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.
Biography
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 2010This 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 2009The 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, USAThis 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, USARoberts 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
