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TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11
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  • 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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Every planar graph has a proper vertex colouring with at most six colours

Statement

Every planar graph has a proper vertex colouring with at most six colours (Proper vertex colourings and chromatic number). Vertex deletion is from Vertex and edge deletion, edge contraction, graph minors, subdivisions and topological minors, and the proof is finite induction The principle of mathematical induction.

Facts & Assumptions

Given: A finite simple planar graph G.

[L1]

Every nonnull simple planar graph has a vertex of degree at most five (Every nonnull simple planar graph has a vertex of degree at most five).

[F1]

A proper k-vertex-colouring is a function c:V→k such that c(u)≠c(v) whenever {u,v}∈E (Proper vertex colourings and chromatic number).

Proof

technique · induction
1.1

The null graph has the empty proper colouring.

baseF1
1.2

For a nonnull graph choose by [L1] a vertex v of degree at most five. The planar graph G−v has a proper six-colouring by the induction hypothesis.

ihL1
2.1

At most five colours appear on the neighbours of v, so one of the six colours is absent there. Give v that colour. Edges not incident with v remain proper, and every edge incident with v has differently coloured endpoints by construction.

step 1.2F1
3.1

This extends the induction colouring at every nonnull stage, so every planar graph is six-colourable.

step 1.1step 2.1discharge-induction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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