The Four Colour Theorem
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A Brief History of Edge-Colorings — with Personal Reminiscences
Discrete Mathematics Letters Discrete Math. Lett. 6 (2021) 38–46 www.dmlett.com DOI: 10.47443/dml.2021.s105 Review Article A brief history of edge-colorings – with personal reminiscences∗ Bjarne Toft1;y, Robin Wilson2;3 1Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark 2Department of Mathematics and Statistics, Open University, Walton Hall, Milton Keynes, UK 3Department of Mathematics, London School of Economics and Political Science, London, UK (Received: 9 June 2020. Accepted: 27 June 2020. Published online: 11 March 2021.) c 2021 the authors. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/). Abstract In this article we survey some important milestones in the history of edge-colorings of graphs, from the earliest contributions of Peter Guthrie Tait and Denes´ Konig¨ to very recent work. Keywords: edge-coloring; graph theory history; Frank Harary. 2020 Mathematics Subject Classification: 01A60, 05-03, 05C15. 1. Introduction We begin with some basic remarks. If G is a graph, then its chromatic index or edge-chromatic number χ0(G) is the smallest number of colors needed to color its edges so that adjacent edges (those with a vertex in common) are colored differently; for 0 0 0 example, if G is an even cycle then χ (G) = 2, and if G is an odd cycle then χ (G) = 3. For complete graphs, χ (Kn) = n−1 if 0 0 n is even and χ (Kn) = n if n is odd, and for complete bipartite graphs, χ (Kr;s) = max(r; s). -
Additive Non-Approximability of Chromatic Number in Proper Minor
Additive non-approximability of chromatic number in proper minor-closed classes Zdenˇek Dvoˇr´ak∗ Ken-ichi Kawarabayashi† Abstract Robin Thomas asked whether for every proper minor-closed class , there exists a polynomial-time algorithm approximating the chro- G matic number of graphs from up to a constant additive error inde- G pendent on the class . We show this is not the case: unless P = NP, G for every integer k 1, there is no polynomial-time algorithm to color ≥ a K -minor-free graph G using at most χ(G)+ k 1 colors. More 4k+1 − generally, for every k 1 and 1 β 4/3, there is no polynomial- ≥ ≤ ≤ time algorithm to color a K4k+1-minor-free graph G using less than βχ(G)+(4 3β)k colors. As far as we know, this is the first non-trivial − non-approximability result regarding the chromatic number in proper minor-closed classes. We also give somewhat weaker non-approximability bound for K4k+1- minor-free graphs with no cliques of size 4. On the positive side, we present additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with addi- tive error 6 under the additional assumption that the graph has no 4-cycles. arXiv:1707.03888v1 [cs.DM] 12 Jul 2017 The problem of determining the chromatic number, or even of just de- ciding whether a graph is colorable using a fixed number c 3 of colors, is NP-complete [7], and thus it cannot be solved in polynomial≥ time un- less P = NP. -
An Update on the Four-Color Theorem Robin Thomas
thomas.qxp 6/11/98 4:10 PM Page 848 An Update on the Four-Color Theorem Robin Thomas very planar map of connected countries the five-color theorem (Theorem 2 below) and can be colored using four colors in such discovered what became known as Kempe chains, a way that countries with a common and Tait found an equivalent formulation of the boundary segment (not just a point) re- Four-Color Theorem in terms of edge 3-coloring, ceive different colors. It is amazing that stated here as Theorem 3. Esuch a simply stated result resisted proof for one The next major contribution came in 1913 from and a quarter centuries, and even today it is not G. D. Birkhoff, whose work allowed Franklin to yet fully understood. In this article I concentrate prove in 1922 that the four-color conjecture is on recent developments: equivalent formulations, true for maps with at most twenty-five regions. The a new proof, and progress on some generalizations. same method was used by other mathematicians to make progress on the four-color problem. Im- Brief History portant here is the work by Heesch, who