph H, in particular, we consider to H be a cycle or a complete graph.