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.
Categories: math.CO