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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). -
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. -
James H. Olsen
James H. Olsen Contact Department of Mathematics Information North Dakota State University Voice: (614) 292-2572 205 Dreese Labs Fax: (614) 292-7596 2015 Neil Avenue E-mail: [email protected] Columbus, OH 43210 USA WWW: math.ndsu.nodak.edu/faculty/olsen/ Research Ergodic theory Interests Education The University of Minnesota, Minneapolis, Minnesota USA Ph.D., Mathematics, 1968 • Dissertation Title: Dominated Estimates of Lp Contractions, 1 < p < ∞. • Advisor: Professor Rafael V. Chacon • Area of Study: Ergodic Theory M.S., Mathematics, 1967 • Advisor: Professor Rafael V. Chacon • Area of Study: Ergodic Theory N.C. State College, Raleigh, North Carolina USA B.S., Electrical Engineering, 1963 Awards and Eta Kappa Nu Honors • Member Scabbard and Blade National Honor Society • First Sergeant Academic and North Dakota State University, Fargo, ND USA Professional Experience Professor of Mathematics 1984 to present Associate Professor of Mathematics 1974 to 1984 Assistant Professor of Mathematics 1970 to 1974 Mathematical Research Division, National Security Agency, Fort George G. Meade, Maryland, USA Research Mathematician 1969 to 1970 • Active duty with the U.S. Army. • Achieved the rank of captain. The University of Minnesota, Minneapolis, Minnesota USA Instructor, School of Mathematics Fall 1968 Nuclear Effects Lab, White Sands Missle Range, New Mexico USA Electronics Engineer Summer 1964 1 of 5 Publications Dominated estimates of positive contractions, (with R.V. Chacon), Proceedings of the American Mathematical Society, 20, 226-271 (1969). Dominated estimates of convex combinations of commuting isometries, Israel Journal of Mathematics, 11, 1-13 (1972). Estimates of convex combinations of commuting isometries, Z. Wahrscheinlicheits- theorie verw. Geb. 26, 317-324 (1973). -
President's Report
Volume 38, Number 4 NEWSLETTER July–August 2008 President’s Report Dear Colleagues: I am delighted to announce that our new executive director is Maeve Lewis McCarthy. I am very excited about what AWM will be able to accomplish now that she is in place. (For more about Maeve, see the press release on page 7.) Welcome, Maeve! Thanks are due to the search committee for its thought and energy. These were definitely required because we had some fabulous candidates. Thanks also to Murray State University, Professor McCarthy’s home institution, for its coopera- tion as we worked out the details of her employment with AWM. The AWM Executive Committee has voted to give honorary lifetime mem- IN THIS ISSUE berships to our founding presidents, Mary Gray and Alice T. Schafer. In my role as president, I am continually discovering just how extraordinary AWM is 7 McCarthy Named as an organization. Looking back at its early history, I find it hard to imagine AWM Executive Director how AWM could have come into existence without the vision, work, and persist- ence of these two women. 10 AWM Essay Contest Among newly elected members of the National Academy of Sciences in the physical and mathematical sciences are: 12 AWM Teacher Partnerships 16 MIT woMen In maTH Emily Ann Carter Department of Mechanical and Aerospace Engineering and the Program in 19 Girls’ Angle Applied and Computational Mathematics, Princeton University Lisa Randal Professor of theoretical physics, Department of Physics, Harvard University Elizabeth Thompson Department of Statistics, University of Washington, Seattle A W M The American Academy of Arts and Sciences has also announced its new members. -
Editorial Board George E
