Random graphs from a minor-closed class

Authors:
Colin Mcdiarmid
Affiliations:
Department of statistics, oxford university (e-mail: cmcd@stats.ox.ac.uk)
Venue:
Combinatorics, Probability and Computing
Year:
2009

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Abstract

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a class of labelled graphs has smooth growth; and, for the random graph Rn sampled uniformly from the n-vertex graphs in , the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, as n → ∞ the probability that Rn is connected tends to 1/A(ρ), where A(x) is the exponential generating function for and ρ is its radius of convergence.