Program of the Sessions, Chattanooga, Volume 48, Number 10
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Introduction to Graph Theory: a Discovery Course for Undergraduates
Introduction to Graph Theory: A Discovery Course for Undergraduates James M. Benedict Augusta State University Contents 1 Introductory Concepts 1 1.1 BasicIdeas ...................................... 1 1.2 Graph Theoretic Equality . 2 1.3 DegreesofVertices .................................. 4 1.4 Subgraphs....................................... 6 1.5 The Complement of a Graph . 7 2 Special Subgraphs 9 2.1 Walks ......................................... 9 2.2 Components...................................... 10 2.3 BlocksofaGraph................................... 11 3 Three Famous Results and One Famous Graph 14 3.1 The Four-Color Theorem . 14 3.2 PlanarGraphs..................................... 16 3.3 ThePetersenGraph ................................. 18 3.4 TraceableGraphs ................................... 18 K(3;3;3) = H1 H2 Removing the edge w1v2 destroys the only C4 v 3 u1 H1: H2: w3 v v2 3 u w1 u 2 u u 2 v 1 3 w 1 w 2 2 w3 u3 w1 v2 v1 Place any edge of H1 into H2 or (vice versa) and a C4 is created. i Remarks By reading through this text one can acquire a familiarity with the elementary topics of Graph Theory and the associated (hopefully standard) notation. The notation used here follows that used by Gary Chartrand at Western Michigan University in the last third of the 20th century. His usage of notation was in‡uenced by that of Frank Harary at the University of Michigan beginning in the early 1950’s. The text’s author was Chartrand’s student at WMU from 1973 to 1976. In order to actually learn any graph theory from this text, one must work through and solve the problems found within it. Some of the problems are very easy. -
Discrete Mathematics a Dedication to Professor Gary Chartrand
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 309 (2009) 5803–5805 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc A dedication to Professor Gary Chartrand Teresa W. Haynes a, Michael A. Henning b, Ping Zhang c,∗ a Department of Mathematics, East Tennessee State University, Johnson City, TN 37614-0002, USA b School of Mathematical Sciences, University of KwaZulu-Natal, Pietermaritzburg, 3209, South Africa c Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA article info Article history: Received 28 May 2008 Accepted 30 May 2008 Available online 2 July 2008 Gary Chartrand was born on August 24, 1936 in the small town of Sault Ste. Marie (pronounced ``Soo Saint Marie'') in the upper peninsula of Michigan, across a river from Canada. Although his parents valued education highly, none of his parents, grandparents, aunts, or uncles ever finished high school, although his father later received his high school diploma through correspondence school. Both parents worked for the telephone company. His father had many interests. He played semi-professional hockey, he played the piano in a band (although he could never read music), and had his own ham radio station (call letters W8PRL). In fact, he built his own radio receiver, even blowing glass for the tubes, and it worked. His father worked part-time for local movie theaters, playing his own music on the piano for silent movies. His parents loved the movies (especially musicals) and took Gary to the movies regularly from the age of 3. -
Graduate Student Handbook
WESTERN MICHIGAN UNIVERSITY DEPARTMENT OF MATHEMATICS GRADUATE STUDENT HANDBOOK (REVISED April 2021) I. FACTS ABOUT WESTERN MICHIGAN UNIVERSITY AND THE MATHEMATICS DEPARTMENT 5 II. ORGANIZATION OF THE DEPARTMENT 7 III. ORGANIZATION OF EVERETT TOWER AND ROOD HALL 8 Departmental Classrooms 8 Departmental Offices 8 Faculty Offices & Conference Rooms 8 IV. RESOURCES AND OPPORTUNITIES 8 Waldo Library 9 University Computing Services 9 Professional Society Memberships 9 Math Club and Math Honors Society 9 Graduate Student Representative 9 Awards 10 V. SEMINARS AND COLLOQUIA 12 Purpose 12 Seminars 12 The Department Colloquium 12 Math Club Talks and Math Honors Society 13 Graduate Student Talk 13 The TA Seminar 13 VI. TALK ATTENDANCE REQUIREMENT 13 VII. PERMANENT PROGRAM OF STUDY AND ANNUAL REVIEW OF GRADUATE STUDENTS 15 Program Planning 15 Program of Study 15 Annual Review of Graduate Students 16 Definition of Good Standing 17 VIII. REGISTERING FOR CLASSES 18 IX. THE PH.D. COMPREHENSIVE EXAMINATIONS 18 Purpose 18 Policies and Procedures for Comprehensive Exams 18 Mathematics Education 18 Mathematics 19 Substituting Coursework and Comprehensive Exam 19 Writing and Proctoring Exams 20 2 Evaluation of the Exam 20 Policy on Non-Program Students 20 X. GUIDELINES FOR THE DIRECTED TEACHING REQUIREMENT FOR DOCTORAL STUDENTS 21 XI. THE PROCEDURES LEADING TO THE COMPLETION OF A DOCTORAL DISSERTATION IN THE DEPARTMENT OF MATHEMATICS 23 The Procedures for the Ph.D. in Mathematics 23 The Procedures for the Ph.D. in Mathematics Education 24 “Stopping the Clock” for Graduate Degree Programs 26 XII. RESEARCH TOOLS REQUIREMENT FOR THE PH.D. DEGREE IN THE DEPARTMENT OF MATHEMATICS 26 Ph.D. -
Hamiltonicity and Connectivity in Distance-Colored Graphs
