A Beginner's Guide to Graph Theory, Second Edition

Graph theory continues to be one of the fastest growing areas of modern mathematics because of its wide applicability in such diverse disciplines as computer science, engineering, chemistry, management science, social science, and resource planning. Graphs arise as mathematical models in these fields, and the theory of graphs provides a spectrum of methods of proof. This concisely written textbook is intended for an introductory course in graph theory for undergraduate mathematics majors or advanced undergraduate and graduate students from the many fields that benefit from graph-theoretic applications. Key features: * Introductory chapters present the main ideas and topics in graph theory—walks, paths and cycles, radius, diameter, eccentricity, cuts and connectivity, trees * Subsequent chapters examine specialized topics and applications * Numerous examples and illustrations * Comprehensive index and bibliography, with suggested literature for more advanced material New to the second edition: * New chapters on labeling and on communications networks and small-worlds * Expanded beginner’s material in the early chapters, including more examples, exercises, hints and solutions to key problems * Many additional changes, improvements, and corrections throughout resulting from classroom use and feedback Striking a balance between a theoretical and practical approach with a distinctly applied flavor, this gentle introduction to graph theory consists of carefully chosen topics to develop graph-theoretic reasoning for a mixed audience. Familiarity with the basic concepts of set theory, along with some background in matrices and algebra, and a little mathematical maturity are the only prerequisites. ——- From a review of the first edition: "Altogether the book gives a comprehensive introduction to graphs, their theory and their application…The use of the text is optimized when the exercises are solved. The obtained skills improve understanding of graph theory as well… It is very useful that the solutions of these exercises are collected in an appendix." —Simulation News Europe Cover......Page 1 Title Page......Page 5 Copyright Page......Page 6 Preface to the Second Edition......Page 9 Preface to the First Edition......Page 11 Outline of Topics......Page 12 Acknowledgments......Page 13 Table of Contents......Page 15 List of Figures......Page 19 1.1 Sets, Binary Relations and Graphs......Page 23 1.2 Some Definitions......Page 28 1.3 Degree......Page 35 2.1 Basic Ideas......Page 41 2.2 Radius, Diameter and Eccentricity......Page 45 2.3 Weighted Distance......Page 47 2.4 Euler Walks......Page 51 2.5 Hamilton Cycles......Page 56 2.6 The Traveling Salesman Problem......Page 61 3.1 Cutpoints and Bridges......Page 65 3.2 Blocks......Page 68 3.3 Connectivity......Page 71 4.1 Characterizations of Trees......Page 75 4.2 Spanning Trees......Page 77 4.3 Minimal Spanning Trees......Page 82 5.1 Finite Fields and Vector Spaces......Page 87 5.2 The Power Set as a Vector Space......Page 88 5.3 The Vector Spaces Associated with a Graph......Page 90 5.4 The Cutset Subspace......Page 92 5.5 Bases and Spanning Trees......Page 94 6.1 Definitions; One-Factorizations......Page 99 6.2 Tournament Applications of One-Factorizations......Page 105 6.3 A General Existence Theorem......Page 107 6.4 Graphs Without One-Factors......Page 111 7.1 Vertex Colorings......Page 115 7.2 Brooks' Theorem......Page 119 7.3 Counting Vertex Colorings......Page 121 7.4 Edge-Colorings......Page 126 7.5 Class 2 Graphs......Page 129 8.1 Representations and Crossings......Page 135 8.2 Euler's Formula......Page 138 8.3 Maps, Graphs and Planarity......Page 141 9.1 Introduction; Graceful Labelings......Page 145 9.2 Edge-Magic Total Labeling......Page 148 9.3 Edge-Magic Labelings of Complete Graphs......Page 154 10.1 The Graphical Case of Ramsey's Theorem......Page 161 10.2 Ramsey Multiplicity......Page 166 10.3 Application of Sum-Free Sets......Page 168 10.4 Bounds on Classical Ramsey Numbers......Page 171 10.5 The General Case of Ramsey's Theorem......Page 174 11.1 Basic Ideas......Page 177 11.2 Orientations and Tournaments......Page 181 11.3 Directed Euler Walks......Page 185 12.1 Activity Digraphs......Page 189 12.2 Critical Path Analysis......Page 192 12.3 Critical Paths Under Uncertainty......Page 198 13.1 Transportation Networks and Flows......Page 203 13.2 Maximal Flows......Page 208 13.3 The Max Flow Min Cut Theorem......Page 214 13.4 The Max Flow Min Cut Algorithm......Page 215 13.5 Supply and Demand Problems......Page 222 14.1 Computation Time......Page 227 14.2 Data Structures......Page 230 14.3 Some Graph Algorithms......Page 231 14.4 Intractability......Page 235 15.1 Preliminaries......Page 239 15.2 Functions on Graphs......Page 240 15.3 Classes of Graphs......Page 242 15.4 Small-World Graphs......Page 244 References......Page 247 Hints......Page 253 Answers and Solutions......Page 257 Index......Page 277 Graph theory continues to be one of the fastest growing areas of modern mathematics because of its wide applicability in such diverse disciplines as computer science, engineering, chemistry, management science, social science, and resource planning. Graphs arise as mathematical models in these fields, and the theory of graphs provides a spectrum of methods of proof. This concisely written textbook is intended for an introductory course in graph theory for undergraduate mathematics majors or advanced undergraduate and graduate students from the many fields that benefit from graph-theoretic applications. Key features: * Introductory chapters present the main ideas and topics in graph theory-walks, paths and cycles, radius, diameter, eccentricity, cuts and connectivity, trees * Subsequent chapters examine specialized topics and applications * Numerous examples and illustrations * Comprehensive index and bibliography, with suggested literature for more advanced material New to the second edition: * New chapters on labeling and on communications networks and small-worlds * Expanded beginner's material in the early chapters, including more examples, exercises, hints and solutions to key problems * Many additional changes, improvements, and corrections throughout resulting from classroom use and feedback Striking a balance between a theoretical and practical approach with a distinctly applied flavor, this gentle introduction to graph theory consists of carefully chosen topics to develop graph-theoretic reasoning for a mixed audience. Familiarity with the basic concepts of set theory, along with some background in matrices and algebra, and a little mathematical maturity are the only prerequisites. -- From a review of the first edition: "Altogether the book gives a comprehensive introduction to graphs, their theory and their application ... The use of the text is optimized when the exercises are solved. The obtained skills improve understanding of graph theory as well ... It is very useful that the solutions of these exercises are collected in an appendix."--Simulation News Europe Concisely written, gentle introduction to graph theory suitable as a textbook or for self-study Graph-theoretic applications from diverse fields (computer science, engineering, chemistry, management science) 2nd ed. includes new chapters on labeling and communications networks and small worlds, as well as expanded beginner's material Many additional changes, improvements, and corrections resulting from classroom use