Random graphs with forbidden vertex degrees
Abstract
We study the random graph G_{n,\lambda/n} conditioned on the event that all vertex degrees lie in some given subset S of the non-negative integers. Subject to a certain hypothesis on S, the empirical distribution of the vertex degrees is asymptotically Poisson with some parameter \mux given as the root of a certain `characteristic equation' of S that maximises a certain function \psis(\mu). Subject to a hypothesis on S, we obtain a partial description of the structure of such a random graph, including a condition for the existence (or not) of a giant component. The requisite hypothesis is in many cases benign, and applications are presented to a number of choices for the set S including the sets of (respectively) even and odd numbers. The random \emph{even} graph is related to the random-cluster model on the complete graph K_n.
Cite
@article{arxiv.0712.0270,
title = {Random graphs with forbidden vertex degrees},
author = {Geoffrey Grimmett and Svante Janson},
journal= {arXiv preprint arXiv:0712.0270},
year = {2007}
}