The Fritsch graph and its dual map. Shown is a partial 4-coloring with the center region uncolored. Because the red–green and blue–green Kempe chains cross, it is not possible to swap the colors in a single Kempe chain in order to make one of the four colors available to the center region, giving a small counterexample to Alfred Kempe's false proof of the four-color theorem.
Own work
David Eppstein