Abstract
We study matchings on sparse random graphs by means of the cavity method. We firstshow how the method reproduces several known results about maximum and perfectmatchings in regular and Erdös–Rényi random graphs. Our main new result is thecomputation of the entropy, i.e. the leading order of the logarithm of the number ofsolutions, of matchings with a given size. We derive both an algorithm to compute thisentropy for an arbitrary graph with a girth that diverges in the large size limit, and ananalytic result for the entropy in regular and Erdös–Rényi random graph ensembles.
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