We consider the expected performance of two greedy matching algorithms on sparse random graphs and also on random trees. In all cases we establish expressions for the mean and variance of the number of edges chosen and establish asymptotic normality.
"The Average Performance of the Greedy Matching Algorithm." Ann. Appl. Probab. 3 (2) 526 - 552, May, 1993. https://doi.org/10.1214/aoap/1177005436