# Counting paths, cycles, and blow‐ups in planar graphs

@inproceedings{Cox2022CountingPC, title={Counting paths, cycles, and blow‐ups in planar graphs}, author={Christopher Cox and Ryan R. Martin}, year={2022} }

For a planar graph H , let N P ( n, H ) denote the maximum number of copies of H in an n -vertex planar graph. In this paper, we prove that N P ( n, P 7 ) ∼ 427 n 4 , N P ( n, C 6 ) ∼ ( n/ 3) 3 , N P ( n, C 8 ) ∼ ( n/ 4) 4 and N P ( n, K 4 { 1 } ) ∼ ( n/ 6) 6 , where K 4 { 1 } is the 1-subdivision of K 4 . In addition, we obtain signiﬁcantly improved upper bounds on N P ( n, P 2 m +1 ) and N P ( n, C 2 m ) for m ≥ 4. For a wide class of graphs H , the key technique developed in this paper…

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