Graphs, Colourings and the Four-Colour Theorem

Graphs, Colourings and the Four-Colour Theorem

by Robert A. Wilson
     
 

ISBN-10: 0198510624

ISBN-13: 9780198510628

Pub. Date: 03/28/2002

Publisher: Oxford University Press, USA

The four-colour theorem is one of the famous problems of mathematics, that frustrated generations of mathematicians from its birth in 1852 to its solution (using substantial assistance from electronic computers) in 1976. The theorem asks whether four colours are sufficient to colour all conceivable maps, in such a way that countries with a common border are

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Overview

The four-colour theorem is one of the famous problems of mathematics, that frustrated generations of mathematicians from its birth in 1852 to its solution (using substantial assistance from electronic computers) in 1976. The theorem asks whether four colours are sufficient to colour all conceivable maps, in such a way that countries with a common border are coloured with different colours. The book discusses various attempts to solve this problem, and some of the mathematics which developed out of these attempts. Much of this mathematics has developed a life of its own, and forms a fascinating part of the subject now known as graph theory. The book is designed to be self-contained, and develops all the graph theoretical tools needed as it goes along. It includes all the elementary graph theory that should be included in an introduction to the subject, before concentrating on specific topics relevant to the four-colour problem. Part I covers basic graph theory, Euler's polyhedral formula, and the first published false proof of the four-colour theorem. Part II ranges widely through related topics, including map-colouring on surfaces with holes, the famous theorems of Kuratowski, Vizing, and Brooks, the conjectures of Hadwiger and Hajos, and much more besides. In Part III we return to the four-colour theorem, and study in detail the methods which finally cracked the problem.

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Product Details

ISBN-13:
9780198510628
Publisher:
Oxford University Press, USA
Publication date:
03/28/2002
Series:
Oxford Science Publications
Pages:
154
Product dimensions:
9.10(w) x 6.10(h) x 0.50(d)

Table of Contents

Pt. IGraphs, Maps and the Four-Colour Problem
1Introduction3
2Basic graph theory6
3Applications of Euler's formula19
4Kempe's approach31
Pt. IIRelated Topics
5Other approaches to the four-colour problem43
6Maps on surfaces with holes58
7Kuratowski's theorem72
8Colouring non-planar graphs88
Pt. IIIHow to Prove the Four-Colour Theorem
9Overview107
10Reducibility111
11Discharging125
Bibliography136
Index139

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