Any planar map can be colored with four colors so that no two regions of the same color touch each other.
This Demonstration uses the following method to four-color each map:
1. A map of rectangles is converted to a cubic graph (not shown).
2. Cycle set
is a non-unique Hamiltonian cycle. Color its edges with two colors.
3. Eliminate one color of
and color the edges that are not in
in a third color. The result is a cycle set
in two colors.
form the boundaries of two regions
5. Four cases leads to four possible colors for a rectangle, according to whether it is inside or outside
The method does not always work, since some cubic graphs exist that are not Hamiltonian. They are still four-colorable, but not by this method.