Matching Covering and Packing

Abstract

Suppose we are given a graph and are asked to find in it as many independent edges as possible. How should we go about this? Will we be able to pair up all its vertices in this way? If not, how can we be sure that this is indeed impossible? Somewhat surprisingly, this basic problem does not only lie at the heart of numerous applications, it also gives rise to some rather interesting graph theory. A set M of independent edges in a graph G = (V, E) is called a matching. M is a matching of U ⊆ V if every vertex in U is incident matching with an edge in M. The vertices in U are then called matched (by M); matched vertices not incident with any edge of M are unmatched. A k-regular spanning subgraph is called a k-factor. Thus, a sub-factor