### Abstract

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.

### License

BSD License

### Topics

- Mathematics/Graph theory

### Session Max-Card-Matching

- Matching

## Cite

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Max-Card-Matching-AFP

    @article{Max-Card-Matching-AFP,
      author  = {Christine Rizkallah},
      title   = {Maximum Cardinality Matching},
      journal = {Archive of Formal Proofs},
      month   = {July},
      year    = {2011},
      note    = {\url{https://isa-afp.org/entries/Max-Card-Matching.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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