developed The Four-Color Problem dates back to 1852 when the two main ingredients needed for the ultimate Francis Guthrie, while trying to color the map of proof—“reducibility” and “discharging”. While the the counties of England, noticed that four colors concept of reducibility was studied by other re- sufficed. He asked his brother Frederick if it was searchers as well, the idea of discharging, crucial true that any map can be colored using four col- for the unavoidability part of the proof, is due to ors in such a way that adjacent regions (i.e., those Heesch, and he also conjectured that a suitable de- sharing a common boundary segment, not just a velopment of this method would solve the Four- point) receive different colors. -
Contemporary Mathematics 78
CONTEMPORARY MATHEMATICS 78 Braids Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Artin's Braid Group held July 13-26. 1986 at the University of California, Santa Cruz, California Joan S. Birman Anatoly Libgober Editors http://dx.doi.org/10.1090/conm/078 Recent Titles in This Series 120 Robert S. Doran, Editor, Selfadjoint and nonselfadjoint operator algebras and operator theory, 1991 119 Robert A. Melter, Azriel Rosenfeld, and Prabir Bhattacharya, Editors, Vision geometry, 1991 118 Yan Shi-Jian, Wang Jiagang, and Yang Chung-chun, Editors, Probability theory and its applications in China, 1991 117 Morton Brown, Editor, Continuum theory and dynamical systems, 1991 116 Brian Harboume and Robert Speiser, Editors, Algebraic geometry: Sundance 1988, 1991 115 Nancy Flournoy an'il Robert K. Tsutakawa, Editors, Statistical multiple integration, 1991 114 Jeffrey C. Lagarias and Michael J. Todd, Editors, Mathematical developments arising from linear programming, 1990 113 Eric Grinberg and Eric Todd Quinto, Editors, Integral geometry and tomography, 1990 112 Philip J. Brown and Wayne A. Fuller, Editors, Statistical analysis of measurement error models and applications, 1990 Ill Earl S. Kramer and Spyros S. Magliveras, Editors, Finite geometries and combinatorial designs, I 990 II 0 Georgia Benkart and J. Marshall Osborn, Editors, Lie algebras and related topics, 1990 109 Benjamin Fine, Anthony Gaglione, and Francis C. Y. Tang, Editors, Combinatorial group theory, 1990 108 Melvyn S. Berger, Editor, Mathematics of nonlinear science, 1990 107 Mario Milman and Tomas Schonbek, Editors, Harmonic analysis and partial differential equations, 1990 I 06 Wilfried Sieg, Editor, Logic and computation, 1990 I 05 Jerome Kaminker, Editor, Geometric and topological invariants of elliptic operators, 1990 I 04 Michael Makkai and Robert Pare, Accessible categories: The foundations of categorical model theory, 1989 I 03 Steve Fisk, Coloring theories, 1989 I 02 Stephen McAdam, Primes associated to an ideal, 1989 101 S.-Y. -
Mathematical Concepts Within the Artwork of Lewitt and Escher
A Thesis Presented to The Faculty of Alfred University More than Visual: Mathematical Concepts Within the Artwork of LeWitt and Escher by Kelsey Bennett In Partial Fulfillment of the requirements for the Alfred University Honors Program May 9, 2019 Under the Supervision of: Chair: Dr. Amanda Taylor Committee Members: Barbara Lattanzi John Hosford 1 Abstract The goal of this thesis is to demonstrate the relationship between mathematics and art. To do so, I have explored the work of two artists, M.C. Escher and Sol LeWitt. Though these artists approached the role of mathematics in their art in different ways, I have observed that each has employed mathematical concepts in order to create their rule-based artworks. The mathematical ideas which serve as the backbone of this thesis are illustrated by the artists' works and strengthen the bond be- tween the two subjects of art and math. My intention is to make these concepts accessible to all readers, regardless of their mathematical or artis- tic background, so that they may in turn gain a deeper understanding of the relationship between mathematics and art. To do so, we begin with a philosophical discussion of art and mathematics. Next, we will dissect and analyze various pieces of work by Sol LeWitt and M.C. Escher. As part of that process, we will also redesign or re-imagine some artistic pieces to further highlight mathematical concepts at play within the work of these artists. 1 Introduction What is art? The Merriam-Webster dictionary provides one definition of art as being \the conscious use of skill and creative imagination especially in the production of aesthetic object" ([1]). -
