Download or read book Map Color Theorem written by G. Ringel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book The Four Color Theorem written by Rudolf Fritsch and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 269 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Graph Coloring Problems written by Tommy R. Jensen and published by John Wiley & Sons. This book was released on 2011-10-24 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains a wealth of information previously scattered in research journals, conference proceedings and technical reports. Identifies more than 200 unsolved problems. Every problem is stated in a self-contained, extremely accessible format, followed by comments on its history, related results and literature. The book will stimulate research and help avoid efforts on solving already settled problems. Each chapter concludes with a comprehensive list of references which will lead readers to original sources, important contributions and other surveys.

Download or read book Discrete Mathematics written by Oscar Levin and published by Createspace Independent Publishing Platform. This book was released on 2018-07-30 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: Note: This is a custom edition of Levin's full Discrete Mathematics text, arranged specifically for use in a discrete math course for future elementary and middle school teachers. (It is NOT a new and updated edition of the main text.)This gentle introduction to discrete mathematics is written for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the "introduction to proof" course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this.Four main topics are covered: counting, sequences, logic, and graph theory. Along the way proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs.While there are many fine discrete math textbooks available, this text has the following advantages: - It is written to be used in an inquiry rich course.- It is written to be used in a course for future math teachers.- It is open source, with low cost print editions and free electronic editions.

Download or read book The Four color Problem written by Thomas L. Saaty and published by . This book was released on 1986 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Every Planar Map is Four Colorable written by Kenneth I. Appel and published by American Mathematical Soc.. This book was released on 1989 with total page 760 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Map Coloring Polyhedra and the Four color Problem written by David Barnette and published by MAA Press. This book was released on 1983 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Map Coloring Polyhedra and the Four color Problem written by David Wilmot Barnette and published by . This book was released on 1983 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book History of Topology written by I.M. James and published by Elsevier. This book was released on 1999-08-24 with total page 1067 pages. Available in PDF, EPUB and Kindle. Book excerpt: Topology, for many years, has been one of the most exciting and influential fields of research in modern mathematics. Although its origins may be traced back several hundred years, it was Poincaré who "gave topology wings" in a classic series of articles published around the turn of the century. While the earlier history, sometimes called the prehistory, is also considered, this volume is mainly concerned with the more recent history of topology, from Poincaré onwards.As will be seen from the list of contents the articles cover a wide range of topics. Some are more technical than others, but the reader without a great deal of technical knowledge should still find most of the articles accessible. Some are written by professional historians of mathematics, others by historically-minded mathematicians, who tend to have a different viewpoint.

Download or read book Map Coloring Polyhedra and the FourColor Problem written by David W. Barnette and published by American Mathematical Soc.. This book was released on 1984-03-01 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book How To Color A Map written by Sankaranarayanan Veeramoni Mythili and published by . This book was released on 2014 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the maximum differential coloring problem, where an n-vertex graph must be colored with colors numbered 1, 2 ... n such that the minimal difference between the two colors of any edge is maximized. This problem is motivated by coloring maps in which not all countries are contiguous. Since it is known that this problem is NP-hard for general graphs; we consider planar graphs and subclasses thereof. In Chapter 1 we introduce the topic of this thesis and in Chapter 2 we review relevant definitions and basic results. In Chapter 3 we prove that the maximum differential coloring problem remains NP-hard even for planar graphs. Then, we present tight bounds for regular caterpillars and spider graphs and close-to-optimal differential coloring algorithms for general caterpillars and biconnected triangle-free outer-planar graphs. In Chapter 4 we introduce the (d, kn)-differential coloring problem. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NP-complete, in this chapter we provide a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the ([2n/3], 2n)-differential coloring problem, even when input graph is planar. In Chapter 5 we experimentally evaluate and compare several algorithms for coloring a map. Motivated by different application scenarios, we classify our approaches into two categories, depending on the dimensionality of the underlying color space. To cope with the one dimensional color space (e.g., gray-scale colors), we employ the (d, kn)-differential coloring. In Chapter 6 we describe a practical approach for visualizing multiple relationships defined on the same dataset using a geographic map metaphor, where clusters of nodes form countries and neighboring countries correspond to nearby clusters. The aim is to provide a visualization that allows us to compare two or more such maps. In the case where we are considering multiple relationships we also provide an interactive tool to visually explore the effect of combining two or more such relationships. Our method ensures good readability and mental map preservation, based on dynamic node placement with node stability, dynamic clustering with cluster stability, and dynamic coloring with color stability. Finally in Chapter 7 we discuss future work and open problems.

