The Poussin graph is the 15-node planar graph illustrated above that tangles the Kempe chains in Kempe's algorithm
and thus provides an example of how Kempe's supposed proof of the four-color
theorem fails.
The Fritsch graph and Soifer
graph provide smaller (and in fact the smallest possible) counterexamples.
Gethner, E. and Springer, W. M. II. "How False Is Kempe's Proof of the Four-Color Theorem?" Congr. Numer.164,
159-175, 2003.Kempe, A. B. "On the Geographical Problem of
Four-Colors." Amer. J. Math.2, 193-200, 1879.
