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El. knyga: Topics in Chromatic Graph Theory

Edited by (Purdue University, Indiana), Edited by (The Open University, Milton Keynes)
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Chromatic graph theory is a thriving area that uses various ideas of 'colouring' (of vertices, edges, and so on) to explore aspects of graph theory. It has links with other areas of mathematics, including topology, algebra and geometry, and is increasingly used in such areas as computer networks, where colouring algorithms form an important feature. While other books cover portions of the material, no other title has such a wide scope as this one, in which acknowledged international experts in the field provide a broad survey of the subject. All fifteen chapters have been carefully edited, with uniform notation and terminology applied throughout. Bjarne Toft (Odense, Denmark), widely recognized for his substantial contributions to the area, acted as academic consultant. The book serves as a valuable reference for researchers and graduate students in graph theory and combinatorics and as a useful introduction to the topic for mathematicians in related fields.

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A broad survey written by acknowledged international experts in the field.
Foreword xiii
Bjarne Toft
Preface xv
Preliminaries 1(12)
Lowell W. Beineke
Robin J. Wilson
1 Graph theory
1(8)
2 Graph colourings
9(4)
1 Colouring graphs on surfaces
13(23)
Bojan Mohar
1 Introduction
13(1)
2 Planar graphs are 4-colourable and 5-choosable
14(4)
3 Heawood's formula
18(2)
4 Colouring with few colours
20(3)
5 Grotzsch's theorem and its generalizations
23(2)
6 Colouring--How duality
25(4)
7 The acyclic chromatic number
29(1)
8 Degenerate colourings
30(1)
9 The star chromatic number
31(1)
10 Summary
32(4)
2 Brooks's theorem
36(20)
Michael Stiebitz
Bjarne Toft
1 Introduction
36(1)
2 Proofs of Brooks's theorem
37(4)
3 Critical graphs with few edges
41(4)
4 Bounding Χ by Δ and ω
45(3)
5 Graphs with Χ close to Δ
48(2)
6 Notes
50(6)
3 Chromatic polynomials
56(17)
Bill Jackson
1 Introduction
56(1)
2 Definitions and elementary properties
57(2)
3 Log concavity and other inequalities
59(1)
4 Chromatic roots
60(4)
5 Related polynomials
64(9)
4 Hadwiger's conjecture
73(21)
Ken-Ichi Kawarabayashi
1 Introduction
73(1)
2 Complete graph minors: early results
74(1)
3 Contraction-critical graphs
75(4)
4 Algorithmic aspects of the weak conjecture
79(2)
5 Algorithmic aspects of the strong conjecture
81(1)
6 The odd conjecture
82(3)
7 Independent sets and Hadwiger's conjecture
85(1)
8 Other variants of the conjecture
86(3)
9 Open problems
89(5)
5 Edge-colourings
94(20)
Jessica McDonald
1 Introduction
94(2)
2 Elementary sets and Kempe changes
96(1)
3 Tashkinov trees and upper bounds
97(4)
4 Towards the Goldberg-Seymour conjecture
101(2)
5 Extreme graphs
103(2)
6 The classification problem and critical graphs
105(3)
7 The dichotomy of edge-colouring
108(1)
8 Final thoughts
109(5)
6 List-colourings
114(23)
Michael Stiebitz
Margit Voigt
1 Introduction
114(4)
2 Orientations and list-colourings
118(3)
3 Planar graphs
121(7)
4 Precolouring extensions
128(1)
5 Notes
129(8)
7 Perfect graphs
137(24)
Nicolas Trotignon
1 Introduction
137(2)
2 Lovasz's perfect graph theorem
139(2)
3 Basic graphs
141(1)
4 Decompositions
142(4)
5 The strategy of the proof
146(2)
6 Book from the Proof
148(3)
7 Recognizing perfect graphs
151(1)
8 Berge trigraphs
152(2)
9 Even pairs: a shorter proof of the SPGT
154(1)
10 Colouring perfect graphs
155(6)
8 Geometric graphs
161(20)
Alexander Soifer
1 The chromatic number of the plane
161(1)
2 The polychromatic number: lower bounds
162(3)
3 The dc Bruijn-Erdos reduction to finite sets
165(2)
4 The polychromatic number: upper bounds
167(2)
5 The continuum of 6-colourings
169(2)
