Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

König's theorem: ν(G)=τ(G) for every finite bipartite graph

Statement

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

Facts & Assumptions

Given: A finite bipartite graph G and a maximum matching M.

[L1]

Alternating reachability from a maximum matching constructs a vertex cover with exactly ∣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∣ vertices, because distinct edges of the matching M require distinct cover vertices.

given
1.2

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

L1
2.1

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

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources