  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.
