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.
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The Five Color Theorem is a result in mathematical graph theory that states that any planar graph can be colored with at most five colors such that no two adjacent vertices share the same color. This theorem is a weakening of the famous Four Color Theorem, which asserts that four colors are sufficient to color any planar graph.