Advances in Mathematics http://www.ees.elsevier.com/aim Elsevier Editorial Office 525 B Street, Suite 1900 San Diego, CA 92101-4495, USA E-mail: [email protected] Telephone: (619) 699-6854 Fax: (619) 699-6700 Edited by Michael J. Hopkins Tomasz S. Mrowka Harvard University Massachusetts Institute of Technology Cambridge, MA 02138, USA Cambridge, MA 02139-4307, USA Gang Tian Princeton University Princeton, New Jersey 08544, USA Editorial Board George E. Andrews Charles Fefferman Pennsylvania State University Princeton University University Park, PA 16802, USA Princeton, NJ 08544, USA Alexandra Bellow Daniel S. Freed Northwestern University University of Texas at Austin Evanston, IL 60201, USA Austin, TX 78712, USA David Ben-Zvi Mathematics Department Adriano Garsia University of Texas at Austin University of California, San Diego Austin, TX 78712-0257, USA La Jolla, CA 92037, USA Roman Bezrukavnikov I.M. Gelfand Massachusetts Institute of Technology Rutgers University Massachusetts, USA New Brunswick, NJ 08903, USA Sara C. Billey Department of Mathematics Mark Hovey University of Washington Wesleyan University Seattle, WA 98195-4350, USA Middletown, CT 06459, USA Felix Browder Vaughan Jones Hill Center, Bush Campus University of California, Berkeley Rutgers University Berkeley, CA 94720, USA New Brunswick, NJ 08903, USA Luis Caffarelli Gil Kalai RLM 8.100 Institute of Mathematics University of Texas at Austin Hebrew University, Givat Ram Austin, TX 78712, USA Jerusalem 91904, Israel Alain Connes Ludmil Katzarkov Institut Hautes Études Scientifiques University of California, Irvine 35, Rue de Chartres Irvine, CA 92697, USA 91440 Bûres-sur-Yvette, France Andreas Dress Takahiro Kawai Research Institute for Mathematical Sciences Postfach 8640 Kyoto University Universität Bielefeld D-4800 Bielefeld 1, Germany Kitashirakawa, Sakyo-ku Kyoto 60601, Japan Kenneth Falconer H. -
Comments on [5], [14], [22], [30], and [31]
Comments on [5], [14], [22], [30], and [31] Donald L. Burkholder Department of Mathematics University of Illinois Urbana, IL 61801 [email protected] During the two decades after the publication of the pi- oneering papers Proof of the ergodic theorem and Proof of the quasi-ergodic hypothesis by Birkhoff [Bi] and von Neumann [vN], but before the publication of A general ergodic theorem by Calder´onin 1953, there were new proofs and generalizations of the results of Birkhoff and von Neumann given by many mathematicians. These in- clude, among others, Khintchine, Hopf, Kolmogorov, F. Riesz, Yoshida, Kakutani, Wiener, Dunford, Pitt, Doob, and Zygmund. Much of this substantial body of work is referenced in [Ho2], [Ka], and [Kr]. Explorations in many new directions have continued since then and have greatly expanded the connections and applications of er- godic theory to other parts of mathematics and beyond. Calder´on's1953 paper [5] played an important role in this development. In the paper, Calder´onstudies the behavior of averages of the form 1 Z F (gx) dg: (1) Nt j j Nt Here (Nt)t>0 is a family of compact symmetric neigh- 1 borhoods of the identity of a locally compact group G of one-to-one measure-preserving transformations g on a measure space E of finite measure, Nt is the left in- j j variant Haar measure of Nt with dg denoting an element of this measure, and F is an integrable function on E such that (g; x) F (gx) has appropriate measurabil- ity. It is not assumed7! that G is abelian but it is as- sumed that G is ergodic in the sense that there exists a family (Nt)t>0 as above satisfying NtNs Nt+s and ⊂ N2t < α Nt with the choice of the constant α not de- pendingj j onj tj. -
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. -
Improving IMS Meetings
Volume 37 • Issue 9 IMS Bulletin November 2008 Improving IMS Meetings CONTENTS Xuming He writes: Each year, the IMS sponsors, and co-sponsors, a healthy number 1 Improving IMS Meetings of conferences, meetings, workshops or symposiums in probability and statistics. The IMS is a major sponsor of the Joint Statistical Meetings (JSM), each August in North 2 Members’ News: Jeff Wu; Sally Morton America. Every other year (odd years), the IMS Annual Meeting takes place at JSM: the 2009 IMS Annual Meeting will be embedded in the JSM to be held in Washington Journal News 3 DC, August 1–6, 2009. Other significant meetings to take place next year include 4 Meetings: Workshop for the ENAR/IMS Spring Meetings in San Antonio, Texas (March 15–19), the WNAR/ Women in Probability; IMS Meeting in Portland, Oregon (June 14–19), and the first IMS Asia Pacific Rim International Workshop in Meeting in Seoul, Korea (June 28 – July 1). Meeting announcements can be found in Applied Probability each issue of the Bulletin, and also online at http://www.imstat.org/meetings/. 