The Four-Color Theorem: History, Topological Foundations, and Idea of Proof
This elegant little book discusses a famous problem that helped to define the field now known as graph theory: what is the minimum number of colors required to print a map such that no two adjoining countries have the same color, no matter how convoluted their boundaries are. Many famous mathematicians have worked on the problem, but the proof eluded formulation until the 1970s, when it was finally cracked with a brute-force approach using a computer. The Four-Color Theorem begins by discussing the history of the problem up to the new approach given in the 1990s (by Neil Robertson, Daniel Sanders, Paul Seymour, and Robin Thomas). The book then goes into the mathematics, with a detailed discussion of how to convert the originally topological problem into a combinatorial one that is both elementary enough that anyone with a basic knowledge of geometry can follow it and also rigorous enough that a mathematician can read it with satisfaction. The authors discuss the mathematics and point to the philosophical debate that ensued when the proof was announced: just what is a mathematical proof, if it takes a computer to provide one - and is such a thing a proof at all?
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admissible 4-coloring Appel and Haken assume Augustus de Morgan belong Birkhoff diamond block decomposition border point boundary coloring bounding circuit C-reducible circuit edge closed chain closed Jordan curve color-pair choice combinatorial graph common borderline configuration consists contains Corollary Definition degree denote distinct edges of G end points essential boundary essential boundary coloring exactly exists final edges finitely Four-Color Problem Four-Color Theorem graph G Guthrie Heawood Heesch Heinrich Heesch Hence homeomorphism inner vertices interior domain interior points intersect joined Jordan curve theorem Kempe chain Kempe interchange Kempe sectors least legs Lemma lies entirely line segments Math mathematical minimal criminal minimal triangulation neutral point neutrality set nonempty normal graph notion obtain open set outer vertices pairwise planar plane graph point in common Proof Let reducibility regular map ring saturated graph Schoenflies theorem subdivided subset three vertices topological triangle unavoidable set unbounded country vertices of G vertices xi
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Essays on the Foundations of Mathematics and Logic, Volume 1
Limited preview - 2005