Turán Number of the Family Consisting of a Blow-up of a Cycle and a Blow-up of a Star

Let F={H_(1),...,H_(k)}(k≥1)be a family of graphs.The Tur´an number of the family F is the maximum number of edges in an n-vertex{H_(1),...,H_(k)}-free graph,denoted by ex(n,F)or ex(n,{H_(1),H_(2),...,H_(k)}).The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different.In this paper we determine the Tur´an number of the family consisting of a blow-up of a cycle and a blow-up of a star in terms of the Tur´an number of the family consisting of a cycle,a star and linear forests with k edges.