Alphabeta Math
CounterexampleConstruction: 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 graph on n vertices with n−1 edges need not be a tree

Statement refuted

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

Facts & Assumptions

Given: The graph G=C3⊔K1.

[F1]

C3 has three vertices and three edges, while K1 has one vertex and no edges (Empty and complete graphs, complete bipartite graphs, and the convention that Pn and Cn have n 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=4 vertices and 3=n−1 edges.

F1
1.2

It is disconnected and also contains the cycle C3, 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 · two levels

13 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