CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

An acyclic graph need not be a tree

Statement refuted

Every acyclic graph is a tree.

Facts & Assumptions

Given: The edgeless graph $G=\overline K_2$ .

[F1]

An edgeless graph contains no cycle (Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices).

[F2]

A tree must be connected as well as acyclic (Trees, forests, leaves and isolated vertices).

Counterexample

technique · direct

1.1

The graph $\overline K_2$ is acyclic because it has no edges.

1.2

Its two vertices lie in different components, so it is disconnected and therefore is not a tree.

2.1

Thus acyclicity alone does not imply the tree property.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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