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

Abstract

Let \({\cal F} = \{ {H_1}, \ldots ,{H_k}\} \,\,(k \ge 1)\) be a family of graphs. The Turán number of the family \({\cal F}\) is the maximum number of edges in an n-vertex {H₁, …, H_k}-free graph, denoted by ex(n, \({\cal F}\)) or ex(n, {H₁,H₂, … 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 Turán number of the family consisting of a blow-up of a cycle and a blow-up of a star in terms of the Turán number of the family consisting of a cycle, a star and linear forests with k edges.