Kempe Chain

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.

REFERENCES

Gethner, E. and Springer, W. M. II. "How False Is Kempe's Proof of the Four-Color Theorem?" Congr. Numer. 164, 159-175, 2003.

Hutchinson, J. P. and Wagon, S. "Kempe Revisited." Amer. Math. Monthly 105, 170-174, 1998.Wagon, S. Mathematica in Action, 2nd ed. New York: Springer-Verlag, pp. 535-536, 1999.