Densities of Minor-Closed Graph Families

TL;DR

This paper investigates the set of limiting densities in minor-closed graph families, revealing its structure, order type, and characterizing densities up to certain cluster points, with specific results for multigraphs.

Contribution

It characterizes the set of limiting densities, proves its countability and order type, and identifies densities of low-density minimal graphs, including for multigraphs.

Findings

The set of limiting densities is countable, well-ordered, and closed.

All limiting densities up to the first two cluster points are identified.

For multigraphs, limiting densities are only integers and ratios i/(i+1).

Abstract

We define the limiting density of a minor-closed family of simple graphs F to be the smallest number k such that every n-vertex graph in F has at most kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting densities. This set of numbers is countable, well-ordered, and closed; its order type is at least {\omega}^{\omega}. It is the closure of the set of densities of density-minimal graphs, graphs for which no minor has a greater ratio of edges to vertices. By analyzing density-minimal graphs of low densities, we find all limiting densities up to the first two cluster points of the set of limiting densities, 1 and 3/2. For multigraphs, the only possible limiting densities are the integers and the superparticular ratios i/(i+1).