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arXiv · 2009.13482

Densities of minor-closed graph classes are rational

Rohan Kapadia·Sergey Norin
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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.

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Rohan Kapadia, Sergey Norin. 2020-09-28. Densities of minor-closed graph classes are rational. https://arxiv.org/abs/2009.13482

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