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

Abstract Let denote the maximum number of copies of in an vertex planar graph. The problem of bounding this function for various graphs has been extensively studied since the 70's. A special case that received a lot of attention recently is when is the path on vertices, denoted . Our main result in this paper is that This improves upon the previously best known bound by a factor , which is best possible up to the hidden constant, and makes a significant step toward resolving conjectures of Ghosh et al. and of Cox and Martin. The proof uses graph theoretic arguments together with (simple) arguments from the theory of convex optimization.