For 66 years, research on the four-color theorem was dominated by Tait's Hamiltonian graph conjecture: any cubic polyhedral graph has a Hamiltonian cycle. In a graph, cubic means that every vertex is incident with exactly three edges. Any planar graph can be made cubic by drawing a small circle around any vertex with valence greater than three and eliminating the original vertex. Tutte, in 1946, found the first counterexample to Tait's conjecture.
Tait's method turns a Hamiltonian cycle into a four-coloring.
1. Alternately color the edges of the cycle blue and purple. Color the other edges red. 2. Throw out red edges, and color the resulting polygon blue. 3. Throw out blue edges, and color the resulting polygon(s) red. 4. Overlay the polygons.
The method used in this Demonstration can four-color any planar map that conforms to Tait's conjecture.