APPLIEDCOMBINATORICSTHIRD EDITION

ALAN TUCKERDepartment of Applied Mathematics and StatisticsState University of New York at Stony Brook

JOHN WILEY & SONS, INC.

NEW YORK CHICHESTER BRISBANE TORONTO SINGAPORE

CONTENTS

PART ONE GRAPH THEORY

CHAPTER 1 ELEMENTS OF GRAPH THEORY1.1 Graph Models 31.2 Isomorphism 191.3 Edge Counting 281.4 Planar Graphs 351.5 Summary and References 48Supplement I: Representing Graphs Inside a Computer 49Supplement II: Supplementary Exercises 51

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CHAPTER 2 COVERING CIRCUITS ANDGRAPH COLORING 57

2.1 Euler Circuits 572.2 Hamilton Circuits 652.3 Graph Coloring 76.2.4 Coloring Theorems 852.5 Summary and References 91

CHAPTER 3 TREES AND SEARCHING

3.1 Properties of Trees 933.2 Depth-First and Breadth-First Search 1033.3 Spanning Trees 1103.4 The Traveling Salesperson Problem 1183.5 Tree Analysis of Sorting Algorithms 1273.6 Summary and References 131

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CHAPTER 4 NETWORK ALGORITHMS 133

4.1 Shortest Paths 1334.2 Network Flows 1374.3 Algorithmic Matching 1564.4 Summary and References 166

PART TWO ENUMERATION 169

CHAPTER 5 GENERATING COUNTINGMETHODS FOR SELECTIONSAND ARRANGEMENTS 171

5.1 Two Basic Counting Principles 1715.2 Simple Arrangements and Selections 1805.3 Arrangements and Selections with Repetition 1955.4 Distributions 2025.5 Binomial Identities 2135.6 Generating Permutations and Combinations and

Programming Projects 2245.7 Summary and References 231Supplement: Selected Solutions to Problems in Chapter 5 232

CHAPTER 6 GENERATING FUNCTIONS 243

6.1 Generating Function Models 2436.2 Calculating Coefficients of Generating Functions 2506.3 Partitions 2606.4 Exponential Generating Functions 2666.5 A Summation Method 2726.6 Summary and References 276

CHAPTER 7 RECURRENCE RELATIONS 279

7.1 Recurrence Relation Models 2797.2 Divide-and-Conquer Relations 2937.3 Solution of Linear Recurrence Relations 2977.4 Solution of Inhomogeneous Recurrence Relations 3017.5 Solutions with Generating Functions 3067.6 Summary and References 315

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CHAPTER 8 INCLUSION-EXCLUSION 317

8.1 Counting with Venn Diagrams 3178.2 Inclusion-Exclusion Formula 3258.3 Restricted Positions and Rook Polynomials 3378.4 Summary and Reference 347

PART THREE ADDITIONAL TOPICS 349

CHAPTER 9 POLYA'S ENUMERATIONFORMULA 351

9.1 Equivalence and Symmetry Groups 3519.2 Burnside's Theorem 3599.3 The Cycle Index 3659.4 Polya's Formula 3729.5 Summary and References 379

CHAPTER 10 GAMES WITH GRAPHS 381

10.1 Graph Model for Instant Insanity 38110.2 Progressively Finite Games 38710.3 Nim-Type Games 39610.4 Summary and References 403

APPENDIX 405

A.I Set Theory and Logic 405A. 2 Mathematical Induction 411A.3 A Little Probability 415A.4 The Pigeonhole Principle 419

GLOSSARY OF COUNTING AND GRAPHTHEORY TERMS 423

BIBLIOGRAPHY 429SOLUTIONS TO ODD-NUMBERED PROBLEMS 431INDEX 457