A matching in a graph is a set of edges such that no two edges share a common vertex. If a matching has k edges, then the matching is simply called a k-matching. There are a variety of applications concerning k-matchings.
In this presentation, one such application will be discussed. Then two general results on k-matchings will be given. The first gives necessary and sufficient conditions on the number of edges in a graph that will guarantee the existence of a k-matching and the second will characterize graphs in which every 2-matching is critical.