Hakimi and Schmeichel considered the problem of maximizing the number of cycles of a given length in an n-vertex planar graph. They determined this number exactly for triangles and 4cycles and conjectured the solution to the problem for 5-cycles. We confirm their conjecture.

Let f (n, H) denote the maximum number of copies of H possible in an n-vertex planar graph. The function f (n, H) has been determined when H is a cycle of length 3 or 4 by Hakimi and Schmeichel and when H is a complete bipartite graph with smaller part of size 1 or 2 by Alon and Caro. We determine f (n, H) exactly in the case when H is a path of length 3.

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Finding the maximum number of induced cycles of length k in a graph on n vertices has been one of the most intriguing open problems of Extremal Graph Theory. Recently Balogh, Hu, Lidický and Pfender answered the question in the case k = 5. In this paper we show that an n-vertex planar graph contains at most n 2 3 + O(n) induced C 5 's, which is asymptotically tight.

In a generalized Turán problem, we are given graphs H and F and seek to maximize the number of copies of H in an F -free graph of order n. We consider generalized Turán problems where the host graph is planar. In particular we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most 2ℓ, for all ℓ, ℓ ≥ 1. We obtain the order of magnitude of the maximum number of cycles of a given length in a planar C 4 -free graph. An exact result is given for the maximum number of 5-cycles in a C 4 -free planar graph. Multiple conjectures are also introduced.