This long-awaited textbook is the most comprehensive introduction to a broad swath of combinatorial and discrete mathematics. The text covers enumeration, graphs, sets, and methods, and it includes both classical results and more recent developments. Assuming no prior exposure to combinatorics, it explains the basic material for graduate-level students in mathematics and computer science. Optional more advanced material also makes it valuable as a research reference. Suitable for a one-year course or a one-semester introduction, this textbook prepares students to move on to more advanced material. It is organized to emphasize connections among the topics, and facilitate instruction, self-study, and research, with more than 2200 exercises (many accompanied by hints) at various levels of difficulty. Consistent notation and terminology are used throughout, allowing for a discussion of diverse topics in a unified language. The thorough bibliography, containing thousands of citations, makes this a valuable source for students and researchers alike.
Author(s): Douglas B. West
Publisher: Cambridge University Press
Year: 2020
Language: English
Pages: 988
Cover
Half-title page
Title page
Copyright page
Dedication
Contents
Preface
Chapter 0 – Introduction
Part I — Enumeration
Chapter 1 – Combinatorial Arguments.
1.1. Classical Models.
1.2. Identities
1.3. Applications
Chapter 2 – Recurrence Relations
2.1. Obtaining Recurrences
2.2. Elementary Solution Methods
2.3. Further Topics
Chapter 3 – Generating Functions
3.1. Ordinary Generating Functions
3.2. Coefficients and Applications
3.3. Exponential Generating Functions
3.4. Partitions of Integers
Chapter 4 – Further Topics
4.1. The Inclusion-Exclusion Principle
4.2. P´olya–Redfield Counting
4.3. Permutations and Tableaux
Part II — Graphs
Chapter 5 – First Concepts for Graphs
5.1. Definitions and Examples
5.2. Vertex Degrees
5.3. Connection and Decomposition
5.4. Trees and Distance
Chapter 6 – Matchings
6.1. Matching in Bipartite Graphs
6.2. Matching in General Graphs
6.3. Algorithmic Aspects
Chapter 7 – Connectivity and Cycles
7.1. Connectivity Parameters
7.2. Properties of k-Connected Graphs
7.3. Spanning Cycles
Chapter 8 – Coloring
8.1. Vertex Coloring
8.2. Structural Aspects
8.3. Edge-Coloring and Perfection
Chapter 9 – Planar Graphs
9.1. Embeddings and Euler ’s Formula
9.2. Structure of Planar Graphs
9.3. Coloring of Planar Graphs
Part III — Sets
Chapter 10 – Ramsey Theory
10.1. The Pigeonhole Principle
10.2. Ramsey’s Theorem
10.3. Further Topics
Chapter 11 – Extremal Problems
11.1. Forced Subgraphs
11.2. Families of Sets
11.3. Matroids
Chapter 12 – Partially Ordered Sets
12.1. Structure of Posets
12.2. Symmetric Chains and LYM Orders
12.3. Linear Extensions and Dimension
12.4. Special Families of Posets
Chapter 13 – Combinatorial Designs
13.1. Arrangements.
13.2. Projective Planes
13.3. Further Constructions
Part IV — Methods
Chapter 14 – The Probabilistic Method
14.1. Existence and Expectation
14.2. Refinements of Basic Methods
14.3. Moments and Thresholds
14.4. Concentration Inequalities
Chapter 15 – Linear Algebra
15.1. Dimension and Polynomials
15.2. Matrices
15.3. Eigenvalues
Chapter 16 – Geometry and Topology
16.1. Graph Drawings
16.2. Combinatorial Topology
16.3. Volumes and Containment
Hints to Selected Exercises
References
Author Index
Glossary of Notation
Subject Index