Download or read book Four Colors Suffice written by Robin J. Wilson and published by Princeton University Press. This book was released on 2002 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: On October 23, 1852, Professor Augustus De Morgan wrote a letter to a colleague, unaware that he was launching one of the most famous mathematical conundrums in history--one that would confound thousands of puzzlers for more than a century. This is the amazing story of how the "map problem" was solved. The problem posed in the letter came from a former student: What is the least possible number of colors needed to fill in any map (real or invented) so that neighboring counties are always colored differently? This deceptively simple question was of minimal interest to cartographers, who saw little need to limit how many colors they used. But the problem set off a frenzy among professional mathematicians and amateur problem solvers, among them Lewis Carroll, an astronomer, a botanist, an obsessive golfer, the Bishop of London, a man who set his watch only once a year, a California traffic cop, and a bridegroom who spent his honeymoon coloring maps. In their pursuit of the solution, mathematicians painted maps on doughnuts and horseshoes and played with patterned soccer balls and the great rhombicuboctahedron. It would be more than one hundred years (and countless colored maps) later before the result was finally established. Even then, difficult questions remained, and the intricate solution--which involved no fewer than 1,200 hours of computer time--was greeted with as much dismay as enthusiasm. Providing a clear and elegant explanation of the problem and the proof, Robin Wilson tells how a seemingly innocuous question baffled great minds and stimulated exciting mathematics with far-flung applications. This is the entertaining story of those who failed to prove, and those who ultimately did prove, that four colors do indeed suffice to color any map.

Download or read book Four Colours Suffice written by Robin J. Wilson and published by Allan Lane. This book was released on 2002 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: A puzzlers delight for over a century, the four-colour problem was one of the most famous conundrums in mathematics, if not the most famous, and many thousands of puzzlers - amateur problem-solvers and professional mathematicians alike - have struggled to answer it. The problem is simply stated, and involves the colouring of maps: Can every map be coloured with no more than four colours so that neighbouring countries are coloured differently?

Download or read book R Graphics Cookbook written by Winston Chang and published by "O'Reilly Media, Inc.". This book was released on 2013 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Practical recipes for visualizing data"--Cover.

Download or read book Four Colors Suffice written by Robin Wilson and published by Princeton University Press. This book was released on 2013-11-10 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: First published: London: Allen Lane, 2002, under title Four colours suffice.

Download or read book The Four Color Problem written by and published by Academic Press. This book was released on 2011-08-29 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Four-Color Problem

Download or read book Chromatic Graph Theory written by Gary Chartrand and published by CRC Press. This book was released on 2019-11-28 with total page 503 pages. Available in PDF, EPUB and Kindle. Book excerpt: With Chromatic Graph Theory, Second Edition, the authors present various fundamentals of graph theory that lie outside of graph colorings, including basic terminology and results, trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. Readers will see that the authors accomplished the primary goal of this textbook, which is to introduce graph theory with a coloring theme and to look at graph colorings in various ways. The textbook also covers vertex colorings and bounds for the chromatic number, vertex colorings of graphs embedded on surfaces, and a variety of restricted vertex colorings. The authors also describe edge colorings, monochromatic and rainbow edge colorings, complete vertex colorings, several distinguishing vertex and edge colorings. Features of the Second Edition: The book can be used for a first course in graph theory as well as a graduate course The primary topic in the book is graph coloring The book begins with an introduction to graph theory so assumes no previous course The authors are the most widely-published team on graph theory Many new examples and exercises enhance the new edition