Independence and chromatic densities of graphs

Abstract

We consider graph densities in countably inflnite graphs. The independence density of a flnite graph G of order n is its proportion of independent sets to all subsets of vertices, while the chromatic density is its proportion of proper n-colourings to all mappings from vertices of G to f1;2;:::;ng. For both densities, we extend their deflnition to countable graphs via limits of chains of flnite graphs. We show that independence densities exist for all chains, and are unique regardless of which limiting chain is used. We prove that independence densities are always rational; in fact, the closure of the set of possible values is contained in the rationals. In contrast, we show that the inflnite random graph contains chains realizing all real numbers in [0;1] as a chromatic density.