Advanced graph theory and combinatorics

Content: 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     247    10.6. Comments 248    Bibliography 249    Index  263