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An update on the fourcolor theorem
 Notices of the AMS
, 1998
"... 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) receive 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 ye ..."
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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) receive 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 FourColor 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 colors 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 appeared; its incorrectness was pointed out by Heawood 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 fivecolor theorem (Theorem 2 below) and discovered what became known as Kempe chains, and Tait found an equivalent formulation of the FourColor Theorem in terms of edge 3coloring, 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 fourcolor conjecture is true for maps with at most twentyfive regions. The same method was used by other mathematicians to make progress on the fourcolor problem. Important 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 researchers 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 development of this method would solve the FourColor Problem. This was confirmed by Appel and Haken (abbreviated A&H) when they published their proof of the FourColor Theorem in two 1977 papers, the second one joint with Koch. An expanded version of the proof was later reprinted in
Invariants de Vassiliev pour les entrelacs dans S³ et dans les variétés de dimension trois
, 1998
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An octonion model for physics
 in Proc. of ECHO IV
, 2000
"... The nozerodivisor division algebra of highest possible dimension over the reals is taken as a model for various physical and mathematical phenomena mostly related to the Four Color Conjecture. A geometric form of associativity is the common thread. ..."
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The nozerodivisor division algebra of highest possible dimension over the reals is taken as a model for various physical and mathematical phenomena mostly related to the Four Color Conjecture. A geometric form of associativity is the common thread.
Graph planarity and related topics
 Graph Drawing (Proc. GD ’99), volume 1731 of LNCS
, 1999
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TemperleyLieb Algebras And The FourColor Theorem
, 2000
"... The TemperleyLieb algebra Tn with parameter 2 is the associative algebra over Q generated by 1, e0, el,..., en, where the generators satisfy the rela 2 = 2el, eiejei = ei if [i j[ = 1 and eiej = ejei if [i j[ > 2. We tions i  use the Four Color Theorem to give a necessary and sufficient ..."
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The TemperleyLieb algebra Tn with parameter 2 is the associative algebra over Q generated by 1, e0, el,..., en, where the generators satisfy the rela 2 = 2el, eiejei = ei if [i j[ = 1 and eiej = ejei if [i j[ > 2. We tions i  use the Four Color Theorem to give a necessary and sufficient condition for certain elements of Tn to be nonzero. It turns out that the characterization is, in fact, equivalent to the Four Color Theorem.