The Heawood four-color graph is the 25-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 Heawood four-color graph is implemented in the Wolfram Language as GraphData["HeawoodFourColorGraph"].
The Fritsch graph and Soifer graph provide smaller (and in fact the smallest possible) counterexamples.
