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

TL;DR

This paper establishes a tight upper bound on the maximum number of odd paths in planar graphs, using a novel combination of graph theory and convex optimization techniques, advancing understanding of graph path enumeration.

Contribution

The paper introduces a new bound on odd path counts in planar graphs, improving previous results by a factor exponential in the path length, and employs convex optimization in the proof.

Findings

Bound $N_{ ext{planar}}(n, P_{2m+1})=O(m^{-m} n^{m+1})$

Improves previous bounds by a factor $e^{m}$

Uses convex optimization alongside graph theoretic methods

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.