It Appears That Four Colors Suffice: A Historical Overview of the Four-Color Theorem

Certainly any mathematical theorem concerning the coloring of maps would be relevant and widely applicable to modern-day cartography. As for the FourColor Theorem, nothing could be further from the truth. Kenneth May, a twentieth century mathematics historian, explains that “books on cartography and the history of map-making do not mention the four-color property, though they often discuss various other problems relating to the coloring of maps.... The four-color conjecture cannot claim either origin or application in cartography” [WIL2]. In addition, there are virtually no other direct applications of this theorem outside of mathematics; however, mathematicians still continue to search for an elegant proof to this day. In Map Coloring, Polyhedra, and the Four-Color Problem, David Barnette guarantees that these efforts have not been wasted. He points out that many advances in graph theory were made during the process of proving the Four-Color Theorem. These advances provide justification for the 150+ years spent proving a theorem with rather useless results [BAR]. In this paper, the historical progress of the Four-Color Theorem will be examined along with the work of some of its contributors. Without doubt, the Four-Color Theorem is one of the few mathematical problems in history whose origin can be dated precisely. Francis Guthrie (183199), a student in London, first posed the conjecture in October, 1852, while he was coloring the regions on a map of England. He noticed that he could color the map using only four colors in such a way so that no two countries sharing a common border receive the same color. Guthrie wanted to know if this were true of all maps and could it be proven mathematically. He explained his conjecture in a letter to his younger brother Frederick, who was a student at University College in London. Frederick could not solve his brother’s problem and he forwarded it to his instructor Augustus DeMorgan (1806-71), a famous mathematician of the nineteenth century. DeMorgan was very impressed with the conjecture and, although he could not solve it himself, he presented the problem to his colleague and friend Sir William Rowan Hamilton (1805-65), the inventor of quaternions, in a famous letter dated October 23, 1852. This final stage in the “Four-Color Conjecture chain letter” marks the birth of the Four-Color Theorem.