Applied Combinatorics

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1 Applied Combinatorics SECOND EDITION FRED S. ROBERTS BARRY TESMAN LßP) CRC Press VV^ J Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group an informa business A CHAPMAN & HALL BOOK

Contents Preface Notation xvii xxvii 1 What Is Combinatorics? 1 1.1 The Three Problems of Combinatorics 1 1.

2 Contents Preface Notation xvii xxvii 1 What Is Combinatorics? The Three Problems of Combinatorics The History and Applications of Combinatorics 8 References for Chapter 1 13 PART I The Basic Tools of Combinatorics 15 2 Basic Counting Rules The Product Rule The Sum Rule Permutations Complexity of Computation r-permutations Subsets r-combinations Probability Sampling with Replacement Occupancy Problems The Types of Occupancy Problems Case 1: Distinguishable Balls and Distinguishable Cells Case 2: Indistinguishable Balls and Distinguishable Cells Case 3: Distinguishable Balls and Indistinguishable Cells Case 4: Indistinguishable Balls and Indistinguishable Cells Examples Multinomial Coefficients Occupancy Problems with a Specified Distribution Permutations with Classes of Indistinguishable Objects Complete Digest by Enzymes 64 Vll

Vlll Contents 2.13 Permutations with Classes of Indistinguishable Objects Revisited.. 68 2.14 The Binomial Expansion 70 2.15 Power in Simple Games 73 2.15.1 Examples of Simple Games 73 2.15.2 The Shapley-Shubik Power Index 75 2.

3 Vlll Contents 2.13 Permutations with Classes of Indistinguishable Objects Revisited The Binomial Expansion Power in Simple Games Examples of Simple Games The Shapley-Shubik Power Index The U.N. Security Council Bicameral Legislatures Cost Allocation Characteristic Functions Generating Permutations and Combinations An Algorithm for Generating Permutations An Algorithm for Generating Subsets of Sets An Algorithm for Generating Combinations Inversion Distance Between Permutations and the Study of Mutations Good Algorithms Asymptotic Analysis NP-Complete Problems Pigeonhole Principle and Its Generalizations The Simplest Version of the Pigeonhole Principle Generalizations and Applications of the Pigeonhole Principle Ramsey Numbers 106 Additional Exercises for Chapter 2 Ill References for Chapter Introduction to Graph Theory Fundamental Concepts Some Examples Definition of Digraph and Graph Labeled Digraphs and the Isomorphism Problem Connectedness Reaching in Digraphs Joining in Graphs Strongly Connected Digraphs and Connected Graphs Subgraphs Connected Components Graph Coloring and Its Applications Some Applications Planar Graphs Calculating the Chromatic Number Colorable Graphs 155

Contents ix 3.3.5 Graph-Coloring Variants 159 3.4 Chromatic Polynomials 172 3.4.1 Definitions and Examples 172 3.4.2 Reduction Theorems 175 3.4.3 Properties of Chromatic Polynomials 179 3.

4 Contents ix Graph-Coloring Variants Chromatic Polynomials Definitions and Examples Reduction Theorems Properties of Chromatic Polynomials Trees Definition of a Tree and Examples Properties of Trees Proof of Theorem Spanning Trees Proof of Theorem 3.16 and a Related Result Chemical Bonds and the Number of Trees Phylogenetic Tree Reconstruction Applications of Rooted Trees to Searching, Sorting, and Phylogeny Reconstruction Definitions Search Trees Proof of Theorem Sorting The Perfect Phylogeny Problem Representing a Graph in the Computer Ramsey Numbers Revisited 224 References for Chapter Relations Relations Binary Relations Properties of Relations/Patterns in Digraphs Order Relations and Their Variants Defining the Concept of Order Relation The Diagram of an Order Relation Linear Orders Weak Orders Stable Marriages Linear Extensions of Partial Orders Linear Extensions and Dimension Chains and Antichains Interval Orders Lattices and Boolean Algebras Lattices Boolean Algebras 276 References for Chapter 4 282

X Contents PART II The Counting Problem 285 5 Generating Functions and Their Applications 285 5.1 Examples of Generating Functions 285 5.1.1 Power Series 286 5.1.2 Generating Functions 288 5.