Mite Beometrles Ana Corn Binatorial Designs
AMERICAN MATHEMATICAL SOCIETY mite beometrles ana Corn binatorial Designs Proceedings of the AMS Special Session in Finite Geometries and Combinatorial Designs held October 29-November 1, 1987 http://dx.doi.org/10.1090/conm/111 Titles in This Series Volume 1 Markov random fields and their 19 Proceedings of the Northwestern applications, Ross Kindermann and homotopy theory conference, Haynes J. Laurie Snell R. Miller and Stewart B. Priddy, Editors 2 Proceedings of the conference on 20 Low dimensional topology, Samuel J. integration, topology, and geometry in Lomonaco, Jr., Editor linear spaces, William H. Graves, Editor 21 Topological methods in nonlinear 3 The closed graph and P-closed functional analysis, S. P. Singh, graph properties in general topology, S. Thomaier, and B. Watson, Editors T. R. Hamlett and L. L. Herrington 22 Factorizations of b" ± 1, b = 2, 4 Problems of elastic stability and 3, 5, 6, 7, 10, 11, 12 up to high vibrations, Vadim Komkov, Editor powers, John Brillhart, D. H. Lehmer, 5 Rational constructions of modules J. L. Selfridge, Bryant Tuckerman, and for simple Lie algebras, George B. S. S. Wagstaff, Jr. Seligman 23 Chapter 9 of Ramanujan's second 6 Umbral calculus and Hopf algebras, notebook-Infinite series identities, Robert Morris, Editor transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi 7 Complex contour integral representation of cardinal spline 24 Central extensions, Galois groups, and functions, Walter Schempp ideal class groups of number fields, A. Frohlich 8 Ordered fields and real algebraic geometry, D. W. Dubois and T. Recio, 25 Value distribution theory and its Editors applications, Chung-Chun Yang, Editor 9 Papers in algebra, analysis and 26 Conference in modern analysis statistics, R. -
The 150 Year Journey of the Four Color Theorem
The 150 Year Journey of the Four Color Theorem A Senior Thesis in Mathematics by Ruth Davidson Advised by Sara Billey University of Washington, Seattle Figure 1: Coloring a Planar Graph; A Dual Superimposed I. Introduction The Four Color Theorem (4CT) was stated as a conjecture by Francis Guthrie in 1852, who was then a student of Augustus De Morgan [3]. Originally the question was posed in terms of coloring the regions of a map: the conjecture stated that if a map was divided into regions, then four colors were sufficient to color all the regions of the map such that no two regions that shared a boundary were given the same color. For example, in Figure 1, the adjacent regions of the shape are colored in different shades of gray. The search for a proof of the 4CT was a primary driving force in the development of a branch of mathematics we now know as graph theory. Not until 1977 was a correct proof of the Four Color Theorem (4CT) published by Kenneth Appel and Wolfgang Haken [1]. Moreover, this proof was made possible by the incremental efforts of many mathematicians that built on the work of those who came before. This paper presents an overview of the history of the search for this proof, and examines in detail another beautiful proof of the 4CT published in 1997 by Neil Robertson, Daniel 1 Figure 2: The Planar Graph K4 Sanders, Paul Seymour, and Robin Thomas [18] that refined the techniques used in the original proof. In order to understand the form in which the 4CT was finally proved, it is necessary to under- stand proper vertex colorings of a graph and the idea of a planar graph. -
The Four Color Theorem
