26. Proof of Hall's Marriage Theorem
The proof of Hall's Marriage Theorem is established, demonstrating the necessary and sufficient conditions for a complete matching exists between two subsets in a bipartite graph. The theorem states that if the number of neighbors of any subset of one partition is at least as large as the size of the subset itself, a complete matching from one partition to another is possible. Both necessary and sufficient conditions have been proven with detailed explanations along with the inductive proof strategy.
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What we have learnt
- Hall's Marriage Theorem provides necessary and sufficient conditions for a complete matching in bipartite graphs.
- A complete matching exists if, for every subset, the number of neighbors is at least equal to the subset size.
- The proof involves both a direct proof for the necessary condition and an existential proof via induction for the sufficiency condition.
Key Concepts
- -- Hall's Marriage Theorem
- A theorem that gives necessary and sufficient conditions for the existence of a complete matching in bipartite graphs.
- -- Bipartite Graph
- A graph whose vertices can be divided into two disjoint sets such that no two graph vertices within the same set are adjacent.
- -- Complete Matching
- A matching in which every vertex of a graph is incident to exactly one edge.
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