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3 cards from this section. Quick, good the night before a test.
Define a bipartite graph.
Consider how the vertices are grouped.
What is Hall's Marriage Theorem?
Think about what the neighbors of a vertex imply.
What condition is required for a complete matching in Hall's Theorem?
Recall the fundamental statement of the theorem.
True or False: If there exists a complete matching, the neighbor condition must hold.
Think about examples where this relationship applies.
Prove that in any bipartite graph where |V1| = |V2|, if |N(A)| ≥ |A| holds for all subsets, a complete matching exists.
Break down the proof into base and inductive cases, illustrating graph changes.
Given a bipartite graph situation, create a scenario where Hall's condition fails and discuss the implications on matchings.
Visualize what happens in your example and analyze neighborhood counts.
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