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.

$τ (C_{n}) = n$ for every $n \geq 3$

Statement

For every $n \geq 3$ , the cycle graph $C_{n}$ has exactly $n$ spanning trees.

Facts & Assumptions

Given: $n \geq 3$ and the cycle graph $C_{n}$ .

[F1]

$C_{n}$ has $n$ edges and deleting an arbitrary edge gives the path $P_{n}$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[L1]

A spanning tree on $n$ vertices has $n - 1$ edges (Equivalent characterisations of a nonempty tree by unique paths, edge count, minimal connectivity and maximal acyclicity).

[F2]

$τ$ counts spanning trees (The spanning-tree number $τ (G)$ , Spanning trees of a graph).

Verification

technique · direct

1.1

Deleting any one edge of $C_{n}$ gives a connected acyclic spanning graph, hence a spanning tree. The $n$ choices yield distinct trees.

F1F2

1.2

Conversely, every spanning tree uses $n - 1$ of the $n$ cycle edges, so it is obtained by deleting exactly one edge.

L1F1

2.1

Therefore $τ (C_{n}) = n$ .

step 1.1step 1.2F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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)

τ(Cn)=nτ(Cn​)=n for every n≥3n≥3