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.

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.

Reviewer: J.Fiamčik (Prešov)

##### 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 |