6 Special circumstances
171(1)
7 Space explorations
172(1)
8 Rational spaces
173(2)
9 One odd graph
175(1)
10 Influence of set theory axioms
175(2)
11 Predicting the future
177(4)
9 Integer flows and orientations
181(18)
Hongjian Lai
Rong Luo
Cun-Quan Zhang
1 Introduction
181(2)
2 Basic properties
183(2)
3 4-flows
185(1)
4 3-fiows
185(2)
5 5-flows
187(1)
6 Bounded orientations and circular flows
188(2)
7 Modulo orientations and (2 + 1/t-flows
190(1)
8 Contractible configurations
191(3)
9 Related problems
194(5)
10 Colouring random graphs
199(31)
Ross J. Kang
Colin Mcdiarmid
1 Introduction
199(3)
2 Dense random graphs
202(6)
3 Sparse random graphs
208(6)
4 Random regular graphs
214(3)
5 Random geometric graphs
217(2)
6 Random planar graphs and related classes
219(3)
7 Other colourings
222(8)
11 Hypergraph colouring
230(25)
Csilla Bujtas
Zsolt Tuza
Vitaly Voloshin
1 Introduction
230(4)
2 Proper vertex- and edge-colourings
234(4)
3 C-colourings
238(5)
4 Colourings of mixed hypergraphs
243(4)
5 Colour-bounded and stably bounded hypergraphs
247(4)
6 Conclusion
251(4)
12 Chromatic scheduling
255(22)
Dominique De Werra
Alain Hertz
1 Introduction
255(1)
2 Colouring with weights on the vertices
256(2)
3 List-colouring
258(2)
4 Mixed graph colouring
260(1)
5 Co-colouring
261(1)
6 Colouring with preferences
262(2)
7 Bandwidth colouring
264(2)
8 Edge-colouring
266(2)
9 Sports scheduling
268(1)
10 Balancing requirements
269(4)
11 Compactness
273(1)
12 Conclusion
274(3)
13 Graph colouring algorithms
277(27)
Thore Husfeldt
1 Introduction
277(2)
2 Greedy colouring
279(5)
3 Recursion
284(3)
4 Subgraph expansion
287(2)
5 Local augmentation
289(3)
6 Vector colouring
292(4)
7 Reductions
296(5)
8 Conclusion
301(3)
14 Colouring games
304(23)
Zsolt Tuza
Xuding Zhu
1 Introduction
304(3)
2 Marking games
307(5)
3 Greedy colouring games
312(1)
4 Playing on the edge-set
312(1)
5 Oriented and directed graphs
313(2)
6 Asymmetric games
315(1)
7 Relaxed games
316(1)
8 Paintability
316(5)
9 Achievement and avoidance games
321(1)
10 The acyclic orientation game
322(5)
15 Unsolved graph colouring problems
327(31)
Tommy Jensen
Bjarne Toft
1 Introduction
327(1)
2 Complete graphs and chromatic numbers
328(5)
3 Graphs on surfaces
333(6)
4 Degrees and colourings
339(5)
5 Edge-colourings
344(4)
6 Flow problems
348(2)
7 Concluding remarks
350(8)
Notes on contributors 358(5)
Index 363
Lowell W. Beineke is Schrey Professor of Mathematics at Indiana University-Purdue University, Fort Wayne (IPFW), where he has worked since receiving his PhD from the University of Michigan under the guidance of Frank Harary. His graph theory interests include topological graph theory, line graphs, tournaments, decompositions and vulnerability. He has published over 100 papers in graph theory and has served as editor of the College Mathematics Journal. With Robin Wilson he has co-edited five books in addition to the three earlier volumes in this series. Recent honours include an award instituted in his name by the College of Arts and Sciences at IPFW and a Certificate of Meritorious Service from the Mathematical Association of America. Robin J. Wilson is Emeritus Professor of Pure Mathematics at the Open University, UK, and Emeritus Professor of Geometry at Gresham College, London. After graduating from Oxford, he received his PhD in number theory from the University of Pennsylvania. He has written and co-edited many books on graph theory and the history of mathematics, including Introduction to Graph Theory, Four Colors Suffice and Combinatorics: Ancient and Modern. His combinatorial research interests formerly included graph colourings and now focus on the history of combinatorics. An enthusiastic populariser of mathematics, he has won two awards for his expository writing from the Mathematical Association of America.
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