The number of matchings in random graphs
Disordered Systems and Neural Networks
2011-11-09 v2 Computational Complexity
Combinatorics
Abstract
We study matchings on sparse random graphs by means of the cavity method. We first show how the method reproduces several known results about maximum and perfect matchings in regular and Erdos-Renyi random graphs. Our main new result is the computation of the entropy, i.e. the leading order of the logarithm of the number of solutions, of matchings with a given size. We derive both an algorithm to compute this entropy for an arbitrary graph with a girth that diverges in the large size limit, and an analytic result for the entropy in regular and Erdos-Renyi random graph ensembles.
Keywords
Cite
@article{arxiv.cond-mat/0603350,
title = {The number of matchings in random graphs},
author = {Lenka Zdeborová and Marc Mézard},
journal= {arXiv preprint arXiv:cond-mat/0603350},
year = {2011}
}
Comments
17 pages, 6 figures, to be published in Journal of Statistical Mechanics