The number of matchings in random graphs

Details The number of matchings in random graphs The number of matchings in random graphs

2006-03-13

arxiv.org/abs/cond-mat/0603350v2

J. Stat. Mech. (2006) P05003

Physics

Condensed Matter

Disordered Systems and Neural Networks

17 pages, 6 figures, to be published in Journal of Statistical Mechanics

Scientific paper

10.1088/1742-5468/2006/05/P05003

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.

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