This theorem does not apply when exclaves are present, and neighbors that share only a single point do not meet this theorem's definition of adjacency.
The most recent proof of this theorem was provided by Georges Gonthier using the Coq assistant, while P. G.  Tait showed that it is equivalent to the statement that no snark is planar. The crucial concepts of reducibility and unavoidability were introduced in Kempe's incorrect proof of this theorem, which can be extended to polyhedra by the Heawood conjecture. This theorem does not apply when exclaves are present, and neighbors that share only a single point do not meet this theorem's definition of adjacency. Proven by Appel and Haken using a computer to test 1,936  distinct cases, name this theorem which gives an upper bound on the maximum number of colors required to label a contiguous map.
The most recent proof of this theorem was provided by Georges Gonthier using the Coq assistant, while P. G.  Tait showed that it is equivalent to the statement that no snark is planar.
four-color theorem