9780130144003

Introduction to Graph Theory

by
  • ISBN13:

    9780130144003

  • ISBN10:

    0130144002

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 8/22/2000
  • Publisher: Pearson
  • View Upgraded Edition

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

Purchase Benefits

  • Free Shipping On Orders Over $59!
    Your order must be $59 or more to qualify for free economy shipping. Bulk sales, PO's, Marketplace items, eBooks and apparel do not qualify for this offer.
  • Get Rewarded for Ordering Your Textbooks! Enroll Now
  • We Buy This Book Back!
    In-Store Credit: $5.25
    Check/Direct Deposit: $5.00
List Price: $193.80 Save up to $66.83
  • Buy Used
    $126.97
    Add to Cart Free Shipping

    USUALLY SHIPS IN 2-3 BUSINESS DAYS

Supplemental Materials

What is included with this book?

  • The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.
  • The Used copy of this book is not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

Summary

This book fills a need for a thorough introduction to graph theory that features both the understanding and writing of proofs about graphs. Verification that algorithms work is emphasized more than their complexity.An effective use of examples, and huge number of interesting exercises, demonstrate the topics of trees and distance, matchings and factors, connectivity and paths, graph coloring, edges and cycles, and planar graphs.For those who need to learn to make coherent arguments in the fields of mathematics and computer science.

