Alphabeta Math
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.

A graph on nn vertices with n−1n-1 edges need not be a tree

Statement refuted

Every graph on nn vertices with n−1n-1 edges is a tree.

Facts & Assumptions

Given: The graph G=C3⊔K1G=C_3\sqcup K_1.

[F1]

C3C_3 has three vertices and three edges, while K1K_1 has one vertex and no edges (Empty and complete graphs, complete bipartite graphs, and the convention that PnP_n and CnC_n have nn vertices).

[L1]

The edge-count condition characterizes trees only together with connectedness or acyclicity (Equivalent characterisations of a nonempty tree by unique paths, edge count, minimal connectivity and maximal acyclicity).

Counterexample

technique · direct
1.1

The disjoint union has n=4n=4 vertices and 3=n−13=n-1 edges.

F1
1.2

It is disconnected and also contains the cycle C3C_3, so it is not a tree.

F1L1
2.1

Hence the edge count alone is insufficient.

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: 25 results over 11 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