Book Description
Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation, and has taught, influenced and inspired generations of students and researchers in mathematics.This volume summarizes and
reviews the consistent themes from his work through a series of articles written by renowned experts.These articles contain original research work, set in a broader context by the inclusion of review material.As a reference text in its own right, this book will be valuable to academic
researchers, research students, and others seeking an introduction to the relevant contemporary aspects of these fields. ... Read more

Book Description
This gradual, systematic introduction to the main concepts of combinatorics is the ideal text for advanced undergraduate and early graduate courses in this subject. Each of the book's three sections—Existence, Enumeration, and Construction—begins with a simply stated first principle, which is then developed step by step until it leads to one of the three major achievements of combinatorics: Van der Waerden's theorem on arithmetic progressions, Pólya's graph enumeration formula, and Leech's 24-dimensional lattice.

Along the way, Professor Martin J. Erickson introduces fundamental results, discusses interconnection and problem-solving techniques, and collects and disseminates open problems that raise new and innovative questions and observations. His carefully chosen end-of-chapter exercises demonstrate the applicability of combinatorial methods to a wide variety of problems, including many drawn from the William Lowell Putnam Mathematical Competition. Many important combinatorial methods are revisited several times in the course of the text—in exercises and examples as well as theorems and proofs. This repetition enables students to build confidence and reinforce their understanding of complex material.

Mathematicians, statisticians, and computer scientists profit greatly from a solid foundation in combinatorics. Introduction to Combinatorics builds that foundation in an orderly, methodical, and highly accessible manner. ... Read more

Book Description
Graphs, Combinatorics, Algorithms and Applications: The research papers contributed by leading experts in their respective field discusses current areas of research in graph theory such as: · Graphoidal covers · Hyper graphs · Domination in graph · Signed graphs · Graph labelings and · Theoretical computer science This volume will serve as an excellent reference for experts and research scholars working in Graph Theory and related topics. ... Read more

Book Description
With the advent of computers, algorithmic principles play an ever increasing role in mathematics. Algorithms have to exploit the structure of the underlying mathematical object; and properties exploited by algorithms are often closely tied to classical structural analysis in mathematics. This connection between algorithms and structure is in particular apparent in discrete mathematics, where proofs are often constructive, and can be turned into algorithms more directly. The principle of greediness plays a fundamental role both in the design of continuous algorithms (where it is called the steepest descent or gradient method) and of discrete algorithms. The discrete structure most closely related to greediness is a matroid; in fact, matroids may be characterized axiomatically as those independence systems for which the greedy solution is optimal for certain optimization problems (e.g. linear objective functions, bottleneck functions). This book is an attempt to unify different approaches and to lead the reader from fundamental results in matroid theory to the current borderline of open research problems. The monograph begins by reviewing classical concepts from matroid theroy and extending them to greedoids. It then proceeds to the discussion of subclasses like interval greedoids, antimatroids or convex geometries, greedoids on partically ordered sets and greedoid intersections. Emphasis is placed on optimization problems in greedois. An algorithmic characterization of greedoids in terms of the greedy algorithm is derived, the behaviour with respect to linear functions is investigated, the shortest path problem for graphs is extended to a class of greedoids, linear descriptions of antimatroid polyhedra and complexity results are given and the Rado-Hall theorem on transversals is generalized. The self-contained volume which assumes only a basic familarity with combinatorial optimization ends with a chapter on topological results in connection with greedoids. ... Read more

Book Description

This comprehensive textbook on combinatorial optimization puts special emphasis on theoretical results and algorithms with provably good performance, in contrast to heuristics. It has arisen as the basis of several courses on combinatorial optimization and more special topics at graduate level. Since the complete book contains enough material for at least four semesters (4 hours a week), one usually selects material in a suitable way. The book contains complete but concise proofs, also for many deep results, some of which did not appear in a book before. Many very recent topics are covered as well, and many references are provided. Thus this book represents the state of the art of combinatorial optimization. This third edition contains a new chapter on facility location problems, an area which has been extremely active in the past few years. Furthermore there are several new sections and further material on various topics. New exercises and updates in the bibliography were added.

