CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-11

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.

$K_{4}$ is planar but has chromatic number four, so the five-colour bound cannot be lowered to three

Statement refuted

The conclusion of Five colour theorem: every planar graph has chromatic number at most five can be strengthened to say that every planar graph is three-colourable.

Facts & Assumptions

Given: The complete graph $K_{4}$ of Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices.

[F1]

A proper vertex colouring assigns distinct colours to adjacent vertices (Proper vertex colourings and chromatic number).

[F2]

Every two distinct vertices of $K_{4}$ are adjacent (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

Counterexample

technique · direct

1.1

Draw three vertices as a triangle, place the fourth inside it, and join that vertex to the three corners. The six edges meet only at their common endpoints, so this is a plane embedding of $K_{4}$ .

F2construct

2.1

By [F1] and [F2], the four vertices must receive pairwise distinct colours in any proper colouring. Assigning a different colour to each vertex is proper, so $χ (K_{4}) = 4$ . Thus a planar graph need not be three-colourable, although Five colour theorem: every planar graph has chromatic number at most five supplies five colours for every planar graph.

F1F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

R. Diestel, Graph Theory, 6th ed., Chapter 5 (standard reference, not scraped)

K4K4​ is planar but has chromatic number four, so the five-colour bound cannot be lowered to three