Counting paths, cycles and blow-ups in planar graphs

For a planar graph H , let NP(n,H) denote the maximum number of copies of H in an n-vertex planar graph. In this paper, we prove that NP(n, P7) ∼ 4 27n4, NP(n,C6) ∼ (n/3), NP(n,C8) ∼ (n/4) and NP(n,K4{1}) ∼ (n/6), where K4{1} is the 1-subdivision of K4. In addition, we obtain significantly improved upper bounds on NP(n, P2m+1) and NP(n,C2m) for m ≥ 4. For a wide class of graphs H , the key technique developed in this paper allows us to bound NP(n,H) in terms of an optimization problem over weighted graphs.