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The four-color theorem. History, topological foundations, and idea of proof. Transl. from the German by Julie Peschke. (English) Zbl 0908.05041
New York, NY: Springer. xvi, 260 p. (1998).
The book discusses a well-known problem: what is the minimum number of colours to colour the countries of a map that no two countries with common border have the same colour. By using a IBM 360 computer Appel and Haken achieved that four colours suffice. In 1976 in this way the complete proof of the four-colour theorem (4CT) was given. More than one hundred years this prolem stimulated the highly interesting new field now known as graph theory.
The book gives a short biography of researchers who posed or finally settled this problem with complete bibliography. In Chapter 5, a purely combinatorial formulation of 4CT without any reference to geometry or topology is given. The last two chapters discuss the crucial steps in the proof of 4CT: reducibility of certain configurations, and discharging procedures. The improvements to the proof of 4CT by N. Robertson, D. Sanders, P. Seymour, and R. Thomas [J.Comb. Theory, Ser. B 70, No. 1, 2-44 (1997; Zbl 0883.05056)]can be viewed in internet web site.

MSC:
05C15 Coloring of graphs and hypergraphs
05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05-03 History of combinatorics
54H99 Connections of general topology with other structures, applications
57M15 Relations of low-dimensional topology with graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory
01A55 History of mathematics in the 19th century
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