# four-color conjecture

The four-color conjecture was a long-standing problem posed by Francis Guthrie to his professor Augustus De Morgan in 1852, while coloring a map of England. The conjecture states that every map on a plane or a sphere can be colored using only four colors such that no two adjacent countries are assigned the same color. There are maps that need four colors to be colored as the figure below shows. This conjecture equivalent to the statement that chromatic number of every planar graph is no more than four.

After many unsuccessful attempts the conjecture was proven by Appel and Haken in 1976 with the aid of a computer. Before it was known that every planar map can be five-colored by the work of Heawood in 1890.

Interestingly, the seemingly harder problem of determining the maximal number of colors needed for all surfaces other than the sphere was solved long before the four-color conjecture was settled. This is the Heawood number of the surface unless the surface is the Klein bottle in which case it is $6$.

## References

• 1 Kenneth O. May. The origin of the four-color conjecture. Isis, 56(3):346–348, 1965. http://links.jstor.org/sici?sici=0021-1753%28196523%2956%3A3%3C346%3ATOOTFC%3E2.0.CO%3B2-ZAvailable online at http://www.jstor.orgJSTOR.
• 2 Thomas L. Saaty and Paul C. Kainen. The Four-Color Problem: Assaults and Conquest. Dover, 1986. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0463.05041Zbl 0463.05041.
 Title four-color conjecture Canonical name FourcolorConjecture Date of creation 2013-03-22 13:21:18 Last modified on 2013-03-22 13:21:18 Owner bbukh (348) Last modified by bbukh (348) Numerical id 19 Author bbukh (348) Entry type Theorem Classification msc 05C10 Classification msc 05C15 Synonym Appel-Haken theorem Synonym 4-color conjecture Related topic PlanarGraph Related topic ChromaticNumber Related topic HeawoodNumber