5 X Contents PART II The Counting Problem Generating Functions and Their Applications Examples of Generating Functions Power Series Generating Functions Operating on Generating Functions Applications to Counting Sampling Problems A Comment on Occupancy Problems The Binomial Theorem Exponential Generating Functions and Generating Functions for Permutations Definition of Exponential Generating Function Applications to Counting Permutations ' Distributions of Distinguishable Balls into Indistinguishable Cells Probability Generating Functions The Coleman and Banzhaf Power Indices 333 References for Chapter Recurrence Relations Some Examples Some Simple Recurrences Fibonacci Numbers and Their Applications Derangements Recurrences Involving More than One Sequence The Method of Characteristic Roots The Case of Distinct Roots Computation of the kth Fibonacci Number The Case of Multiple Roots Solving Recurrences Using Generating Functions The Method Derangements Simultaneous Equations for Generating Functions Some Recurrences Involving Convolutions The Number of Simple, Ordered, Rooted Trees The Ways to Multiply a Sequence of Numbers in a Computer Secondary Structure in RNA 389

Contents XI 6.4.4 Organic Compounds Built Up from Benzene Rings 391 References for Chapter 6 400 7 The Principle of Inclusion and Exclusion 403 7.1 The Principle and Some of Its Applications 403 7.1.1 Some Simple Examples 403 7.

6 Contents XI Organic Compounds Built Up from Benzene Rings 391 References for Chapter The Principle of Inclusion and Exclusion The Principle and Some of Its Applications Some Simple Examples Proof of Theorem Prime Numbers, Cryptography, and Sieves The Probabilistic Case The Occupancy Problem with Distinguishable Balls and Cells Chromatic Polynomials Derangements Counting Combinations Rook Polynomials The Number of Objects Having Exactly m Properties The Main Result and Its Applications Proofs of Theorems 7.4 and References for Chapter The Polya Theory of Counting Equivalence Relations Distinct Configurations and Databases Definition of Equivalence Relations Equivalence Classes Permutation Groups Definition of a Permutation Group The Equivalence Relation Induced by a Permutation Group Automorphisms of Graphs 453 8"3 Burnside's Lemma Statement of Burnside's Lemma Proof of Burnside's Lemma Distinct Colorings Definition of a Coloring Equivalent Colorings Graph Colorings Equivalent under Automorphisms The Case of Switching Functions The Cycle Index Permutations as Products of Cycles A Special Case of Polya's Theorem Graph Colorings Equivalent under Automorphisms Revisited 475

xii Contents 8.5.4 The Case of Switching Functions 476 8.5.5 The Cycle Index of a Permutation Group 476 8.5.6 Proof of Theorem 8.6 477 8.6 Polya's Theorem 480 8.6.1 The Inventory of Colorings 480 8.6.2 Computing the Pattern Inventory 482 8.

7 xii Contents The Case of Switching Functions The Cycle Index of a Permutation Group Proof of Theorem Polya's Theorem The Inventory of Colorings Computing the Pattern Inventory The Case of Switching Functions Proof of Polya's Theorem 485 References for Chapter PART III The Existence Problem Combinatorial Designs Block Designs Latin Squares Some Examples Orthogonal Latin Squares Existence Results for Orthogonal Families Proof of Theorem Orthogonal Arrays with Applications to Cryptography Finite Fields and Complete Orthogonal Families of Latin Squares Modular Arithmetic Modular Arithmetic and the RSA Cryptosystem The Finite Fields GF(p*) Construction of a Complete Orthogonal Family ofnxn Latin Squares if n Is a Power of a Prime Justification of the Construction of a Complete Orthogonal Family if n = p k Balanced Incomplete Block Designs (b,v,r, fc,a)-designs Necessary Conditions for the Existence of (b,v,r, k, A)-Designs Proof of Fisher's Inequality Resolvable Designs Steiner Triple Systems Symmetric Balanced Incomplete Block Designs Building New (b, v, r, &, A)-Designs from Existing Ones Group Testing and Its Applications Steiner Systems and the National Lottery Finite Projective Planes Basic Properties 549

Contents хш 9.5.2 Projective Planes, Latin Squares, and (v, k, A)-Designs... 553 References for Chapter 9 558 10 Coding Theory 561 10.1 Information Transmission 561 10.2 Encoding and Decoding 562 10.

