Small maximal matchings in random graphs

Abstract

We look at the minimal size of a maximal matching in general, bipartite and d-regular random graphs. We prove that with high probability 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 d-regular 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 number of systems of linear differential equations.