Customer Reviews (1)

Useful yet dense!
This is the most comprehensive compilation on combinatorial optiomization I have seen so far.
Usually, Papadimitriou's book is a good place for this material - but in many cases, looking for proofs and theorems - I had to use several books:
(*) Combinatorial Optimization Algorithms and Complexity by Papadimitriou and Steiglitz.
(*) Integer and Combinatorial Optimization by Nemhauser and Wolsey
(*) Theory of linear and integer programming by Schrijver
(*) Combinatorial Optimization by Cook, Cunningham, Pulleyblank and Schrijver
(*)Combinatorial Algorithms by Kreher and Stinson

This book, on the other hand, contains so much information and so many proved theorems - it's the richest resuorce in this topic, in my humble opinion.

Using it as a graduate level textbook for an *introduction* to combinatorial optimization is kind of hard - as although it's richness, some topics are described without enough detail or examples (like the topics on network flow and bipartite graphs) - yet the authors probably assumed some previous knowledge in those topics.

I prefer using this book as a reference rather than and intoduction.

The heavy mathematical notations in this book might scare some readers, but no-fear! You quickly get used to it, and appreciate the greatness in the notations, as they make the theorems more short and to the point. On the other hand - getting back to this book for a quick review on some subject might forceyou to flip pages for a fwe minutes, just to remember the notation again.

The authors intended this book to be a graduaet level textbook or an up-to-date referencework for current research. I believe they accomplished both targets! ... Read more

Book Description
mathematical gem--freshly cleaned and polished
This book is intended to be used as the text for a first course in combinatorics. the text has been shaped by two goals, namely, to make complex mathematics accessible to students with a wide range of abilities, interests, and motivations; and to create a pedagogical tool, useful to the broad spectrum of instructors who bring a variety of perspectives and expectations to such a course.

Features retained from the first edition:
* Lively and engaging writing style
* Timely and appropriate examples
* Numerous well-chosen exercises
* Flexible modular format
* Optional sections and appendices
Highlights of Second Edition enhancements:
* Smoothed and polished exposition, with a sharpened focus on key ideas
* Expanded discussion of linear codes
* New optional section on algorithms
* Greatly expanded hints and answers section
* Many new exercises and examples ... Read more

Customer Reviews (3)

The best combinatorics text I have seen so far
There is no question that combinatorics is becoming more important in the practice of computer science. While some of the most important techniques are covered in discrete mathematics courses, there is not enough breadth or depth in the coverage. As a consequence, one of my goals this upcoming year is to convince my colleagues at Mount Mercy to add a course in combinatorics to the curriculum. As part of the proposal, I need to create a list of resources and a sample schedule of topics to be covered. To begin this process, I studied several combinatorics texts and decided that this one was the best.
There are six chapters, and the titles are:

 Chapter 1 The mathematics of choice
 Chapter 2 Thecombinatorics of finite functions
 Chapter 3 Polya's theory of enumeration
 Chapter 4 Generating functions
 Chapter 5 Enumeration in graphs
 Chapter 6 Codes and designs

The coverage is thorough and the sequence has been structured to make the last four chapters independent. This necessitated the duplication of some definitions and theorems, but it is a sensible strategy. There is more than enough material for a one-semester course, so this increases the options available to an instructor. I often skip chapters in a textbook in order to cover a topic I consider critical and find it very helpful when the author repeats key definitions and theorems.
The style is mathematically rigorous, the approach is that of a mathematics textbook with the sequence being: theorem, proof, example(s). A large number of exercises are given at the end of each section and solutions/hints to the odd numbered ones are included. It is a sound combinatorics text and the one that I would most likely use if I ever teach such a course.

Published in Journal of Recreational Mathematics, reprinted with permission.

Author Responds
It is unfortunate that Amazon.com has attached to the second edition of this book a review of the first edition.

