TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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

Statement

A matching $M$ in a finite graph is maximum if and only if no $M$ -augmenting path exists.

Facts & Assumptions

Given: A matching $M$ of a finite graph $G$ .

[L1]

Switching along an augmenting path produces a matching with exactly one more edge (Switching along an augmenting path increases a matching by one edge).

[L2]

The nontrivial components of the symmetric difference of two matchings are alternating paths and even cycles (The symmetric difference of two matchings is a disjoint union of alternating paths and even cycles).

Proof

technique · direct

1.1

If an $M$ -augmenting path exists, [L1] gives a matching larger than $M$ , so $M$ is not maximum.

1.2

Conversely, suppose $M$ is not maximum and choose a matching $N$ with $|N|>|M|$ ; decompose $M\triangle N$ using [L2].

1.3

Every alternating cycle and every alternating path with equally many $M$ - and $N$ -edges has zero contribution to $|N|-|M|$ , so some path component has one more $N$ -edge than $M$ -edge.

2.1

Such a path begins and ends with $N$ -edges, so both endpoints are $M$ -exposed and it is an $M$ -augmenting path.

step 1.3

3.1

Steps 1.1 and 1.2--2.1 prove both directions of the equivalence.

step 1.1step 2.1∎

Remarks

Finiteness is used only to ensure that a nonmaximum matching has a larger matching and to make the component count finite.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 7 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)