On the critical densities of minor-closed classes

TL;DR

This paper studies the maximum edge-to-vertex ratios in minor-closed graph classes, characterizes small values, shows the set of these ratios is densely populated at large scales, and addresses open questions from prior research.

Contribution

It determines small critical densities, proves the density of the set of all such densities, and resolves several open questions in the field.

Findings

Identified all critical densities up to 2.

Proved the set of critical densities is asymptotically dense.

Answered key open questions posed by Eppstein.

Abstract

Given a minor-closed class $\mathcal{A}$ of graphs, let $\beta_{\mathcal{A}}$ denote the supremum over all graphs in $\mathcal{A}$ of the ratio of edges to vertices. We investigate the set $B$ of all such values $\beta_{\mathcal{A}}$, taking further the project begun by Eppstein. Amongst other results, we determine the small values in $B$ (those up to 2); we show that $B$ is `asymptotically dense'; and we answer some questions posed by Eppstein.