This book discusses a famous problem that helped to define the field now known as topology: What is the minimum number of colors required to print a map so that no two adjoining countries have the same color? This problem remained unsolved until the 1950s, when it was finally cracked using a computer. This book discusses the history and mathematics of the problem, as well as the philosophical debate which ensued, regarding the validity of computer generated proofs.
Explore a variety of fascinating concepts relating to the four-color theorem with an accessible introduction to related concepts from basic graph theory. From a clear explanation of Heawood's disproof of Kempe's argument to novel features like quadrilateral switching, this book by Chris McMullen, Ph.D., is packed with content. It even includes a novel handwaving argument explaining why the four-color theorem is true. What is the four-color theorem? Why is it common to work with graphs instead of maps? What are Kempe chains? What is the problem with Alfred Kempe's attempted proof? How does Euler's formula relate the numbers of faces, edges, and vertices? What are Kuratowski's theorem and Wagner's theorem? What is the motivation behind triangulation? What is quadrilateral switching? What is vertex splitting? What is the three-edges theorem? Is there an algorithm for four-coloring a map or graph? What is a Hamiltonian cycle? What is a separating triangle? How is the four-color theorem like an ill-conditioned logic puzzle? Why is the four-color theorem true? What makes the four-color theorem so difficult to prove by hand?
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.
The four-colour problem was one of the most famous and controversial conundrums ever known, and stumped thousands of puzzlers for over a century. It sounded simple- what is the least number of colours needed to fill in any map, so that neighbouring countries are always coloured differently? However, it would take over a hundred years for amateur problem-solvers and mathematicians alike to answer the question first posed by Francis Guthrie in 1852. And, even when a solution was finally found using computers, debate raged over whether this technology could ever provide the proof that traditional pen-and-paper calculations could. This is the gripping story of the race to solve the riddle - a tale of dedicated puzzlers, mind-boggling maps, human ingenuity and the great rhombicuboctahedron
In this volume, the authors present their 1972 proof of the celebrated Four Color Theorem in a detailed but self-contained exposition accessible to a general mathematical audience. An emended version of the authors' proof of the theorem, the book contains the full text of the supplements and checklists, which originally appeared on microfiche. The thiry-page introduction, intended for nonspecialists, provides some historical background of the theorem and details of the authors' proof. In addition, the authors have added an appendix which treats in much greater detail the argument for situations in which reducible configurations are immersed rather than embedded in triangulations. This result leads to a proof that four coloring can be accomplished in polynomial time.
Robert Wilson discusses the four-colour theorem and some of the mathematics which developed out of attempts to solve it. He covers basic graph theory, Euler's polyhedral formula and the first published false solution of the four-colour problem.
Graph theory is one of the fastest growing branches of mathematics. Until recently, it was regarded as a branch of combinatorics and was best known by the famous four-color theorem stating that any map can be colored using only four colors such that no two bordering countries have the same color. Now graph theory is an area of its own with many deep results and beautiful open problems. Graph theory has numerous applications in almost every field of science and has attracted new interest because of its relevance to such technological problems as computer and telephone networking and, of course, the internet. In this new book in the Johns Hopkins Studies in the Mathematical Science series, Bojan Mohar and Carsten Thomassen look at a relatively new area of graph theory: that associated with curved surfaces. Graphs on surfaces form a natural link between discrete and continuous mathematics. The book provides a rigorous and concise introduction to graphs on surfaces and surveys some of the recent developments in this area. Among the basic results discussed are Kuratowski's theorem and other planarity criteria, the Jordan Curve Theorem and some of its extensions, the classification of surfaces, and the Heffter-Edmonds-Ringel rotation principle, which makes it possible to treat graphs on surfaces in a purely combinatorial way. The genus of a graph, contractability of cycles, edge-width, and face-width are treated purely combinatorially, and several results related to these concepts are included. The extension by Robertson and Seymour of Kuratowski's theorem to higher surfaces is discussed in detail, and a shorter proof is presented. The book concludes with a survey of recent developments on coloring graphs on surfaces.
In 1890 P. J. Heawood [35] published a formula which he called the Map Colour Theorem. But he forgot to prove it. Therefore the world of mathematicians called it the Heawood Conjecture. In 1968 the formula was proven and therefore again called the Map Color Theorem. (This book is written in California, thus in American English. ) Beautiful combinatorial methods were developed in order to prove the formula. The proof is divided into twelve cases. In 1966 there were three of them still unsolved. In the academic year 1967/68 J. W. T. Youngs on those three cases at Santa Cruz. Sur invited me to work with him prisingly our joint effort led to the solution of all three cases. It was a year of hard work but great pleasure. Working together was extremely profitable and enjoyable. In spite of the fact that we saw each other every day, Ted wrote a letter to me, which I present here in shortened form: Santa Cruz, March 1, 1968 Dear Gerhard: Last night while I was checking our results on Cases 2, 8 and 11, and thinking of the great pleasure we had in the afternoon with the extra ordinarily elegant new solution for Case 11, it seemed to me appropriate to pause for a few minutes and dictate a historical memorandum. We began working on Case 8 on 10 October 1967, and it was settled on Tuesday night, 14 November 1967.
