LemmaStatement: 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 symmetric difference of two matchings is a disjoint union of alternating paths and even cycles

Statement

For matchings $M, N$ of a finite graph $G$ , every nontrivial connected component of the spanning subgraph with edge set $M △ N$ is an alternating path or an even alternating cycle. The components are edge-disjoint.

Facts & Assumptions

Given: Matchings $M, N$ in a finite graph $G$ .

[F1]

A matching is a set of edges no two of which share an endpoint (Matchings, saturated vertices, maximal and maximum matchings, perfect matchings and $ν (G)$ ).

Proof

technique · direct

1.1

Each vertex is incident with at most one edge of $M$ and at most one edge of $N$ , hence has degree at most two in the spanning subgraph on $M △ N$ .

1.2

A nontrivial connected finite graph whose degrees are at most two is a path or a cycle; here its consecutive edges alternate between $M ∖ N$ and $N ∖ M$ .

2.1

A cyclic component has even length, because alternation returns to the initial edge-type only after an even number of edges.

step 1.2

3.1

The nontrivial components partition the edge set $M △ N$ , while degree-zero vertices contribute no edge, which proves the stated disjoint union.

step 1.2step 2.1∎

Depends on

Used by

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

Dependency tree · two levels

9 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)