Graph colouring and the probabilistic method:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg ; New York
Springer
[2002]
|
| Series: | Algorithms and combinatorics
23 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 326 Seiten 19 Diagramme |
| ISBN: | 3540421394 |
Staff View
MARC
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| 100 | 1 | |a Molloy, Michael S. |d 1967- |e Verfasser |0 (DE-588)1334536767 |4 aut | |
| 245 | 1 | 0 | |a Graph colouring and the probabilistic method |c Michael Molloy, Bruce Reed |
| 264 | 1 | |a Berlin ; Heidelberg ; New York |b Springer |c [2002] | |
| 300 | |a XIV, 326 Seiten |b 19 Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Record in the Search Index
| _version_ | 1822603439470804992 |
|---|---|
| adam_text |
Contents
Part I. Preliminaries
1. Colouring Preliminaries 3
1.1 The Basic Definitions 3
1.2 Some Classical Results 5
1.3 Fundamental Open Problems 7
1.4 A Point of View 9
1.5 A Useful Technical Lemma 10
1.6 Constrained Colourings
and the List Chromatic Number 11
1.7 Intelligent Greedy Colouring 12
Exercises 13
2. Probabilistic Preliminaries 15
2.1 Finite Probability Spaces 15
2.2 Random Variables and Their Expectations 17
2.3 One Last Definition 19
2.4 The Method of Deferred Decisions 20
Exercises 21
Part II. Basic Probabilistic Tools
3. The First Moment Method 27
3.1 2 Colouring Hypergraphs 28
3.2 Triangle Free Graphs with High Chromatic Number 29
3.3 Bounding the List Chromatic Number as a Functione
of the Colouring Number 31
3.3.1 An Open Problem 33
3.4 The Cochromatic Number 34
Exercises 36
X Contents
4. The Lovasz Local Lemma 39
4.1 Constrained Colourings
and the List Chromatic Number 41
Exercises 42
5. The Chernoff Bound 43
5.1 Hajos's Conjecture 44
Exercises 46
Part III. Vertex Partitions
6. Hadwiger's Conjecture 49
6.1 Step 1: Finding a Dense Subgraph 50
6.2 Step 2: Finding a Split Minor 50
6.3 Step 3: Finding the Minor 52
Exercises 53
7. A First Glimpse of Total Colouring 55
8. The Strong Chromatic Number 61
Exercises 65
9. Total Colouring Revisited 67
9.1 The Idea 67
9.2 Some Details 70
9.3 The Main Proof 74
Exercises 75
Part IV. A Naive Colouring Procedure
10. Talagrand's Inequality
and Colouring Sparse Graphs 79
10.1 Talagrand's Inequality 79
10.2 Colouring Triangle Free Graphs 83
10.3 Colouring Sparse Graphs 86
10.4 Strong Edge Colourings 87
Exercises 89
11. Azuma's Inequality and a Strengthening
of Brooks' Theorem 91
11.1 Azuma's Inequality 91
11.2 A Strengthening of Brooks' Theorem 94
11.3 The Probabilistic Analysis 98
Contents XI
11.4 Constructing the Decomposition 100
Exercises 103
Part V. An Iterative Approach
12. Graphs with Girth at Least Five 107
12.1 Introduction 107
12.2 A Wasteful Colouring Procedure 109
12.2.1 The Heart of The Procedure 109
12.2.2 The Finishing Blow Ill
12.3 The Main Steps of the Proof 112
12.4 Most of the Details 115
12.5 The Concentration Details 120
Exercises 123
13. Triangle Free Graphs 125
13.1 An Outline 126
13.1.1 A Modified Procedure 126
13.1.2 Fluctuating Probabilities 128
13.1.3 A Technical Fiddle 130
13.1.4 A Complication 131
13.2 The Procedure 131
13.2.1 Dealing with Large Probabilities 131
13.2.2 The Main Procedure 132
13.2.3 The Final Step 132
13.2.4 The Parameters 133
13.3 Expectation and Concentration 136
Exercises 138
14. The List Colouring Conjecture 139
14.1 A Proof Sketch 140
14.1.1 Preliminaries 140
14.1.2 The Local Structure 140
14.1.3 Rates of Change 141
14.1.4 The Preprocessing Step 142
14.2 Choosing Reserve,, 144
14.3 The Expected Value Details 145
14.4 The Concentration Details 149
14.5 The Wrapup 151
14.6 Linear Hypergraphs 152
Exercises 153
XII Contents
Part VI. A Structural Decomposition
15. The Structural Decomposition 157
15.1 Preliminary Remarks 157
15.2 The Decomposition 157
15.3 Partitioning the Dense Sets 160
15.4 Graphs with x Near A 165
15.4.1 Generalizing Brooks' Theorem 165
15.4.2 Blowing Up a Vertex 166
Exercises 167
16. u , A and x 169
16.1 The Modified Colouring Procedure 171
16.2 An Extension of Talagrand's Inequality 172
16.3 Strongly Non Adjacent Vertices 173
16.4 Many Repeated Colours 175
16.5 The Proof of Theorem 16.5 179
16.6 Proving the Harder Theorems 181
16.7 Two Proofs 182
Exercises 184
17. Near Optimal Total Colouring I: Sparse Graphs 185
17.1 Introduction 185
17.2 The Procedure 187
17.3 The Analysis of the Procedure 188
17.4 The Final Phase 191
18. Near Optimal Total Colouring II: General Graphs 195
18.1 Introduction 195
18.2 Phase I: An Initial Colouring 198
18.2.1 Ornery Sets 198
18.2.2 The Output of Phase I 200
18.2.3 A Proof Sketch 201
18.3 Phase II: Colouring the Dense Sets 206
18.3.1 Tt is Non Empty 207
18.3.2 Our Distribution is Nearly Uniform 208
18.3.3 Completing the Proof 209
18.4 Phase III: The Temporary Colours 210
18.4.1 Step 1: The Kernels of the Ornery Sets 211