6 Contribute Even with rising costs, enthusiasm about meetings never diminishes in our profes- 7 IMS Awards sion, because we all benefit from face-to-face meetings. We exchange ideas, disseminate new research outcomes, and promote collaborations. So how do we maximize our Laha Awardees 8 rewards each time we take several days off and travel to speak at, or attend, a meeting? 9 Making the Most of Almost all of our meetings are organized by volunteers in our own societies. -
Ergodic Theory in the Perspective of Functional Analysis
Ergodic Theory in the Perspective of Functional Analysis 13 Lectures by Roland Derndinger, Rainer Nagel, GÄunther Palm (uncompleted version) In July 1984 this manuscript has been accepted for publication in the series \Lecture Notes in Mathematics" by Springer-Verlag. Due to a combination of unfortunate circumstances none of the authors was able to perform the necessary ¯nal revision of the manuscript. TÄubingen,July 1987 1 2 I. What is Ergodic Theory? The notion \ergodic" is an arti¯cial creation, and the newcomer to \ergodic theory" will have no intuitive understanding of its content: \elementary ergodic theory" neither is part of high school- or college- mathematics (as does \algebra") nor does its name explain its subject (as does \number theory"). Therefore it might be useful ¯rst to explain the name and the subject of \ergodic theory". Let us begin with the quotation of the ¯rst sentence of P. Walters' introductory lectures (1975, p. 1): \Generally speaking, ergodic theory is the study of transformations and flows from the point of view of recurrence properties, mixing properties, and other global, dynamical, properties connected with asymptotic behavior." Certainly, this de¯nition is very systematic and complete (compare the beginning of our Lectures III. and IV.). Still we will try to add a few more answers to the question: \What is Ergodic Theory ?" Naive answer: A container is divided into two parts with one part empty and the other ¯lled with gas. Ergodic theory predicts what happens in the long run after we remove the dividing wall. First etymological answer: ergodhc=di±cult. Historical answer: 1880 - Boltzmann, Maxwell - ergodic hypothesis 1900 - Poincar¶e - recurrence theorem 1931 - von Neumann - mean ergodic theorem 1931 - Birkho® - individual ergodic theorem 1958 - Kolmogorov - entropy as an invariant 1963 - Sinai - billiard flow is ergodic 1970 - Ornstein - entropy classi¯es Bernoulli shifts 1975 - Akcoglu - individual Lp-ergodic theorem Naive answer of a physicist: Ergodic theory proves that time mean equals space mean. -
A Complete Bibliography of Publications in Zeitschrift Für Wahrscheinlichkeitstheorie Und Verwandte Gebiete
A Complete Bibliography of Publications in Zeitschrift f¨ur Wahrscheinlichkeitstheorie und verwandte Gebiete Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 13 October 2017 Version 1.03 Title word cross-reference (r; p) [Tak84]. 0 [JK69, R¨os77a]. 0 <p≤ 1 [Her74b]. 1 [HQ85, Hel82, JK69, R¨os77a, Sla72, Tan63, Tuo76]. 1=2∆u + qu = 0 [Fal83a]. 1=n [Bor70a]. 2 [Mic72b, Roy75b]. 3 [Don65]. 3 × 3[Fry80,JR79]. ∗ [Hig82]. + [BF83]. p [Ber80, FS74]. α [Acz64, Che68, Cre75, DJ74]. B [Mor81, Mor82]. B + L [EP82]. B0 [Tie62].ρ ¯ [Gun79a]. C [Sch71b]. C(S) 1 [Gin76, Hei77, Led82]. C[0; 1] [Kue76]. C [Fab66]. = f(nkx) ^ [Ber76a, Ber76b]. chi2 [ERV65, R¨us65, ERV65, R¨us65]. D [DH78, Hah78, Web80a, BKS82, Wsc82]. ∆ [R¨us80a]. E [TM71b]. e(λ) 2 1 [Tor76]. Ek(X; Y ) [CSS76]. [Uus76]. EX1 = [Sto79]. f [KP66, Wat70]. γ [Fie71a]. [CGH78]. H [Kue73]. Hγ [Kle63]. H1 [CHP73]. Hp 2 [Kaz79, Pro78]. Hp [Her74b]. I [Bol78]. 1 [KL65]. J [GO76]. K [Bla76b, Bur79, Emc74, ABH84, Bil66, Deh83d, DD84, GR84a, ST77]. 1 2 k ≤ c log n [DD84]. L [Dav71, O'C79, iSY78]. L(Rd) [HS80]. L1 2 p [SA75, Eng76, Sch77]. L [B¨uh63, Sch69c, Sul84]. L [RS77, Sj¨o82]. L0 [MP77]. L1 [DW80, Tra75]. L2 [FD81a, HS76]. L1 [Hor72c]. L1[0; 1] [Kim72b].