8 Contents хш Projective Planes, Latin Squares, and (v, k, A)-Designs References for Chapter Coding Theory Information Transmission Encoding and Decoding Error-Correcting Codes Error Correction and Hamming Distance The Hamming Bound The Probability of Error Consensus Decoding and Its Connection to Finding Patterns in Molecular Sequences Linear Codes Generator Matrices Error Correction Using Linear Codes Hamming Codes The Use of Block Designs to Find Error-Correcting Codes Hadamard Codes Constructing Hadamard Designs The Richest (n,d)-codes Some Applications 602 References for Chapter Existence Problems in Graph Theory Depth-First Search: A Test for Connectedness Depth-First Search The Computational Complexity of Depth-First Search A Formal Statement of the Algorithm Testing for Connectedness of Truly Massive Graphs The One-Way Street Problem Robbins' Theorem A Depth-First Search Algorithm Efficient One-Way Street Assignments Efficient One-Way Street Assignments for Grids Annular Cities and Communications in Interconnection Networks Eulerian Chains and Paths The Königsberg Bridge Problem An Algorithm for Finding an Eulerian Closed Chain Further Results about Eulerian Chains and Paths Applications of Eulerian Chains and Paths The "Chinese Postman" Problem 640

xiv Contents 11.4.2 Computer Graph Plotting 642 11.4.3 Street Sweeping 642 11.4.4 Finding Unknown RNA/DNA Chains 645 11.4.5 A Coding Application 648 11.4.6 De Bruijn Sequences and Telecommunications 650 11.

9 xiv Contents Computer Graph Plotting Street Sweeping Finding Unknown RNA/DNA Chains A Coding Application De Bruijn Sequences and Telecommunications Hamiltonian Chains and Paths Definitions Sufficient Conditions for the Existence of a Hamiltonian ' Circuit in a Graph Sufficient Conditions for the Existence of a Hamiltonian Cycle in a Digraph Applications of Hamiltonian Chains and Paths Tournaments Topological Sorting Scheduling Problems in Operations Research Facilities Design Sequencing by Hybridization 673 References for Chapter PART IV Combinatorial Optimization Matching and Covering Some Matching Problems Some Existence Results: Bipartite Matching and Systems of Distinct Representatives Bipartite Matching Systems of Distinct Representatives The Existence of Perfect Matchings for Arbitrary Graphs Maximum Matchings and Minimum Coverings Vertex Coverings Edge Coverings Finding a Maximum Matching M-Augmenting Chains Proof of Theorem An Algorithm for Finding a Maximum Matching Matching as Many Elements of X as Possible Maximum-Weight Matching The "Chinese Postman" Problem Revisited An Algorithm for the Optimal Assignment Problem (Maximum-Weight Matching) Stable Matchings Gale-Shapley Algorithm 726

Contents xv 12.8.2 Numbers of Stable Matchings 727 12.8.3 Structure of Stable Matchings 729 12.8.4 Stable Marriage Extensions 731 References for Chapter 12 735 13 Optimization Problems for Graphs and Networks 737 13.

10 Contents xv Numbers of Stable Matchings Structure of Stable Matchings Stable Marriage Extensions 731 References for Chapter Optimization Problems for Graphs and Networks Minimum Spanning Trees Kruskal's Algorithm Proof of Theorem Prim's Algorithm The Shortest Route Problem The Problem Dijkstra's Algorithm Applications to Scheduling Problems Network Flows The Maximum-Flow Problem Cuts A Faulty Max-Flow Algorithm Augmenting Chains The Max-Flow Algorithm A Labeling Procedure for Finding Augmenting Chains Complexity of the Max-Flow Algorithm Matching Revisited Menger's Theorems Minimum-Cost Flow Problems Some Examples 785 References for Chapter Appendix: Answers to Selected Exercises 797 Author Index 833 Subject Index 841

Introductory Combinatorics

Introductory Combinatorics Third Edition KENNETH P. BOGART Dartmouth College,. " A Harcourt Science and Technology Company San Diego San Francisco New York Boston London Toronto Sydney Tokyo xm CONTENTS