Let there be a matching $M'$ of greater cardinality than $M$. We consider the symmetric difference $Q = M \oplus M'$. The subgraph $Q$ is no longer necessarily a matching.

Any vertex in $Q$ has a maximum degree of $2$, which means that all connected components in it are one of the three -

* an isolated vertex

* a (simple) path whose edges are alternately from $M$ and $M'$

* a cycle of even length whose edges are alternately from $M$ and $M'$

Since $M'$ has a cardinality greater than $M$, $Q$ has more edges from $M'$ than $M$. By the Pigeonhole principle, at least one connected component will be a path having

more edges from $M'$ than $M$. Because any such path is alternating, it will have initial and final vertices unsaturated by $M$, making it an augmenting path for $M$,

- [Assignment problem. Solution using min-cost-flow in O (N^5)](graph/Assignment-problem-min-flow.md)

- **Matchings and related problems**

- [Bipartite Graph Check](graph/bipartite-check.md)

- **Miscellaneous**

- [Topological Sorting](graph/topological-sort.md)

- [Edge connectivity / Vertex connectivity](graph/edge_vertex_connectivity.md)