Let be a planar graph whose vertices have been properly colored and suppose is colored . Define the -Kempe chain containing to be the maximal connected component of that

1. Contains , and

2. Contains only vertices that are colored with elements from

(Gethner and Springer 2003).

A number of small graphs (with the vertex count) that tangle the chains in Kempe's algorithm and so provide examples of how Kempe's supposed proof of the four-color theorem fails are illustrated above and summarized in the following table.

graph name | |

9 | Fritsch graph |

9 | Soifer graph |

15 | Poussin graph |

17 | Errera graph |

23 | Kittell graph |

25 | Heawood four-color graph |

Interestingly, a number of these examples (though not the Soifer graph, which contains a 4-cycle) correspond to the skeletons of (possibly degenerate) deltahedra (E. Weisstein, Mar. 7, 2022). In particular, the Fritsch graph is the skeleton of the triaugmented triangular prism and the Errera graph is the skeleton of two pentagon-adjoined gyroelongated pentagonal pyramids.