Abstract

In the past decades the Gn,p model of random graphs, introduced by Erdős and Rényi in the 60’s, has led to numerous beautiful and deep theorems. A key feature that is used in basically all proofs is that edges in Gn,p appear independently. The independence of the edges allows, for example, to obtain extremely tight bounds on the number of edges of Gn,p and its degree sequence by straightforward applications of Chernoff bounds. This situation changes dramatically if one considers graph classes with structural side constraints. For example, in a random planar graph Rn (a graph drawn uniformly at random from the class of all labeled planar graphs on n vertices) the edges are obviously far from being independent. Consequently, so far basically all results about properties of random graphs with structural side constraints rely on completely different methods, mostly from analytic combinatorics. In this paper we show that recent progress in the construction of so-called Boltzmann samplers by Duchon, Flajolet, Louchard, and Schaeffer (Combinatorics, Probability and Computing 13, 2004) and Fusy (International Conference on Analysis of Algorithms ’05) can be used to reduce the study of degree sequences and subgraph counts to properties of sequences of independent and identically distributed random variables – to which we can then again apply Chernoff bounds to obtain extremely tight results. We elaborate our ideas by studying random dissections and triangulations of a labeled convex n-gon. For both we obtain the degree sequence and the number of induced copies of given fixed graphs. The degree sequence for triangulations was already obtained previously by Gao and Wormald (Combinatorica 23, 2003) using deep methods from analytic combinatorics. We do, however, get better probabilities for the tails of the distributions. 1

Citations