The maximum number of paths of length four in a planar graph

Let f(n,H) denote the maximum number of copies of H in an n-vertex planar graph. The order of magnitude of f(n, Pk), where Pk is a path on k vertices, is n ⌊ k−1 2 . In this paper we determine the asymptotic value of f(n, P5) and give conjectures for longer paths.

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