The density of a sparse graph class is the supremum of the ratio $|E(G)|/|V(G)|$ taken over all non-null graphs $G$ in the class. The densities of graph classes determined by forbidding a single minor have been extensively studied, in part due to connections to the Hadwiger’s conjecture.
We will survey recent results on densities of minor-closed graph classes, including a proof of the conjecture of Eppstein that the densities of minor-closed graph classes are rational, joint with Rohan Kapadia, and joint work with Kevin Hendrey, Bruce Reed, Andrew Thomason and David Wood on densities of classes which forbid a single sparse minor.