Western Michigan University ScholarWorks at WMU Dissertations Graduate College 1-2011 Hamiltonicity and Connectivity in Distance-Colored Graphs Kyle C. Kolasinski Western Michigan University Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Applied Mathematics Commons, and the Mathematics Commons Recommended Citation Kolasinski, Kyle C., "Hamiltonicity and Connectivity in Distance-Colored Graphs" (2011). Dissertations. 431. https://scholarworks.wmich.edu/dissertations/431 This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. Hamiltonicity and Connectivity in Distance-Colored Graphs by Kyle C. Kolasinski A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment for the requirement for the Degree of Doctor of Philosophy Department of Mathematics Advisor: Ping Zhang, Ph.D. Western Michigan University Kalamazoo, Michigan June 2011 Hamiltonicity and Connectivity in Distance-Colored Graphs Kyle Kolasinski, Ph.D. Western Michigan University, 2011 For a connected graph G and a positive integer fc, the fcth power Gk of G is the graph with V(Gk) — V(G) where uv E E(Gk) if the distance dc{u^v) between u and v is at most fc. The edge coloring of Gk defined by assigning each edge uv of Gk the color da(u,v) produces an edge-colored graph Gk called a distance-colored graph. A distance-colored graph Gk is Hamiltonian-colored if Gk contains a properly colored Hamiltonian cycle. -
How to Define an Irregular Graph
How to Define an Irregular Graph Gary Chartrand Paul Erdös Ortrud R. Oellermann I have lived in Michigan nearly all my life, having been brought up in Sault Ste. Marie, educated at Michigan State University, studying at the University of Michigan, and em- ployed by Western Michigan University . I've enjoyed graph theory from the day I heard my eventual thesis advisor E. A. Nordhaus speak on it. I've had eleven doctoral students, the most recent being Ortrud Oellermann, and the two of us are now very pleased to have Erdös number 1 . I enjoy musicals and many sports. The misfortune of birth overtook me on March 26, 1913. My parents were both mathematicians, and I learned a great deal from them. I got my Ph.D. at the University of Budapest in 1934. 1 spent the years 1938-1948 in Manchester, En- gland, and I have been in the United States since 1954. 1 travel constantly around the world: Hungary, Israel, the U.S.A., England, etc. My subjects are number theory, combi- natorics, set theory and probability. I was born in Vryheid, South Africa, and have been inter- ested in mathematics since childhood. After receiving my predoctoral degrees from the University of Natal in Durban, I came to the United States to study graph theory at Western Michigan University, where I received my Ph .D. in 1986, and where I am now on the faculty. I enjoy playing the piano, violin and recorder. I speak German, Afrikaans and Zulu, in addition to English . There are several curious real-life puzzles and problems that can be translated into the language of graph theory, where they are often found to have interesting solutions. -
Autobiographical Notes∗ School Life Bachelor's Degree
Discrete Mathematics Letters Discrete Math. Lett. 1 (2019) 83–92 www.dmlett.com Autobiographical notes∗ Gary Chartrand Professor Emeritus of Mathematics, Western Michigan University, Kalamazoo, MI 49008–5248, USA (Published online: 18 May 2019.) c 2019 the author. This is an open access article under the CC BY (International 4.0) license (https://creativecommons.org/licenses/by/4.0/). School life I was born in Sault Ste. Marie, Michigan, in 1936 (the same year that the first book on graph theory, written by Denes´ Konig,˝ was published). As a youngster, I attended many movies with my parents. In fact, my parents started taking me to movies when I was three years old. My father especially liked movie musicals, which I did too. When he was young, he played the piano in movie theaters for silent movies. (I remember very often, after coming home from watching a musical at the movies, my father would play some of the songs he just heard on the piano.) Having lived through the depression, he worked hard to support his family. Through him I learned the importance of accomplishing something, contributing to society, and not doing something just for money. I was especially interested in comic books when I was a kid. My parents would often buy comic books for me. Because of that, I learned to read before I went to Kindergarten. When I thought the stories could have been better, I started writing my own comic books but I was terrible at drawing. I lived in Sault Ste. Marie until I was 13 years old, when my parents moved to Lansing, Michigan. -
Graph Theory, an Antiprism Graph Is a Graph That Has One of the Antiprisms As Its Skeleton
Graph families From Wikipedia, the free encyclopedia Chapter 1 Antiprism graph In the mathematical field of graph theory, an antiprism graph is a graph that has one of the antiprisms as its skeleton. An n-sided antiprism has 2n vertices and 4n edges. They are regular, polyhedral (and therefore by necessity also 3- vertex-connected, vertex-transitive, and planar graphs), and also Hamiltonian graphs.[1] 1.1 Examples The first graph in the sequence, the octahedral graph, has 6 vertices and 12 edges. Later graphs in the sequence may be named after the type of antiprism they correspond to: • Octahedral graph – 6 vertices, 12 edges • square antiprismatic graph – 8 vertices, 16 edges • Pentagonal antiprismatic graph – 10 vertices, 20 edges • Hexagonal antiprismatic graph – 12 vertices, 24 edges • Heptagonal antiprismatic graph – 14 vertices, 28 edges • Octagonal antiprismatic graph– 16 vertices, 32 edges • ... Although geometrically the star polygons also form the faces of a different sequence of (self-intersecting) antiprisms, the star antiprisms, they do not form a different sequence of graphs. 1.2 Related graphs An antiprism graph is a special case of a circulant graph, Ci₂n(2,1). Other infinite sequences of polyhedral graph formed in a similar way from polyhedra with regular-polygon bases include the prism graphs (graphs of prisms) and wheel graphs (graphs of pyramids). Other vertex-transitive polyhedral graphs include the Archimedean graphs. 1.3 References [1] Read, R. C. and Wilson, R. J. An Atlas of Graphs, Oxford, England: Oxford University Press, 2004 reprint, Chapter 6 special graphs pp. 261, 270. 2 1.4. EXTERNAL LINKS 3 1.4 External links • Weisstein, Eric W., “Antiprism graph”, MathWorld.