Western Washington University Western CEDAR WWU Honors Program Senior Projects WWU Graduate and Undergraduate Scholarship Spring 2012 The Four Color Theorem Patrick Turner Western Washington University Follow this and additional works at: https://cedar.wwu.edu/wwu_honors Part of the Computer Sciences Commons, and the Mathematics Commons Recommended Citation Turner, Patrick, "The Four Color Theorem" (2012). WWU Honors Program Senior Projects. 299. https://cedar.wwu.edu/wwu_honors/299 This Project is brought to you for free and open access by the WWU Graduate and Undergraduate Scholarship at Western CEDAR. It has been accepted for inclusion in WWU Honors Program Senior Projects by an authorized administrator of Western CEDAR. For more information, please contact [email protected]. Western WASHINGTON UNIVERSITY ^ Honors Program HONORS THESIS In presenting this Honors paper in partial requirements for a bachelor’s degree at Western Washington University, I agree that the Library shall make its copies freely available for inspection. I further agree that extensive copying of this thesis is allowable only for scholarly purposes. It is understood that anv publication of this thesis for commercial purposes or for financial gain shall not be allowed without mv written permission. Signature Active Minds Changing Lives Senior Project Patrick Turner The Four Color Theorem The history of mathematics is pervaded by problems which can be stated simply, but are difficult and in some cases impossible to prove. The pursuit of solutions to these problems has been an important catalyst in mathematics, aiding the development of many disparate fields. While Fermat’s Last theorem, which states x ” + y ” = has no integer solutions for n > 2 and x, y, 2 ^ is perhaps the most famous of these problems, the Four Color Theorem proved a challenge to some of the greatest mathematical minds from its conception 1852 until its eventual proof in 1976. -
A Historical Approach to Understanding Explanatory Proofs Based on Mathematical Practices
University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School February 2019 A Historical Approach to Understanding Explanatory Proofs Based on Mathematical Practices Erika Oshiro University of South Florida, [email protected] Follow this and additional works at: https://scholarcommons.usf.edu/etd Part of the Philosophy Commons Scholar Commons Citation Oshiro, Erika, "A Historical Approach to Understanding Explanatory Proofs Based on Mathematical Practices" (2019). Graduate Theses and Dissertations. https://scholarcommons.usf.edu/etd/7882 This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. A Historical Approach to Understanding Explanatory Proofs Based on Mathematical Practices by: Erika Oshiro A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Philosophy College of Arts and Sciences University of South Florida Co-Major Professor: Alexander Levine, Ph.D. Co-Major Professor: Douglas Jesseph, Ph.D. Roger Ariew, Ph.D. William Goodwin, Ph.D. Date of Approval: December 5, 2018 Keywords: Philosophy of Mathematics, Explanation, History, Four Color Theorem Copyright © 2018, Erika Oshiro DEDICATION To Himeko. ACKNOWLEDGMENTS I would like to express my gratitude to Dr. Alexander Levine, who was always very encouraging and supportive throughout my time at USF. I would also like to thank Dr. Douglas Jesseph, Dr. Roger Ariew, and Dr. William Goodwin for their guidance and insights. I am grateful to Dr. Dmitry Khavinson for sharing his knowledge and motivation during my time as a math student. -
Map Coloring, Polyhedra, and the Four-Color Problem
AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 8 8 Map Coloring, Polyhedra, and the Four-Color Problem and the Four-Color Map Coloring, Polyhedra, Map Coloring, Polyhedra, and the Four-Color Problem DAVID BARNETTE BARNETTE AMS/ MAA PRESS 4-Color Cover 178 page • 3/8” • 50lb stock • finish size: 5 1/2” x 8 1/2” 10.1090/dol/008 MAP COLORING, POLYHEDRA, AND THE FOUR-COLOR PROBLEM By DAVID BARNETTE 'rHI~ DOLeIANI MA~rH.EMA'rI('AL EXPOSITIONS Published hv THE MATHEMATICAL AssociA'rION OF AMERICA Committee on Publications EDWIN F. BECKENBACH, Chairman Subcommittee on Dolciani Mathematical Expositions ROSS HONSBERGER, Chairman D. J. ALBERS H.L.ALDER G.L.ALEXANDERSON R. P. BOAS H. EVES J. MALKEVITCH K. R. REBMAN The Dolciani Mathematical Expositions NUMBER EIGHT MAP COLORING, POLYHEDRA, AND THE FOUR-COLOR PROBLEM By DAVID BARNETTE University of California, Davis Published and distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA © 1983 by The Mathematical Association ofAmerica (Incorporated) Library of Congress Catalog Card Number 82-062783 Complete Set ISBN 0-88385-300-0 Vol. 8 ISBN 0-88385-309-4 Printed in the United States ofAmerica Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathe matical Association of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City University of New York. In making this gift, Professor Dolciani, herself an exceptionally talented and successful expositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. -
