Small Maximal Matchings in Random Graphs

We look at the minimal size of a maximal matching in general, bipartite and d-regular random graphs. We prove that the ratio between the sizes of any two maximal matchings approaches one in dense random graphs and random bipartite graphs. Weaker bounds hold for sparse random graphs and random dregular graphs. We also describe an algorithm that with high probability finds a matching of size strictly less than n=2 in a cubic graph. The result is based on approximating the algorithm dynamics by a system of linear differential equations.