A matching in a graph G is a subset M of the edges of G such that no two share an endpoint. A matching has maximum cardinality if its cardinality is at least as large as that of any other matching. An odd-set cover OSC of a graph G is a labeling of the nodes of G with integers such that every edge of G is either incident to a node labeled 1 or connects two nodes labeled with the same number i ≥ 2.
This article proves Edmonds theorem:
Let M be a matching in a graph G and let OSC be an odd-set cover of G. For any i ≥ 0, let n(i) be the number of nodes labeled i. If |M| = n(1) + ∑i ≥ 2(n(i) div 2), then M is a maximum cardinality matching.