Who is the audience?
I used this book as a text for a junior-level course.The first two chapters are well-written, with useful exercises, and appropriate for the audience; later chapters are far more difficult, with uneven coverage, and the exercises are far less useful (I had to use other books for exercises, hand out extensive notes, etc.)These later chapters might be good for professional review, but not for students.
This is not just my perception; several students commented that the early part of the book was much better than the later part. ... Read more

Book Description
R.A. Bailey covers in this study the mathematics of association schemes--an area lying between pure mathematics and statistics that relates to the optimal design of scientific experiments. The book is accessible to mathematicians as well as statisticians.Arising from a graduate course taught by the author, it appeals to students as well as researchers as a valuable reference work from which to learn about the statistical/combinatorial aspects of their work. ... Read more

Book Description
A matroid is an abstract mathematical structure that captures combinatorial properties of matrices. This book offers a unique introduction to matroid theory, emphasizing motivations from matrix theory and applications to systems analysis.
This book serves also as a comprehensive presentation of the theory and application of mixed matrices, developed primarily by the present author in the last decade. A mixed matrix is a convenient mathematical tool for systems analysis, compatible with the physical observation that "fixed constants" and "system parameters" are to be distinguished in the description of engineering systems.
This book will be extremely useful to graduate students and researchers in engineering, mathematics and computer science. ... Read more

Book Description
A gentle introduction to the highly sophisticated world ofdiscrete mathematics, Mathematical Problems and Proofs presentstopics ranging from elementary definitions and theorems to advancedtopics -- such as cardinal numbers, generating functions, propertiesof Fibonacci numbers, and Euclidean algorithm. This excellent primerillustrates more than 150 solutions and proofs, thoroughly explainedin clear language. The generous historical references and anecdotesinterspersed throughout the text create interesting intermissions thatwill fuel readers' eagerness to inquire further about the topics andsome of our greatest mathematicians. The author guides readers throughthe process of solving enigmatic proofs and problems, and assists themin making the transition from problem solving to theorem proving.
At once a requisite text and an enjoyable read, MathematicalProblems and Proofs is an excellent entrée to discretemathematics for advanced students interested in mathematics,engineering, and science. ... Read more

Customer Reviews (1)

A Basketful of Gems
This is a very interesting book.If you have mastered the bare essentials of set theory (through the upside down A for "for all" and the backwards E for "there exists"), AND either 1) the bare essentials of combinatorics (through Pascal's triangle), or 2) the bare essentials of number theory (through the definition of the Moebius function and the statement of the Chinese Remainder theorem), or 3) the bare essentials of geometry (through the law of cosines), AND if you are very talented in mathematics, then this book is a "MUST READ".It matters not whether you are a high school student or a professional mathematician.You will find new and fruitful insights and quite a few interesting problems in this book.For the beginner, there are several tantalizing (but somewhat oversimplified) references to advanced topics such as Paul Cohen's proof of the independence of the continuum hypotheses and Wiles' proof of Fermat's last theorem. For the professional there are footnotes with references to little known and suprising results obtained in the 20th century.But, unlike the claims in the "editorial review", this book neither prepares you to read the literaure nor is it a store house of exercises which will help you take your problem solving abilities to the next level.The "editorial review" is "off", but the book is "right on".Only the title is unfortunate.Where it will help the career mathematician is in the "beer hall" or "coffee house" or "tea time" discussions with other mathematicians.It is just chock full of beautiful little "gems" which can be shared with one's friends.This is the kind of beautiful stuff that makes mathematics truly interesting and exciting and though I have searched for a book like this for the last 35 years, this is by far the best in its class which I have found.Dr. John Aiken, Jan 5, 2003. ... Read more

Book Description
Providing a unified exposition of the theory of symmetric designs with emphasis on recent developments, this volume covers the combinatorial aspects of the theory, giving particular attention to the construction of symmetric designs and related objects. The last five chapters are devoted to balanced generalized weighing matrices, decomposable symmetric designs, subdesigns of symmetric designs, non-embeddable quasi-residual designs, and Ryser designs. The book concludes with a comprehensive bibliography of over 400 entries. Detailed proofs and a large number of exercises make it suitable as a text for an advanced course in combinatorial designs. ... Read more

Book Description
This volume is comprised of the invited lectures given at the 14th British Combinatorial Conference.The lectures survey many topical areas of current research activity in combinatorics and its applications, and also provide a valuable overview of the subject, for both mathematicians and computer scientists. ... Read more

