Extremal graphs for the k ‐flower

The Turán number of a graph H , e x ( n , H ) , is the maximum number of edges in any graph of order n that does not contain an H as a subgraph. A graph on 2 k + 1 vertices consisting of k triangles that intersect in exactly one common vertex is called a k ‐fan, and a graph consisting of k cycles that intersect in exactly one common vertex is called a k ‐flower. In this article, we determine the Turán number of any k ‐flower containing at least one odd cycle and characterize all extremal graphs provided n is sufficiently large. Erdős, Füredi, Gould, and Gunderson determined the Turán number for the k ‐fan. Our result is a generalization of their result. The addition aim of this article is to draw attention to a powerful tool, the so‐called progressive induction lemma of Simonovits.