Growth constants of minor-closed classes of graphs

Abstract

A minor-closed class of graphs is a set of labelled graphs which is closed under isomorphism and under taking minors. For a minor-closed class G , let g n be the number of graphs in G which have n vertices. The growth constant of G is γ = lim sup ( g n / n ! ) 1 / n . We study the properties of the set Γ of growth constants of minor-closed classes of graphs. Among other results, we show that Γ does not contain any number in the interval [ 0 , 2 ] , besides 0, 1, ξ and 2, where ξ ≈ 1.76 . An infinity of further gaps is found by determining all the possible growth constants between 2 and δ ≈ 2.25159 . Our results give in fact a complete characterization of all the minor-closed classes with growth constant at most δ in terms of their excluded minors.