18.4.2 Step 2: The Remaining Temporary Colours 215
18.5 Phase IV Finishing the Sparse Vertices 216
18.6 The Ornery Set Lemmas 217
Contents XIII
Part VII. Sharpening our Tools
19. Generalizations of the Local Lemma 221
19.1 Non Uniform Hypergraph Colouring 222
19.2 More Frugal Colouring 224
19.2.1 Acyclic Edge Colouring 225
19.3 Proofs 226
19.4 The Lopsided Local Lemma 228
Exercises 229
20. A Closer Look at Talagrand's Inequality 231
20.1 The Original Inequality 231
20.2 More Versions 234
Exercises 236
Part VIII. Colour Assignment via Fractional Colouring
21. Finding Fractional Colourings and Large Stable Sets 239
21.1 Fractional Colouring 239
21.2 Finding Large Stable Sets in Triangle Free Graphs 242
21.3 Fractionally, X y±f±1 244
Exercises 246
22. Hard Core Distributions on Matchings 247
22.1 Hard Core Distributions 247
22.2 Hard Core Distributions from Fractional Colourings 249
22.3 The Mating Map 252
22.4 An Independence Result 254
22.5 More Independence Results 260
23. The Asymptotics of Edge Colouring Multigraphs 265
23.1 Assigning the Colours 265
23.1.1 Hard Core Distributions
and Approximate Independence 266
23.2 The Chromatic Index 267
23.3 The List Chromatic Index 270
23.3.1 Analyzing an Iteration 272
23.3.2 Analyzing a Different Procedure 274
23.3.3 One More Tool 277
23.4 Comparing the Procedures 279
23.4.1 Proving Lemma 23.9 282
XIV Contents
Part IX. Algorithmic Aspects
24. The Method of Conditional Expectations 287
24.1 The Basic Ideas 287
24.2 An Algorithm 288
24.3 Generalized Tic Tac Toe 289
24.4 Proof of Lemma 24.3 291
25. Algorithmic Aspects of the Local Lemma 295
25.1 The Algorithm 296
25.1.1 The Basics 296
25.1.2 Further Details 299
25.2 A Different Approach 300
25.3 Applicability of the Technique 301
25.3.1 Further Extensions 303
25.4 Extending the Approach 304
25.4.1 3 Uniform Hypergraphs 305
25.4.2 fc Uniform Hypergraphs with k 4 308
25.4.3 The General Technique 310
Exercises 312
References 314
Index 323 |
| any_adam_object | 1 |
| author | Molloy, Michael S. 1967- Reed, Bruce A. 1962- |
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| dewey-full | 514.223 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 514 - Topology |
| dewey-raw | 514.223 |
| dewey-search | 514.223 |
| dewey-sort | 3514.223 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV013970071 |
| illustrated | Not Illustrated |
| indexdate | 2025-01-29T17:01:29Z |
| institution | BVB |
| isbn | 3540421394 |
| language | English |
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| series | Algorithms and combinatorics |
| series2 | Algorithms and combinatorics |
| spelling | Molloy, Michael S. 1967- Verfasser (DE-588)1334536767 aut Graph colouring and the probabilistic method Michael Molloy, Bruce Reed Berlin ; Heidelberg ; New York Springer [2002] XIV, 326 Seiten 19 Diagramme txt rdacontent n rdamedia nc rdacarrier Algorithms and combinatorics 23 Graphfärbung - Kombinatorische Wahrscheinlichkeitstheorie Graphfärbung (DE-588)4472286-2 gnd rswk-swf Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd rswk-swf Graphfärbung (DE-588)4472286-2 s Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 s DE-604 Reed, Bruce A. 1962- Verfasser (DE-588)170807916 aut Erscheint auch als Online-Ausgabe 978-3-642-04016-0 (DE-604)BV042422450 Algorithms and combinatorics 23 (DE-604)BV000617357 23 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009562425&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Molloy, Michael S. 1967- Reed, Bruce A. 1962- Graph colouring and the probabilistic method Algorithms and combinatorics Graphfärbung - Kombinatorische Wahrscheinlichkeitstheorie Graphfärbung (DE-588)4472286-2 gnd Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd |
| subject_GND | (DE-588)4472286-2 (DE-588)4132446-8 |
| title | Graph colouring and the probabilistic method |
| title_auth | Graph colouring and the probabilistic method |
| title_exact_search | Graph colouring and the probabilistic method |
| title_full | Graph colouring and the probabilistic method Michael Molloy, Bruce Reed |
| title_fullStr | Graph colouring and the probabilistic method Michael Molloy, Bruce Reed |
| title_full_unstemmed | Graph colouring and the probabilistic method Michael Molloy, Bruce Reed |
| title_short | Graph colouring and the probabilistic method |
| title_sort | graph colouring and the probabilistic method |
| topic | Graphfärbung - Kombinatorische Wahrscheinlichkeitstheorie Graphfärbung (DE-588)4472286-2 gnd Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd |
| topic_facet | Graphfärbung - Kombinatorische Wahrscheinlichkeitstheorie Graphfärbung Kombinatorische Wahrscheinlichkeitstheorie |
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| volume_link | (DE-604)BV000617357 |
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