CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-02

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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.
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The middle edge of $P_{4}$ is a maximal matching but the two end edges form a larger matching

Counterexample

Let $P_{4}$ have consecutive vertices $1, 2, 3, 4$ and let $M = {23}$ . Then $M$ is maximal but not maximum: ${12, 34}$ is a matching of size two.

Facts & Assumptions

Given: The path $P_{4}$ and its middle-edge matching $M = {23}$ .

[L1]

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

Verification

Verification technique: direct.

1.1

The two remaining edges $12$ and $34$ each meet $23$ , so neither can be added to $M$ ; hence $M$ is maximal.

given

1.2

The path $1, 2, 3, 4$ is $M$ -augmenting, and [L1] therefore also shows that $M$ is not maximum.

2.1

Independently, ${12, 34}$ is a disjoint two-edge matching, so the maximal and maximum notions differ in this finite path.

step 1.1step 1.2∎

Depends on

Used by

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The middle edge of P4P4​ is a maximal matching but the two end edges form a larger matching