|Table of graphs and parameters|
In 1879, Alfred Kempe published a proof of the four color theorem, one of the big conjectures in graph theory. While the theorem is true, Kempe's proof is incorrect. Percy John Heawood illustrated it in 1890 with a counter-example, and de la Vallée-Poussin reached the same conclusion in 1896 with the Poussin graph.
Kempe's (incorrect) proof is based on alternating chains, and as those chains prove useful in graph theory mathematicians remain interested in such counterexamples. More were found later: first, the Errera graph in 1921, then the Kittell graph in 1935, with 23 vertices, and finally two minimal counter-examples (the Soifer graph in 1997 and the Fritsch graph in 1998, both of order 9).
- Kempe, A. B. "On the Geographical Problem of Four-Colors." Amer. J. Math. 2, 193–200, 1879.
- P. J. Heawood, "Map colour theorem", Quart. J. Pure Appl. Math. 24 (1890), 332–338.
- R. A. Wilson, Graphs, colourings and the four-colour theorem, Oxford University Press, Oxford, 2002. MR1888337 Zbl 1007.05002.
- Errera, A. "Du coloriage des cartes et de quelques questions d'analysis situs." Ph.D. thesis. 1921.
- Peter Heinig. Proof that the Errera Graph is a narrow Kempe-Impasse. 2007.
- Kittell, I. "A Group of Operations on a Partially Colored Map." Bull. Amer. Math. Soc. 41, 407–413, 1935.
- A. Soifer, “Map coloring in the victorian age: problems and history”, Mathematics Competitions 10 (1997), 20–31.
- R. Fritsch and G. Fritsch, The Four-Color Theorem, Springer, New York, 1998. MR1633950.
- Gethner, E. and Springer, W. M. II. « How False Is Kempe's Proof of the Four-Color Theorem? » Congr. Numer. 164, 159–175, 2003.