Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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  • AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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König's theorem: ν(G)=τ(G)\nu(G)=\tau(G) for every finite bipartite graph

Statement

For every finite bipartite graph GG, its matching number and vertex-cover number agree: ν(G)=τ(G)\nu(G)=\tau(G).

Facts & Assumptions

Given: A finite bipartite graph GG and a maximum matching MM.

[L1]

Alternating reachability from a maximum matching constructs a vertex cover with exactly ∣M∣|M| vertices (Alternating reachability from unmatched left vertices produces a vertex cover of the same size as a maximum matching).

Proof

technique · direct
1.1

Every vertex cover has at least ∣M∣|M| vertices, because distinct edges of the matching MM require distinct cover vertices.

given
1.2

By [L1], some vertex cover has exactly ∣M∣|M| vertices.

L1
2.1

The lower bound and exhibited cover show τ(G)=∣M∣=ν(G)\tau(G)=|M|=\nu(G).

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 19 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources