Advanced Graph Theory and Combinatorics

Foreword  ix

Introduction xi

Chapter 1. A First Encounter with Graphs  1

1.1. A few definitions  1

1.1.1. Directed graphs  1

1.1.2. Unoriented graphs 9

1.2. Paths and connected components  14

1.2.1. Connected components  16

1.2.2. Stronger notions of connectivity 18

1.3. Eulerian graphs  23

1.4. Defining Hamiltonian graphs 25

1.5. Distance and shortest path  27

1.6. A few applications 30

1.7. Comments 35

1.8. Exercises  37

Chapter 2. A Glimpse at Complexity Theory 43

2.1. Some complexity classes 43

2.2. Polynomial reductions 46

2.3. More hard problems in graph theory 49

Chapter 3. Hamiltonian Graphs 53

3.1. A necessary condition 53

3.2. A theorem of Dirac  55

3.3. A theorem of Ore and the closure of a graph  56

3.4. Chvatal’s condition on degrees  59

3.5. Partition of Kn into Hamiltonian circuits  62

3.6. De Bruijn graphs and magic tricks  65

3.7. Exercises  68

Chapter 4. Topological Sort and Graph Traversals  69

4.1. Trees  69

4.2. Acyclic graphs 79

4.3. Exercises  82

Chapter 5. Building New Graphs from Old Ones  85

5.1. Some natural transformations  85

5.2. Products  90

5.3. Quotients  92

5.4. Counting spanning trees  93

5.5. Unraveling 94

5.6. Exercises  96

Chapter 6. Planar Graphs 99

6.1. Formal definitions 99

6.2. Euler’s formula 104

6.3. Steinitz’ theorem  109

6.4. About the four-color theorem 113

6.5. The five-color theorem  115

6.6. From Kuratowski’s theorem to minors  120

6.7. Exercises  123

Chapter 7. Colorings  127

7.1. Homomorphisms of graphs  127

7.2. A digression: isomorphisms and labeled vertices  131

7.3. Link with colorings  134

7.4. Chromatic number and chromatic polynomial 136

7.5. Ramsey numbers  140

7.6. Exercises  147

Chapter 8. Algebraic Graph Theory  151

8.1. Prerequisites  151

8.2. Adjacency matrix 154

8.3. Playing with linear recurrences  160

8.4. Interpretation of the coefficients 168

8.5. A theorem of Hoffman  169

8.6. Counting directed spanning trees 172

8.7. Comments 177

8.8. Exercises  178

Chapter 9. Perron–Frobenius Theory 183

9.1. Primitive graphs and Perron’s theorem 183

9.2. Irreducible graphs 188

9.3. Applications  190

9.4. Asymptotic properties 195

9.4.1. Canonical form  196

9.4.2. Graphs with primitive components 197

9.4.3. Structure of connected graphs 206

9.4.4. Period and the Perron–Frobenius theorem 214

9.4.5. Concluding examples 218

9.5. The case of polynomial growth  224

9.6. Exercises  231

Chapter 10. Google’s Page Rank  233

10.1. Defining the Google matrix 238

10.2. Harvesting the primitivity of the Google matrix  241

10.3. Computation 246

10.4. Probabilistic interpretation 246

10.5. Dependence on the parameter α  247

10.6. Comments 248

Bibliography 249

Index  263