The proof of the four color problem remained elusive for over a century, until two mathematicians with a computer took a closer look.

Math’s Famous Map Problem: The Four-Color Theorem David Richeson discusses the history and significance of the four color theorem.

In this way, the map coloring problem becomes a graph coloring problem: Color the vertices so neighbors are different colors. The minimum number of colors is called the chromatic number of the graph.

This is called a zero-knowledge proof. Since you only show a part of the map's coloring to your opponent, they don't know the exact coloring method (answer).