Math 2.0

statement theorem

Five colour theorem

Statement

Let GG be a finite planar graph. Then χ(G)≤5\chi(G) \le 5.

Remarks

Proof idea: GG has a vertex vv of degree at most 55, by the edge bound for planar graphs. Colour G−vG - v by induction. If the neighbours of vv use all five colours, a swap of two colours along a Kempe chain frees one colour for vv. The four colour theorem 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.

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Graph colouring, definition, no evidence levelGraph colouringdefinitionPlanar graph, definition, no evidence levelPlanar graphdefinitionK-colourable graph, definition, no evidence levelK-colourable graphdefinitionChromatic number, definition, no evidence levelChromatic numberdefinitionFive colour theorem, statement, no evidence levelFive colour theoremstatement
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  1. Graph colouring definition
  2. Planar graph definition
  3. K-colourable graph definition
  4. Chromatic number definition
  5. Five colour theorem statement

Sources

  1. Web https://en.wikipedia.org/wiki/Five_color_theorem, lead section

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