The Turán number of a graph H, ex(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 2k+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.

The Turán number of a graph H, ex(n, H), is the maximum number of edges in a simple graph of order n which does not contain H as a subgraph. Let k · P 3 denote k disjoint copies of a path on 3 vertices. In this paper, we determine the value ex(n, k ·P 3 ) and characterize all extremal graphs. This extends a result of

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The circumference of a graph is the length of a longest cycle of it. We determine the maximum number of copies of $K_{r,s}$, the complete bipartite graph with classes sizes $r$ and $s$, in 2-connected graphs with circumference less than $k$. As corollaries of our main result, we determine the maximum number of copies of $K_{r,s}$ in $n$-vertex $P_{k}$-free and $M_k$-free graphs for all values of $n$, where $P_k$ is a path on $k$ vertices and $M_k$ is a matching on $k$ edges.