LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

Alternating reachability from unmatched left vertices produces a vertex cover of the same size as a maximum matching

Statement

Let $G$ have bipartition $(X,Y)$ and let $M$ be a maximum matching. From every $M$ -exposed vertex of $X$ , follow alternating paths that begin with a nonmatching edge. Let $Z_X\subseteq X$ and $Z_Y\subseteq Y$ be the reachable vertices in the two parts. Then $C:=(X\setminus Z_X)\cup Z_Y$ is a vertex cover and $|C|=|M|$ .

Facts & Assumptions

Given: A finite bipartite graph $(X,Y)$ and a maximum matching $M$ .

[L1]

A maximum matching has no augmenting path (Berge's theorem: a matching is maximum if and only if it has no augmenting path).

[F1]

A vertex cover meets every edge, and its size is its cardinality (Vertex covers, edge covers and the covering numbers $\tau(G)$ and $\rho(G)$ ).

Proof

technique · direct

1.1

No vertex of $Z_Y$ is $M$ -exposed, for an alternating path from an exposed vertex of $X$ to it would be $M$ -augmenting, contrary to [L1].

1.2

Every edge $xy$ is met by $C$ : if $x\notin Z_X$ it is met by $x$ , while if $x\in Z_X$ , a nonmatching edge reaches $y$ and a matching edge reaches $y$ by the preceding alternating step; hence $y\in Z_Y$ .

L1F1

2.1

The matching edges pair $Z_Y$ with $Z_X$ and pair $X\setminus Z_X$ with $Y\setminus Z_Y$ ; exposed left vertices lie in $Z_X$ , so this gives $|M|=|Z_Y|+|X\setminus Z_X|=|C|$ .

step 1.1

3.1

Step 1.2 makes $C$ a vertex cover and step 2.1 gives its size.

step 1.2step 2.1∎

Remarks

The direction of alternation is essential: left-to-right steps are nonmatching edges and right-to-left steps are matching edges.

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 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

M. Goemans, Lecture notes on bipartite matching (standard reference, not scraped)