Densities of minor-closed graph classes are rational

Abstract

For a graph class $\mathcal{F}$, let $e{x}_{\mathcal{F}}(n)$ 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 $e{x}_{\mathcal{F}}(n)-\Delta n$ is eventually periodic, where $\Delta ={\lim }_{n\to \infty }e{x}_{\mathcal{F}}(n)/n$ 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.

References in corpus (2)

• Islands in minor-closed classes. I. Bounded treewidth and separators
• A generalization of the Grid Theorem