Any planar map can be colored with four colors so that no two regions of the same color touch each other.[more]
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.
4. and form the boundaries of two regions and
5. Four cases leads to four possible colors for a rectangle, according to whether it is inside or outside or .
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.[less]
Ed Pegg Jr, "Math Games: Square Packing," Dec. 1, 2003.