---
tag: 06C0JBE52D
kind: statement
role: theorem
name: Five colour theorem
msc: [05C15, 05C10]
uses: [029JQEK15B, 0C5M7AM6FH]
sources:
  - url: https://en.wikipedia.org/wiki/Five_color_theorem
    locator: lead section
same_as:
  wikidata: Q105222412
  wikipedia: Five color theorem
---

## Statement

Let $G$ be a finite [[planar graph|029JQEK15B]].
Then $\chi(G) \le 5$.

## Remarks

Proof idea: $G$ has a vertex $v$ of degree at most $5$, by the [[edge bound for planar graphs|0SM1REAVYJ]]. Colour $G - v$ by induction.
If the neighbours of $v$ use all five colours, a swap of two colours along a Kempe chain frees one colour for $v$.
The [[four colour theorem|0M406JC8J1]] is stronger, but its proof is much harder.

## History

Percy John Heawood proved the theorem in 1890.
He found an error in the 1879 proof of the four colour theorem by Alfred Kempe, and he used Kempe's ideas to prove five colours.

## Explanations

### L1 human

Every map can be coloured with five colours so that neighbouring countries have different colours.
The proof is short, unlike the proof that four colours are enough.