Five color theorem
= Five color theorem
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= Five colour theorem
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Every finite <planar graph> admits a proper vertex coloring with at most five colors. <Euler formula for a connected planar graph> supplies a vertex of degree at most five. Inductively color its deletion. The only obstruction is five differently colored neighbours in cyclic order. If opposite neighbours 1 and 3 are in different two-color components, interchange one component. If they are connected, planarity prevents opposite neighbours 2 and 4 from being connected in their two-color subgraph, and that interchange frees a color instead. This is a <Kempe chain> proof.