Book:John M. Harris/Combinatorics and Graph Theory/Second Edition

Subject Matter

 * Graph Theory

Contents

 * Preface to the Second Edition
 * Preface to the First Edition


 * 1 Graph Theory
 * 1.1 Introductory Concepts
 * 1.1.1 Graphs and Their Relatives
 * 1.1.2 The Basics
 * 1.1.3 Special Types of Graphs
 * 1.2 Distance in Graphs
 * 1.2.1 Definitions and a Few Properties
 * 1.2.2 Graphs and Matrices
 * 1.2.3 Graph Models and Distance
 * 1.3 Trees
 * 1.3.1 Definitions and Examples
 * 1.3.2 Properties of Trees
 * 1.3.3 Spanning Trees
 * 1.3.4 Counting Trees
 * 1.4 Trails, Circuits, Paths, and Cycles
 * 1.4.1 The Bridges of Königsberg
 * 1.4.2 Eulerian Trails and Circuits
 * 1.4.3 Hamiltonian Paths and Cycles
 * 1.4.4 Three Open Problems
 * 1.5 Planarity
 * 1.5.1 Definitions and Examples
 * 1.5.2 Euler's Formula and Beyond
 * 1.5.3 Regular Polyhedra
 * 1.5.4 Kuratowski's Theorem
 * 1.6 Colorings
 * 1.6.1 Definitions
 * 1.6.2 Bounds on Chromatic Number
 * 1.6.3 The Four Color Problem
 * 1.6.4 Chromatic Polynomials
 * 1.7 Matchings
 * 1.7.1 Definitions
 * 1.7.2 Hall's Theorem and SDRs
 * 1.7.3 The König–Egerváry Theorem
 * 1.7.4 Perfect Matchings
 * 1.8 Ramsey Theory
 * 1.8.1 Classical Ramsey Numbers
 * 1.8.2 Exact Ramsey Numbers and Bounds
 * 1.8.3 Graph Ramsey Theory
 * 1.9 References


 * 2 Combinatorics
 * 2.1 Some Essential Problems
 * 2.2 Binomial Coefficients
 * 2.3 Multinomial Coefficients
 * 2.4 The Pigeonhole Principle
 * 2.5 The Principle of Inclusion and Exclusion
 * 2.6 Generating Functions
 * 2.6.1 Double Decks
 * 2.6.2 Counting with Repetition
 * 2.6.3 Changing Money
 * 2.6.4 Fibonacci Numbers
 * 2.6.5 Recurrence Relations
 * 2.6.6 Catalan Numbers
 * 2.7 Pólya's Theory of Counting
 * 2.7.1 Permutation Groups
 * 2.7.2 Burnside's Lemma
 * 2.7.3 The Cycle Index
 * 2.7.4 Pólya's Enumeration Formula
 * 2.7.5 de Bruijn's Generalization
 * 2.8 More Numbers
 * 2.8.1 Partitions
 * 2.8.2 Stirling Cycle Numbers
 * 2.8.3 Stirling Set Numbers
 * 2.8.4 Bell Numbers
 * 2.8.5 Eulerian Numbers
 * 2.9 Stable Marriage
 * 2.9.1 The Gale–Shapley Algorithm
 * 2.9.2 Variations on Stable Marriage
 * 2.10 Combinatorial Geometry
 * 2.10.1 Sylvester's Problem
 * 2.10.2 Convex Polygons
 * 2.11 References


 * 3 Infinite Combinatorics and Graphs
 * 3.1 Pigeons and Trees
 * 3.2 Ramsey Revisited
 * 3.3 ZFC
 * 3.3.1 Language and Logical Axioms
 * 3.3.2 Proper Axioms
 * 3.3.3 Axiom of Choice
 * 3.4 The Return of der König
 * 3.5 Ordinals, Cardinals, and Many Pigeons
 * 3.5.1 Cardinality
 * 3.5.2 Ordinals and Cardinals
 * 3.5.3 Pigeons Finished Off
 * 3.6 Incompleteness and Cardinals
 * 3.6.1 Gödel's Theorems for PA and ZFC
 * 3.6.2 Inaccessible Cardinals
 * 3.6.3 A Small Collage of Large Cardinals
 * 3.7 Weakly Compact Cardinals
 * 3.8 Infinite Marriage Problems
 * 3.8.1 Hall and Hall
 * 3.8.2 Countably Many Men
 * 3.8.3 Uncountably Many Men
 * 3.8.4 Espousable Cardinals
 * 3.8.5 Perfect Matchings
 * 3.9 Finite Combinatorics with Infinite Consequences
 * 3.10 $k$-critical Linear Orderings
 * 3.11 Points of Departure
 * 3.12 References


 * References
 * Index