LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-02

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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.
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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 $C : = (X ∖ Z_{X}) \cup Z_{Y}$ 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 $τ (G)$ and $ρ (G)$ ).

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 $x y$ 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 $X ∖ Z_{X}$ with $Y ∖ Z_{Y}$ ; exposed left vertices lie in $Z_{X}$ , so this gives $∣ M ∣ = ∣ Z_{Y} ∣ + ∣ X ∖ Z_{X} ∣ = ∣ C ∣$ .

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 · two levels

11 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

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