Extremal functions for sparse minors

TL;DR

This paper investigates the extremal functions for sparse minors, especially for graphs with strongly sublinear separators, providing tight bounds and extending known results for planar graphs and minor-closed families.

Contribution

It establishes asymptotically tight bounds for the extremal function of sparse minors in new graph classes, extending previous work and connecting to open problems like Hadwiger's conjecture.

Findings

Derived tight bounds for planar graphs' extremal functions.

Extended results to graphs with strongly sublinear separators.

Linked extremal function bounds to open problems in graph minor theory.

Abstract

The "extremal function" $c(H)$ of a graph $H$ is the supremum of densities of graphs not containing $H$ as a minor, where the "density" of a graph $G$ is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005), Norin, Reed, Thomason and Wood (2020), and Thomason and Wales (2019) determined the asymptotic behaviour of $c(H)$ for all polynomially dense graphs $H$, as well as almost all graphs $H$ of constant density. We explore the asymptotic behavior of the extremal function in the regime not covered by the above results, where in addition to having constant density the graph $H$ is in a graph class admitting strongly sublinear separators. We establish asymptotically tight bounds in many cases. For example, we prove that for every planar graph $H$, $$c(H) = (1+o(1))\cdot\max\left\{\frac{|V(H)|}{2},|V(H)| - \alpha (H)\right\},$$ extending recent results of Haslegrave, Kim and Liu (2020). We also show that an asymptotically tight bound on the extremal function of graphs in minor-closed families proposed by Haslegrave, Kim and Liu (2020) is equivalent to a well studied open weakening of Hadwiger's conjecture.