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Maximum matching on random graphs

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arxiv cond-mat/0309348 v1 pith:BTC7QOWN submitted 2003-09-15 cond-mat.dis-nn cond-mat.stat-mech

Haijun Zhou , Zhong-can Ou-Yang This is my paper · ORCID
classification cond-mat.dis-nncond-mat.stat-mech
keywords glassygraphsmatchingmax-matchingmaximumphaseproblemrandom
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The maximum matching problem on random graphs is studied analytically by the cavity method of statistical physics. When the average vertex degree \mth{c} is larger than \mth{2.7183}, groups of max-matching patterns which differ greatly from each other {\em gradually} emerge. An analytical expression for the max-matching size is also obtained, which agrees well with computer simulations. Discussion is made on this {\em continuous} glassy phase transition and the absence of such a glassy phase in the related minimum vertex covering problem.

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