Book Description
Resulting from papers from Algebraic, Extremal and Metric Combinatorics 1986 conference held at the University of Montreal, this book represents a comprehensive overview of the present state of progress in three related areas of combinatorics.Topics covered in the articles include association shemes, extremal problems, combinatorial geometries and matroids, and designs.All the papers contain new results and many are extensive surveys of particular areas of research. ... Read more

Product Description
One of the most active areas in mathematics today is the rapidly emerging new topic of "additive combinatorics". Building on Gowers' use of the Freiman-Ruzsa theorem in harmonic analysis (in particular, his proof of Szemerédi's theorem), Green and Tao famously proved that there are arbitrarily long arithmetic progressions of primes, and Bourgain and his co-authors have given non-trivial estimates for hitherto untouchably short exponential sums. There are further important consequences in group theory and in complexity theory and compelling questions in ergodic theory, discrete geometry and many other disciplines. The basis of the subject is not too difficult: it can be best described as the theory of adding together sets of numbers; in particular, understanding the structure of the two original sets if their sum is small. This book brings together key researchers from all of these different areas, sharing their insights in articles meant to inspire mathematicians coming from all sorts of different backgrounds.Titles in this series are co-published with the Centre de Recherches Mathématiques. ... Read more

Book Description
This is the first book on higher dimensional Hadamard matrices and their applications in telecommunications and information security. It is divided into three parts according to the dimensions of the Hadamard matrices treated. The first part stresses the classical 2-dimensional Walsh and Hadamard matrices. Fast algorithms, updated constructions, existence results, and their generalised forms are presented. The second part deals with the lower-dimensional cases, e.g. 3-, 4-, and 6-dimensional Walsh and Hadamard matrices and transforms. One of the aims of this part is to simplify moving smoothly from 2-dimensional cases to the general higher-dimensional cases. This part concentrates on the 3-dimensional Hadamard and Walsh matrices. Constructions based upon direct multiplication, and upon recursive methods, and perfect binary arrays are also introduced. Another important topic is the existence and construction of 3-dimensional Hadamard matrices of orders 4k and 4k + 2 respectively, and a group of transforms based on 2-, 3-, 4-, and 6-dimensional Walsh-Hadamard matrices and their corresponding fast algorithms.
The third part is the key part, which investigates the N-dimensional Hadamard matrices of order 2, which have been proved equivalent to the well-known H-Boolean functions and the perfect binary arrays of order 2. This equivalence motivates a group of perfect results about the enumeration of higher-dimensional Hadamard matrices of order 2. Applications of these matrices to feed forward networking, stream cipher, Bent functions, and error correction codes are presented in turn. After introducing the definitions of the regular, proper, improper, and generalised higher-dimensional Hadamard matrices, many theorems about the existence and constructions are presented. Perfect binary arrays, generalised perfect arrays, and the orthogonal designs are also used to construct new higher-dimensional Hadamard matrices. The many open problems in the study of the theory of higher-dimensional Hadamard matrices, which are also listed in the book, will encourage further research.
Audience: This volume will appeal to researchers and graduate students whose work involves signal processing, coding, information security, and applied discrete mathematics. ... Read more

Book Description
Cuts and metrics are well-known objects that arise - independently, but with many deep and fascinating connections - in diverse fields: in graph theory, combinatorial optimization, geometry of numbers, combinatorial matrix theory, statistical physics, VLSI design etc. This book offers a comprehensive summary together with a global view, establishing both old and new links. Its treatment ranges from classical theorems of Menger and Schoenberg to recent developments such as approximation results for multicommodity flow and max-cut problems, metric aspects of Delaunay polytopes, isometric graph embeddings, and matrix completion problems. The discussion leads to many interesting subjects that cannot be found elsewhere, providing a unique and invaluable source for researchers and graduate students. ... Read more

Customer Reviews (1)

This book should not be out of print
It would be a very useful addition to the existing texts. Everybody who teaches combinatorics to undergraduates knows how important is to have a source of interesting examples and problems. This book is of special interest for those who teaches enumerative combinatorics and whose course is expected not to have much of graph theory (because graph theory is taught separately). For such courses this book is more useful than many others. Also it contains enough theory for third year course taught at many of the USA universities. ... Read more