CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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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.
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$K_4$ 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 $\chi(K_4)=4$ . 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 39 results over 17 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

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

K4K_4K4​ is planar but has chromatic number four, so the five-colour bound cannot be lowered to three