How does the four-colour map problem belong to graph theory?

The four-colour map problem is fundamentally linked to graph theory through the concept of planar graphs.

While the problem originated as a topological question regarding the coloring of regions on a map, it can be translated into an equivalent graph-theoretic form. In this representation, each region of a map is treated as a vertex, and an edge is drawn between two vertices if the corresponding regions share a common boundary. The problem then becomes a question of whether the vertices of a planar graph can be colored using only four colors such that no two vertices connected by an edge share the same color.

This translation allows mathematicians to apply the tools and techniques of graph theory to solve the problem. For instance, the successful 1976 proof by Kenneth Appel and Wolfgang Haken relied on identifying a catalog of "unavoidable" configurations within a graph, demonstrating that the properties of these configurations dictate the colorability of the entire map.12