English: Tangled Kempe chains in the Poussin graph. The map shown has regions whose adjacencies form the Poussin graph. All but one of the regions (the outer region) have been colored, and must be recolored in order to color the whole map. This map and its coloring provides a counterexample to Kempe's erroneous proof of the four color theorem. In this case, because the blue-yellow and blue-green Kempe chains (shown as dashed lines) are connected, Kempe's proof would try to swap the colors on the red-yellow chain on the left, and the red-green chain on the right, freeing the red color for the outer face. But because the blue-yellow and blue-green chains cross each other, Kempe's recoloring would cause the uppermost yellow and green regions to both become red, producing an invalid coloring.