What is a "map" in the four color theorem?

Question

The four color theorem declares that any map in the plane (and, more generally, spheres and so on) can be colored with four colors so that no two adjacent regions have the same colors.

However, it's not clear what constitutes a map, or a region in a map. Is this actually a theorem in graph theory, something about every planar graph with some properties admitting a certain coloring?

Or does one really prove it using regions in the plane (which I guess we take to have non-fractal boundaries, or something). What is the precise definition?

Answer 1

Here is a formal statement. Let $G=(V,E)$ be a finite planar graph with vertices $V$ and edges $E$. It is possible to find a collection $P=\{S_1,S_2,S_3,S_4\}$ of dijoint subsets of $V$ such that $V=S_1\cup S_2\cup S_3\cup S_4$ and for every $a,b\in S_i$, there is no edge in $E$ joining $a$ and $b$.

The map is the graph, and regions are the vertices.

Answer 2

The theorem is a statement about planar graphs. In the application to geography, you have a finite collection of "countries" (disjoint connected open subsets of the plane with, let's say, piecewise-smooth boundaries). Two countries are considered to be adjacent if the intersection of their boundaries has nonzero length. You then consider the graph with vertices corresponding to countries and edges between adjacent countries.