A Walk Through Combinatorics

An Introduction to Enumeration and Graph Theory

This is a textbook for an introductory combinatorics course lasting one or two semesters. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain material not explicitly discussed in the preceding text, so as to provide instructors with extra choices if they want to shift the emphasis of their course.

Just as with the first three editions, the new edition walks the reader through the classic parts of combinatorial enumeration and graph theory, while also discussing some recent progress in the area: on the one hand, providing material that will help students learn the basic techniques, and on the other hand, showing that some questions at the forefront of research are comprehensible and accessible to the talented and hardworking undergraduate. The basic topics discussed are: the twelvefold way, cycles in permutations, the formula of inclusion and exclusion, the notion of graphs and trees, matchings, Eulerian and Hamiltonian cycles, and planar graphs.

New to this edition are the Quick Check exercises at the end of each section. In all, the new edition contains about 240 new exercises. Extra examples were added to some sections where readers asked for them.

The selected advanced topics are: Ramsey theory, pattern avoidance, the probabilistic method, partially ordered sets, the theory of designs, enumeration under group action, generating functions of labeled and unlabeled structures and algorithms and complexity.

The book encourages students to learn more combinatorics, provides them with a not only useful but also enjoyable and engaging reading.

The Solution Manual is available upon request for all instructors who adopt this book as a course text. Please send your request to sales@wspc.com.

The previous edition of this textbook has been adopted at various schools including UCLA, MIT, University of Michigan, and Swarthmore College. It was also translated into Korean.

Contents:

• Basic Methods:

  • Seven is More Than Six. The Pigeon-Hole Principle
  • One Step at a Time. The Method of Mathematical Induction

• Enumerative Combinatorics:

  • There are a Lot of Them. Elementary Counting Problems
  • No Matter How You Slice It. The Binomial Theorem and Related Identities
  • Divide and Conquer. Partitions
  • Not So Vicious Cycles. Cycles in Permutations
  • You Shall Not Overcount. The Sieve
  • A Function is Worth Many Numbers. Generating Functions

• Graph Theory:

  • Dots and Lines. The Origins of Graph Theory
  • Staying Connected. Trees
  • Finding a Good Match. Coloring and Matching
  • Do Not Cross. Planar Graphs

• Horizons:

  • Does It Clique? Ramsey Theory
  • So Hard to Avoid. Subsequence Conditions on Permutations
  • Who Knows What It Looks Like, But It Exists. The Probabilistic Method
  • At Least Some Order. Partial Orders and Lattices
  • As Evenly as Possible. Block Designs and Error Correcting Codes
  • Are They Really Different? Counting Unlabeled Structures
  • The Sooner the Better. Combinatorial Algorithms
  • Does Many Mean More Than One? Computational Complexity

Readership: Upper level undergraduates and graduate students in the field of combinatorics and graph theory.