Graphs, Colourings and the Four-Colour Theorem

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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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Editorial Reviews

From The Critics
This brief monograph considers the attempts to resolve the four-color theorem, and outlines the contributions to graph theory that grew out of those attempts. The book develops the theoretical concepts as it progresses, providing an introduction to elementary graph theory. Chapters discuss Euler's formula, Kenpe's approach, Kuratowski's theorem, reducibility, discharging, and related topics. Wilson teaches at the University of Birmingham. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780198510628
  • Publisher: Oxford University Press, USA
  • Publication date: 1/28/2002
  • Series: Oxford Science Publications
  • Pages: 154
  • Product dimensions: 9.10 (w) x 6.10 (h) x 0.50 (d)

Meet the Author

The University of Birmingham
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Table of Contents

Pt. I Graphs, Maps and the Four-Colour Problem
1 Introduction 3
2 Basic graph theory 6
3 Applications of Euler's formula 19
4 Kempe's approach 31
Pt. II Related Topics
5 Other approaches to the four-colour problem 43
6 Maps on surfaces with holes 58
7 Kuratowski's theorem 72
8 Colouring non-planar graphs 88
Pt. III How to Prove the Four-Colour Theorem
9 Overview 107
10 Reducibility 111
11 Discharging 125
Bibliography 136
Index 139
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