Table of Contents

Preface xi
Fundamental Concepts
1(66)
What Is a Graph?
1(18)
The Definition
1(2)
Graphs as Models
3(3)
Matrices and Isomorphism
6(5)
Decomposition and Special Graphs
11(3)
Exercises
14(5)
Paths, Cycles, and Trails
19(15)
Connection in Graphs
20(4)
Bipartite Graphs
24(2)
Eulerian Circuits
26(5)
Exercises
31(3)
Vertex Degrees and Counting
34(19)
Counting and Bijections
35(3)
Extremal Problems
38(6)
Graphic Sequences
44(3)
Exercises
47(6)
Directed Graphs
53(14)
Definitions and Examples
53(5)
Vertex Degrees
58(2)
Eulerian Digraphs
60(1)
Orientations and Tournaments
61(2)
Exercises
63(4)
Trees and Distance
67(40)
Basic Properties
67(14)
Properties of Trees
68(2)
Distance in Trees and Graphs
70(3)
Disjoint Spanning Trees (optional)
73(2)
Exercises
75(6)
Spanning Trees and Enumeration
81(14)
Enumeration of Trees
81(2)
Spanning Trees in Graphs
83(4)
Decomposition and Graceful Labelings
87(2)
Branchings and Eulerian Digraphs (optional)
89(3)
Exercises
92(3)
Optimization and Trees
95(12)
Minimum Spanning Tree
95(2)
Shortest Paths
97(3)
Trees in Computer Science (optional)
100(3)
Exercises
103(4)
Matchings and Factors
107(42)
Matchings and Covers
107(16)
Maximum Matchings
108(2)
Hall's Matching Condition
110(2)
Min-Max Theorems
112(1)
Independent Sets and Covers
113(3)
Dominating Sets (optional)
116(2)
Exercises
118(5)
Algorithms and Applications
123(13)
Maximum Bipartite Matching
123(2)
Weighted Bipartite Matching
125(5)
Stable Matchings (optional)
130(2)
Faster Bipartite Matching (optional)
132(2)
Exercises
134(2)
Matchings in General Graphs
136(13)
Tutte's 1-factor Theorem
136(4)
f-factors of Graphs (optional)
140(2)
Edmonds' Blossom Algorithm (optional)
142(3)
Exercises
145(4)
Connectivity and Paths
149(42)
Cuts and Connectivity
149(12)
Connectivity
149(3)
Edge-connectivity
152(3)
Blocks
155(3)
Exercises
158(3)
k-connected Graphs
161(15)
2-connected Graphs
161(3)
Connectivity of Digraphs
164(2)
k-connected and k-edge-connected Graphs
166(4)
Applications of Menger's Theorem
170(2)
Exercises
172(4)
Network Flow Problems
176(15)
Maximum Network Flow
176(5)
Integral Flows
181(3)
Supplies and Demands (optional)
184(4)
Exercises
188(3)
Coloring of Graphs
191(42)
Vertex Colorings and Upper Bounds
191(13)
Definitions and Examples
191(3)
Upper Bounds
194(3)
Brooks' Theorem
197(2)
Exercises
199(5)
Structure of k-chromatic Graphs
204(15)
Graphs with Large Chromatic Number
205(2)
Extremal Problems and Turan's Theorem
207(3)
Color-Critical Graphs
210(2)
Forced Subdivisions
212(2)
Exercises
214(5)
Enumerative Aspects
219(14)
Counting Proper Colorings
219(5)
Chordal Graphs
224(2)
A Hint of Perfect Graphs
226(2)
Counting Acyclic Orientations (optional)
228(1)
Exercises
229(4)
Planar Graphs
233(40)
Embeddings and Euler's Formula
233(13)
Drawings in the Plane
233(3)
Dual Graphs
236
Euler's Formula
241(2)
Exercises
243(3)
Characterization of Planar Graphs
246(11)
Preparation for Kuratowski's Theorem
247(1)
Convex Embeddings
248(4)
Planarity Testing (optional)
252(3)
Exercises
255(2)
Parameters of Planarity
257(16)
Coloring of Planar Graphs
257(4)
Crossing Number
261(5)
Surfaces of Higher Genus (optional)
266(3)
Exercises
269(4)
Edges and Cycles
273(46)
Line Graphs and Edge-coloring
273(13)
Edge-colorings
274(5)
Characterization of Line Graphs (optional)
279(3)
Exercises
282(4)
Hamiltonian Cycles
286(13)
Necessary Conditions
287(1)
Sufficient Conditions
288(5)
Cycles in Directed Graphs (optional)
293(1)
Exercises
294(5)
Planarity, Coloring, and Cycles
299(20)
Tait's Theorem
300(2)
Grinberg's Theorem
302(2)
Snarks (optional)
304(3)
Flows and Cycle Covers (optional)
307(7)
Exercises
314(5)
Additional Topics (optional)
319(152)
Perfect Graphs
319(30)
The Perfect Graph Theorem
320(3)
Chordal Graphs Revisited
323(5)
Other Classes of Perfect Graphs
328(6)
Imperfect Graphs
334(6)
The Strong Perfect Graph Conjecture
340(4)
Exercises
344(5)
Matroids
349(29)
Hereditary Systems and Examples
349(5)
Properties of Matroids
354(4)
The Span Function
358(2)
The Dual of a Matroid
360(3)
Matroid Minors and Planar Graphs
363(3)
Matroid Intersection
366(3)
Matroid Union
369(3)
Exercises
372(6)
Ramsey Theory
378(18)
The Pigeonhole Principle Revisited
378(2)
Ramsey's Theorem
380(3)
Ramsey Numbers
383(3)
Graph Ramsey Theory
386(2)
Sperner's Lemma and Bandwidth
388(4)
Exercises
392(4)
More Extremal Problems
396(29)
Encodings of Graphs
397(7)
Branchings and Gossip
404(4)
List Coloring and Choosability
408(5)
Partitions Using Paths and Cycles
413(3)
Circumference
416(6)
Exercises
422(3)
Random Graphs
425(27)
Existence and Expectation
426(4)
Properties of Almost All Graphs
430(2)
Threshold Functions
432(4)
Evolution and Graph Parameters
436(3)
Connectivity, Cliques, and Coloring
439(3)
Martingales
442(6)
Exercises
448(4)
Eigenvalues of Graphs
452(19)
The Characteristic Polynomial
453(3)
Linear Algebra of Real Symmetric Matrices
456(2)
Eigenvalues and Graph Parameters
458(2)
Eigenvalues of Regular Graphs
460(3)
Eigenvalues and Expanders
463(1)
Strongly Regular Graphs
464(3)
Exercises
467(4)
Appendix A Mathematical Background 471(22)
Sets
471(4)
Quantifiers and Proofs
475(4)
Induction and Recurrence
479(4)
Functions
483(2)
Counting and Binomial Coefficients
485(4)
Relations
489(2)
The Pigeonhole Principle
491(2)
Appendix B Optimization and Complexity 493(14)
Intractability
493(3)
Heuristics and Bounds
496(3)
NP-Completeness Proofs
499(6)
Exercises
505(2)
Appendix C Hints for Selected Exercises 507(8)
General Discussion
507(1)
Supplemental Specific Hints
508(7)
Appendix D Glossary of Terms 515(18)
Appendix E Supplemental Reading 533(34)
Appendix F References 567(2)
Author Index 569(6)
Subject Index 575

