CorollaryStatement: 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.

The maximum matching from $X$ has size $∣ X ∣ - \max_{S \subseteq X} (∣ S ∣ - ∣ N (S) ∣)$

Statement

Let $(X, Y)$ be a finite bipartite graph and put $d : = \max_{S \subseteq X} (∣ S ∣ - ∣ N (S) ∣) .$ Then the greatest number of vertices of $X$ saturable by a matching is $∣ X ∣ - d$ .

Facts & Assumptions

Given: A finite bipartite graph with parts $(X, Y)$ and the displayed $d$ .

[L1]

Hall's theorem supplies a matching saturating a finite left part exactly under Hall's inequalities (Hall's marriage theorem for a finite bipartite graph).

Proof

technique · direct

1.1

Any matching saturating $r$ vertices of $X$ leaves at least $∣ S ∣ - ∣ N (S) ∣$ vertices of every $S \subseteq X$ unmatched, so $r \leq ∣ X ∣ - d$ .

given

1.2

Adjoin $d$ new right vertices, adjacent to every $x \in X$ ; then every $S \subseteq X$ has at least $∣ N (S) ∣ + d \geq ∣ S ∣$ neighbours, so [L1] gives a matching saturating $X$ in the enlarged graph.

2.1

At most $d$ of its matching edges use new vertices, so deleting those edges leaves a matching of the original graph saturating at least $∣ X ∣ - d$ vertices of $X$ .

step 1.2

3.1

The upper bound in step 1.1 and lower bound in step 2.1 establish the formula.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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)

The maximum matching from XX has size ∣X∣−max⁡S⊆X(∣S∣−∣N(S)∣)∣X∣−maxS⊆X​(∣S∣−∣N(S)∣)