Fourcolor

Fourcolor, more commonly known as the four-color theorem, is a result in mathematics that asserts every planar map can be colored with at most four colors so that no two regions sharing a boundary of nonzero length have the same color. The concept is usually stated in terms of planar graphs: the question is whether the vertices of any planar graph can be colored with four colors so that adjacent vertices receive different colors.

The theorem was conjectured in 1852 by Francis Guthrie and became a famous problem in the field of graph theory and cartography. It resisted proof for more than a century, in part because its truth hinges on checking a large number of possible configurations. A landmark computer-assisted proof by Kenneth Appel and Wolfgang Haken in 1976 established the result by demonstrating that every planar map contains a reducible configuration that leads to a finite verification. A subsequent, more streamlined computer-assisted proof by Robertson, Seymour, Sanders, and Thomas in 1997 is also widely accepted and is seen as refining the original approach.

The four-color theorem has several implications in graph theory and practical coloring problems. It confirms that a universal bound exists for planar maps and has influenced methods in algorithmic graph coloring, the study of planar graphs, and the development of proof techniques such as the discharging method. While four colors suffice for planar maps, more colors may be required for maps drawn on other surfaces; for example, the maximum number of colors needed on a torus exceeds four.