Excerpts

PrefaceGraph theory is a delightful playground for the exploration of proof techniques in discrete mathematics, and its results have applications in many areas of the computing, social, and natural sciences. The design of this book permits usage in a one-semester introduction at the undergraduate or beginning graduate level, or in a patient two-semester introduction. No previous knowledge of graph theory is assumed. Many algorithms and applications are included, but the focus is on understanding the structure of graphs and the techniques used to analyze problems in graph theory.Many textbooks have been written about graph theory. Due to its emphasis on both proofs and applications, the initial model for this book was the elegant text by J.A. Bondy and US.R. Murty,Graph Theory with Applications(Macmillan/NorthHolland 1976). Graph theory is still young, and no consensus has emerged on how the introductory material should be presented. Selection and order of topics, choice of proofs, objectives, and underlying themes are matters of lively debate. Revising this book dozens of times has taught me the difficulty of these decisions. This book is my contribution to the debate. The Second EditionThe revision for the second edition emphasizes making the text easier for the students to learn from and easier for the instructor to teach from. There have not been great changes in the overall content of the book, but the presentation has been modified to make the material more accessible, especially in the early parts of the book. Some of the changes are discussed in more detail later in this preface; here I provide a brief summary. Optional material within non-optional sections is now designated by (*); such material is not used later and can be skipped. Most of it isintendedto be skipped in a one-semester course. When a subsection is marked "optional", the entire subsection is optional, and hence no individual items are starred. For less-experienced students, Appendix A has been added as a reference summary of helpful material on sets, logical statements, induction, counting arguments, binomial coefficients, relations, and the pigeonhole principle. Many proofs have been reworded in more patient language with additional details, and more examples have been added. More than 350 exercises have been added, mostly easier exercises in Chapters 1-7. There are now more than 1200 exercises. More than 100 illustrations have been added; there are now more than 400. In illustrations showing several types of edges, the switch to bold and solid edges instead of solid and dashed edges has increased clarity. Easier.problems are now grouped at the beginning of each exercise section, usable as warm-ups. Statements of some exercises have been clarified. In addition to hints accompanying the exercise statements, there is now an appendix of supplemental hints. For easier access, terms being defined are in bold type, and the vast majority of them appear in Definition items. For easier access, the glossary of notation has been placed on the inside covers. Material involving Eulerian circuits, digraphs, and Turyn's Theorem has been relocated to facilitate more efficient learning. Chapters 6 and 7 have been switched to introduce the idea of planarity earlier, and the section on complexity has become an appendix. The glossary has been improved to eliminate errors and to emphasize items more directly related to the text. FeaturesVarious features of this book facilitate students' efforts to understand the material. There is discussion of proof techniques, more than 1200 exercises of varying difficulty, more than 400 illustrations, and many examples. Proofs are presented in full in the text.Many undergraduates begin a course in graph theory with little exposure to proof techniques.

Rewards Program

Write a Review