An update on the four-color theorem

very planar map of connected countries can be colored using four colors in such a way that countries with a common boundary segment (not just a point) re-ceive different colors. It is amazing that such a simply stated result resisted proof for one and a quarter centuries, and even today it is not yet fully understood. In this article I concentrate on recent developments: equivalent formulations, a new proof, and progress on some generalizations. Brief History The Four-Color Problem dates back to 1852 when Francis Guthrie, while trying to color the map of the counties of England, noticed that four colors sufficed. He asked his brother Frederick if it was true that any map can be colored using four col-ors in such a way that adjacent regions (i.e., those sharing a common boundary segment, not just a point) receive different colors. Frederick Guthrie then communicated the conjecture to DeMorgan. The first printed reference is by Cayley in 1878. A year later the first “proof ” by Kempe ap-peared; its incorrectness was pointed out by Hea-wood eleven years later. Another failed proof was published by Tait in 1880; a gap in the argument was pointed out by Petersen in 1891. Both failed proofs did have some value, though. Kempe proved the five-color theorem (Theorem 2 below) and discovered what became known as Kempe chains, and Tait found an equivalent formulation of the Four-Color Theorem in terms of edge 3-coloring, stated here as Theorem 3. The next major contribution came in 1913 from G. D. Birkhoff, whose work allowed Franklin to prove in 1922 that the four-color conjecture is true for maps with at most twenty-five regions. The same method was used by other mathematicians to make progress on the four-color problem. Im-portant here is the work by Heesch, who developed the two main ingredients needed for the ultimate proof—“reducibility ” and “discharging”. While the concept of reducibility was studied by other re-searchers as well, the idea of discharging, crucial for the unavoidability part of the proof, is due to Heesch, and he also conjectured that a suitable de-velopment of this method would solve the Four-Color Problem. This was confirmed by Appel and Haken (abbreviated A&H) when they published their proof of the Four-Color Theorem in two 1977 papers, the second one joint with Koch. An ex-panded version of the proof was later reprinted i