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How do planar graphs relate to the four-colour map problem?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In topology and graph theory, the four-colour map problem and planar graphs are fundamentally linked through the translation of map regions into graph structures.

The four-colour map problem asks whether any map can be coloured with at most four colours such that no two adjacent regions share the same colour. This can be expressed in an equivalent graph-theoretic form: one can represent each region of a map as a vertex in a graph, with edges connecting vertices that correspond to adjacent regions. Because these regions exist on a plane (or sphere) without overlapping in a way that forces non-planar connections, the resulting structure is a planar graph. The problem then becomes determining whether the vertices of such a planar graph can always be coloured with four colours so that no two connected vertices share the same colour.1

This relationship is central to the 1976 proof by Kenneth Appel and Wolfgang Haken, which utilized the properties of planar configurations to demonstrate that all such maps are indeed four-colourable.21