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arXiv · 2208.02097

Bounding the number of odd paths in planar graphs via convex optimization

Asaf Cohen Antonir·Asaf Shapira
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Abstract

Let $N_{\mathcal{P}}(n,H)$ denote the maximum number of copies of $H$ in an $n$ vertex planar graph. The problem of bounding this function for various graphs $H$ has been extensively studied since the 70's. A special case that received a lot of attention recently is when $H$ is the path on $2m+1$ vertices, denoted $P_{2m+1}$. Our main result in this paper is that $$ N_{\mathcal{P}}(n,P_{2m+1})=O(m^{-m}n^{m+1})\;.$$ This improves upon the previously best known bound by a factor $e^{m}$, which is best possible up to the hidden constant, and makes a significant step towards resolving conjectures of Gosh et al. and of Cox and Martin. The proof uses graph theoretic arguments together with (simple) arguments from the theory of convex optimization.

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Asaf Cohen Antonir, Asaf Shapira. 2022-08-03. Bounding the number of odd paths in planar graphs via convex optimization. https://arxiv.org/abs/2208.02097

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