Densities of minor-closed graph classes are rational

TL;DR

This paper proves that the limiting densities of proper minor-closed graph classes are rational and that their extremal functions exhibit eventual periodicity, confirming several conjectures in graph theory.

Contribution

It establishes the rationality of limiting densities for proper minor-closed classes and shows their extremal functions are eventually periodic, advancing understanding of graph class densities.

Findings

Limiting densities of proper minor-closed classes are rational.

Extremal functions differ from linear growth by an eventually periodic function.

Every proper minor-closed class contains a bounded pathwidth subclass with the same density.

Abstract

For a graph class $\mathcal{F}$, let $ex_{\mathcal{F}}(n)$ denote the maximum number of edges in a graph in $\mathcal{F}$ on $n$ vertices. We show that for every proper minor-closed graph class $\mathcal{F}$ the function $ex_{\mathcal{F}}(n) - \Delta n$ is eventually periodic, where $\Delta = \lim_{n \to \infty} ex_{\mathcal{F}}(n)/n$ is the limiting density of $\mathcal{F}$. This confirms a special case of a conjecture by Geelen, Gerards and Whittle. In particular, the limiting density of every proper minor-closed graph class is rational, which answers a question of Eppstein. As a major step in the proof we show that every proper minor-closed graph class contains a subclass of bounded pathwidth with the same limiting density, confirming a conjecture of the second author. Finally, we investigate the set of limiting densities of classes of graphs closed under taking topological minors.