Abstract
We define the limiting density of a minor-closed family of simple graphs $\mathcal{F}$ to be the smallest number $k$ such that every $n$-vertex graph in $\mathcal{F}$ 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 $i/(i+1)$.
Highlights
Planar simple graphs with n vertices have at most 3n − 6 edges
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
The set of limiting densities of minor-closed graph families is the closure of the set of densities of a certain family of finite graphs, the densityminimal graphs for which no minor has a greater ratio of edges to vertices (Theorem 20)
Summary
Planar simple graphs with n vertices have at most 3n − 6 edges. Outerplanar graphs have at most 2n − 4 edges. One may apply the theory of graph minors to families of multigraphs allowing multiple edges between the same pair of vertices as well as multiple self-loops connecting a single vertex to itself. In this case the theory of limiting densities and density-minimal graphs is simpler: the only possible limiting densities for minor-closed families of multigraphs are the integers and the superparticular ratios, and the only limit point of the set of limiting densities is the number 1 (Theorem 23)
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