Matching number ν(G): max independent edges. Perfect matching: ν=n/2, every vertex matched. Gallai: α+ν=n (no isolated vertices). König (bipartite): ν=τ (matching = vertex cover). Hall's theorem: perfect matching in bipartite iff |N(S)|≥|S| for all S.
How to Use
Select graph:
- ν: Matching number
- Perfect: ν=n/2?
- Gallai: α+ν=n
König's Theorem
In bipartite graphs: ν(G)=τ(G), max matching = min vertex cover. This is equivalent to max-flow min-cut for bipartite flow networks. Does NOT hold for general graphs (e.g., triangle: ν=1, τ=2).
Hall's Marriage Theorem
A bipartite graph G=(X∪Y,E) has a perfect matching iff |N(S)|≥|S| for every S⊆X. This 'marriage condition' ensures every subset can be matched. Equivalent to ν=|X|=|Y|.
Step-by-Step Instructions
- 1Select graph.
- 2Compute ν(G).
- 3Check perfect.
- 4Apply Gallai.
- 5Try König.