Why it matters

Hall's theorem characterizes matchability. Understanding shapes combinatorics.

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The architecture

For every subset S on one side.

|N(S)| >= |S|.

Iff perfect matching exists.

Hall's theoremSubset Sone sideNeighborhood N(S)on other sideHall condition|N(S)| >= |S|Foundational for bipartite matching theory; algorithmic via König or max flow
Hall's theorem.
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How it works end to end

Necessary + sufficient condition.

Proof via augmenting paths.

Constructive via max-flow.