7,179 research outputs found
The number of matchings in random graphs
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.Comment: 17 pages, 6 figures, to be published in Journal of Statistical
Mechanic
A proof of the Upper Matching Conjecture for large graphs
We prove that the `Upper Matching Conjecture' of Friedland, Krop, and
Markstr\"om and the analogous conjecture of Kahn for independent sets in
regular graphs hold for all large enough graphs as a function of the degree.
That is, for every and every large enough divisible by , a union of
copies of the complete -regular bipartite graph maximizes the
number of independent sets and matchings of size for each over all
-regular graphs on vertices. To prove this we utilize the cluster
expansion for the canonical ensemble of a statistical physics spin model, and
we give some further applications of this method to maximizing and minimizing
the number of independent sets and matchings of a given size in regular graphs
of a given minimum girth
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