Council Congratulates Exxon Education Foundation
from.qxp 4/27/98 3:17 PM Page 1315 From the AMS ics. The Exxon Education Foundation funds programs in mathematics education, elementary and secondary school improvement, undergraduate general education, and un- dergraduate developmental education. —Timothy Goggins, AMS Development Officer AMS Task Force Receives Two Grants The AMS recently received two new grants in support of its Task Force on Excellence in Mathematical Scholarship. The Task Force is carrying out a program of focus groups, site visits, and information gathering aimed at developing (left to right) Edward Ahnert, president of the Exxon ways for mathematical sciences departments in doctoral Education Foundation, AMS President Cathleen institutions to work more effectively. With an initial grant Morawetz, and Robert Witte, senior program officer for of $50,000 from the Exxon Education Foundation, the Task Exxon. Force began its work by organizing a number of focus groups. The AMS has now received a second grant of Council Congratulates Exxon $50,000 from the Exxon Education Foundation, as well as a grant of $165,000 from the National Science Foundation. Education Foundation For further information about the work of the Task Force, see “Building Excellence in Doctoral Mathematics De- At the Summer Mathfest in Burlington in August, the AMS partments”, Notices, November/December 1995, pages Council passed a resolution congratulating the Exxon Ed- 1170–1171. ucation Foundation on its fortieth anniversary. AMS Pres- ident Cathleen Morawetz presented the resolution during —Timothy Goggins, AMS Development Officer the awards banquet to Edward Ahnert, president of the Exxon Education Foundation, and to Robert Witte, senior program officer with Exxon. -
Quantum Interpretations of the Four Color Theorem ∗
Quantum interpretations of the four color theorem ¤ Paul C. Kainen Dept. of Mathematics Georgetown University Washington, D.C. 20057 Abstract Some connections between quantum computing and quantum algebra are described, and the Four Color Theorem (4CT) is interpreted within both contexts. keywords: Temperley-Lieb algebra, Jones-Wenzl idempotents, Bose-Einstein condensate, analog computation, R-matrix 1 Introduction Since their discovery, the implications of quantum phenomena have been considered by computer scientists, physicists, biologists and mathematicians. We are here proposing a synthesis of several of these threads, in the context of the well-known mathematical problem of four-coloring planar maps. \Quantum computing" now refers to all those proposed theories, algorithms and tech- niques for exploiting the unique nature of quantum events to obtain computational ad- vantages. While it is not yet practically feasible and may, to quote Y. Manin, be only \mathematical wishful thinking" [30], quantum computing is taken seriously by a number of academic and corporate research groups. Whether or not quantum techniques will turn out to be implementable for computing and communications, their study throws new light on the nature of computation and information. Meanwhile, within mathematics and physics, work continues in the area known as "quan- tum algebra" (or sometimes \quantum geometry") in which a variety of esoteric mathemat- ical structures are interwoven with physical models like quantum ¯eld theories, algebras of operators on Hilbert space and so on. This variety of algebraic and topological structure is a consequences of the fact that any su±ciently precise logical model, which is physically veridical, will necessarily be non-commutative.