ExampleConstruction: AI-adaptedVerification: 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.

A connected graph with two distinct spanning trees

Statement

The triangle $C_{3}$ is connected and has three distinct spanning trees, obtained by deleting any one edge.

Facts & Assumptions

Given: The cycle graph $C_{3}$ .

[F1]

$C_{3}$ is connected and has three edges (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[F2]

A spanning tree is a connected acyclic spanning subgraph (Spanning trees of a graph).

[L1]

A finite graph is connected if and only if it has a spanning tree (A finite graph is connected if and only if it has a spanning tree).

Verification

technique · direct

1.1

Deleting any edge of $C_{3}$ leaves a two-edge path on all three vertices, so each result is a spanning tree.

F1F2

2.1

The three deleted edges are different, so the three resulting edge sets are distinct. In particular, this connected graph has at least two spanning trees.

step 1.1L1∎

Depends on

Used by

Every edge of a connected graph lies in every spanning tree False statement

Dependency tree · two levels

11 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)