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  "body": "\n## Statement\n\nLet $G$ be a finite [[planar graph|029JQEK15B]].\nThen $\\chi(G) \\le 5$.\n\n## Remarks\n\nProof idea: $G$ has a vertex $v$ of degree at most $5$, by the [[edge bound for planar graphs|0SM1REAVYJ]]. Colour $G - v$ by induction.\nIf the neighbours of $v$ use all five colours, a swap of two colours along a Kempe chain frees one colour for $v$.\nThe [[four colour theorem|0M406JC8J1]] is stronger, but its proof is much harder.\n\n## History\n\nPercy John Heawood proved the theorem in 1890.\nHe 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.\n\n## Explanations\n\n### L1 human\n\nEvery map can be coloured with five colours so that neighbouring countries have different colours.\nThe proof is short, unlike the proof that four colours are enough.\n",
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