2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/22217The 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.7 pages with 2 eps figures included. Use EPL style. Submitted to Europhysics LettersDisordered Systems and Neural NetworksStatistical Mechanicstext