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.

Figure 1: Four mutuallyadjacent countries

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 themaximal number of colors needed for all surfaces other than thesphere was solved long before the four-color conjecture wassettled. This is the Heawood number of the surface unless the surfaceis the Klein bottle in which case it is $6$.