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.

Figure: Four mutually adjacent countries

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 .

References

1

Kenneth O. May.
The origin of the four-color conjecture.
Isis, 56(3):346-348, 1965.
Available online at JSTOR.

2

Thomas L. Saaty and Paul C. Kainen.
The Four-Color Problem: Assaults and Conquest.
Dover, 1986.
Zbl 0463.05041.