888 Pages 309 B/W Illustrations
    by Chapman & Hall

    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


    Basic Counting Rules

    The Product Rule

    The Sum Rule


    Complexity of Computation





    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


    Graph Coloring and Its Applications

    Chromatic Polynomials


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

    Representing a Graph in the Computer

    Ramsey Numbers Revisited



    Order Relations and Their Variants

    Linear Extensions of Partial Orders

    Lattices and Boolean Algebras


    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


    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


    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