LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-01

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.

Every edge outside a spanning tree determines a unique fundamental cycle

Statement

If $T$ is a spanning tree of $G$ and $e = u v \in E (G) ∖ E (T)$ , then $T + e$ has a unique cycle, namely $e$ together with the unique $u$ - $v$ path in $T$ . This is the fundamental cycle of $e$ with respect to $T$ .

Facts & Assumptions

Given: A spanning tree $T$ of $G$ and $e = u v \notin E (T)$ .

[L1]

A tree has a unique path between each two vertices (Equivalent characterisations of a nonempty tree by unique paths, edge count, minimal connectivity and maximal acyclicity).

[L2]

Adding an edge between nonadjacent vertices of a tree creates exactly one cycle (Adding an edge between two nonadjacent vertices of a tree creates exactly one cycle).

[F1]

$T$ contains every vertex of $G$ (Spanning trees of a graph).

Proof

technique · direct

1.1

The endpoints $u, v$ belong to $T$ and are nonadjacent there.

GivenF1

2.1

Let $P$ be the unique $u$ - $v$ path in $T$ . By edge addition, $P + e$ is the unique cycle in $T + e$ .

step 1.1L1L2∎

Depends on

Used by

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

Reinhard Diestel, Graph Theory, Preview Chapter 1 (standard reference, not scraped)