Alphabeta Math

Plane Graphs, Euler's Formula and the Five Colour Theorem — Examples

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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.

✓ 7 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 7 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

Euler's formula checked on a plane tree, a cycle, and the four-face embedding of K₄

Counterexample

K₄ is planar but has chromatic number four, so the five-colour bound cannot be lowered to three

Example

The Petersen graph is nonplanar by an explicit subdivision of K_3,3 after deleting one vertex

Example

Two plane embeddings of one connected planar graph have nonisomorphic dual multigraphs

Example

A degree-five vertex is inserted into a plane graph after one explicit Kempe-chain colour swap

Counterexample

K_3,3 satisfies |E|≤3|V|-6 but is nonplanar, so the planar edge bound is not sufficient

Counterexample

An injective nonpolygonal arc drawing is excluded by the page's finite polygonal plane-graph convention

Sources