eBook

- Presents many examples from the biological, computer, and social sciences as well as many other areas, including disease screening, genome mapping, satellite communication, search engines, telecommunications traffic, web data, smallpox vaccinations, sound systems, oil drilling, dynamic labor markets, data mining, and distributed computing
- Covers list colorings, the inversion distance between permutations and mutations in evolutionary biology, graph coloring, relations, DNA sequence alignment, cryptography, automorphisms of graphs, orthogonal arrays, secret sharing, the RSA cryptosystem, and consensus decoding
- Emphasizes problem solving through a range of exercises that either test routine ideas, introduce new concepts and applications, or challenge readers to use the combinatorial techniques developed
- Includes answers to selected exercises in the appendix

Now with solutions to selected problems, **Applied Combinatorics, Second Edition** presents the tools of combinatorics from an applied point of view. This bestselling textbook offers numerous references to the literature of combinatorics and its applications that enable readers to delve more deeply into the topics.

After introducing fundamental counting rules and the tools of graph theory and relations, the authors focus on three basic problems of combinatorics: counting, existence, and optimization problems. They discuss advanced tools for dealing with the counting problem, including generating functions, recurrences, inclusion/exclusion, and Pólya theory. The text then covers combinatorial design, coding theory, and special problems in graph theory. It also illustrates the basic ideas of combinatorial optimization through a study of graphs and networks.

**What Is Combinatorics?**

The Three Problems of Combinatorics

The History and Applications of Combinatorics

*THE BASIC TOOLS OF COMBINATORICS*

**Basic Counting Rules**

The Product Rule

The Sum Rule

Permutations

Complexity of Computation

*r*-Permutations

Subsets

*r*-Combinations

Probability

Sampling with Replacement

Occupancy Problems

Multinomial Coefficients

Complete Digest by Enzymes

Permutations with Classes of Indistinguishable Objects Revisited

The Binomial Expansion

Power in Simple Games

Generating Permutations and Combinations

Inversion Distance between Permutations and the Study of Mutations

Good Algorithms

Pigeonhole Principle and Its Generalizations

**Introduction to Graph Theory**

Fundamental Concepts

Connectedness

Graph Coloring and Its Applications

Chromatic Polynomials

Trees

Applications of Rooted Trees to Searching, Sorting, and Phylogeny Reconstruction

Representing a Graph in the Computer

Ramsey Numbers Revisited

**Relations**

Relations

Order Relations and Their Variants

Linear Extensions of Partial Orders

Lattices and Boolean Algebras

*THE COUNTING PROBLEM*

**Generating Functions and Their Applications**

Examples of Generating Functions

Operating on Generating Functions

Applications to Counting

The Binomial Theorem

Exponential Generating Functions and Generating Functions for Permutations

Probability Generating Functions

The Coleman and Banzhaf Power Indices

**Recurrence Relations**

Some Examples

The Method of Characteristic Roots

Solving Recurrences Using Generating Functions

Some Recurrences Involving Convolutions

**The Principle of Inclusion and Exclusion**

The Principle and Some of Its Applications

The Number of Objects Having Exactly *m* Properties

**The Pólya Theory of Counting**

Equivalence Relations

Permutation Groups

Burnside’s Lemma

Distinct Colorings

The Cycle Index

Pólya’s Theorem

*THE EXISTENCE PROBLEM*

**Combinatorial Designs**

Block Designs

Latin Squares

Finite Fields and Complete Orthogonal Families of Latin Squares

Balanced Incomplete Block Designs

Finite Projective Planes

**Coding Theory**

Information Transmission

Encoding and Decoding

Error-Correcting Codes

Linear Codes

The Use of Block Designs to Find Error-Correcting Codes

**Existence Problems in Graph Theory**

Depth-First Search: A Test for Connectedness

The One-Way Street Problem

Eulerian Chains and Paths

Applications of Eulerian Chains and Paths

Hamiltonian Chains and Paths

Applications of Hamiltonian Chains and Paths

**COMBINATORIAL OPTIMIZATION**

**Matching and Covering**

Some Matching Problems

Some Existence Results: Bipartite Matching and Systems of Distinct Representatives

The Existence of Perfect Matchings for Arbitrary Graphs

Maximum Matchings and Minimum Coverings

Finding a Maximum Matching

Matching as Many Elements of X as Possible

Maximum-Weight Matching

Stable Matchings

**Optimization Problems for Graphs and Networks**

Minimum Spanning Trees

The Shortest Route Problem

Network Flows

Minimum-Cost Flow Problems

**Appendix: Answers to Selected Exercises**

**Author Index**

**Subject Index**

*References appear at the end of each chapter.*

**Fred S. Roberts** is Professor of Mathematics and Director of DIMACS at Rutgers University.

**Barry Tesman** is Professor of Mathematics at Dickinson College.

The book has been substantially rewritten with more than 200 pages of new materials and many changes in the exercises. There are also many new examples to reflect the new developments in computer science and biology since 1990. … Many important topics are covered and they are done in detail. This book is one of the rare ones that does the job really well. … I strongly endorse this book. It is suitable for motivated math, computer science or engineering sophomores and even beginning graduate students. In fact bright high school students would love this book and if they are exposed early (through reading this book and being guided by their teachers), many of them might end up doing combinatorics for their careers! I really love this book. It is a gem.

—IACR Book Reviews, 2011

… the overall organization is excellent. … Many inviting exercises are included. They cover both theoretical aspects and practical problems from state-of-the-art scientific research in various areas, such as biology and telecommunications. … I can heartily recommend expanding your library with a copy of this work. It is so much fun to just open the book at random and explore the material that jumps out of the pages.

—*Computing Reviews*, March 2010

This is an overwhelmingly complete introductory textbook in combinatorics. It not only covers nearly every topic in the subject, but gives several realistic applications for each topic. … much more breadth than its competitors. …valuable as a source of applications and for enrichment reading.

—*MAA Reviews*, December 2009

The writing style is excellent. … The explanations are detailed enough that the students can follow the arguments readily. The motivating examples are a truly strong point for the text. No other text with which I am familiar comes even close to the number of applications presented here.

—John Elwin, San Diego State University, California, USA

This book is a required textbook for my graduate course in discrete mathematics. Both my students and I have found it to be an excellent resource with interesting application examples from a variety of fields interspersed throughout the text. The book is very well organized and clearly reinforces both the practical and theoretical understanding in a way students are able to correlate. Because the level of difficulty for selected problems range from simple to challenging, it makes an appropriate text for junior, senior, and graduate students alike. I am particularly pleased with the relevancy and inclusion of computer science applications … .

—Dawit Haile, Virginia State University, Petersburg, USA

Roberts and Tesman’s book covers all the major areas of combinatorics in a clear, insightful fashion. But what really sets it apart is its impressive use of applications. I know of no other text which comes close. There are entire sections devoted to subjects like computing voting power, counting organic compounds built up from benzene rings, and the use of orthogonal arrays in cryptography. And in exercises and examples, students test psychic powers, consider the UNIX time problem, plan mail carriers’ routes, and assign state legislators to committees. This really helps them to understand the mathematics and also to see how this field is useful in the real world.

—Thomas Quint, University of Nevada, Reno, USA