Introduction to Graph and Hypergraph Theory

303 Pages·2013·9.203 MB·English

by  Vitaly I. Voloshin

I G
NTRODUCTION TO RAPH
H T
AND YPERGRAPH HEORY
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I G
NTRODUCTION TO RAPH
H T
AND YPERGRAPH HEORY
VITALY I. VOLOSHIN
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LibraryofCongressCataloging-in-PublicationData
Voloshin,VitalyI.(VitalyIvanovich),1954-
Introductiontographandhypergraphtheory/VitalyI.Voloshin.
p. cm.
Includesindex.
ISBN978-1-61470-112-5 (eBook)
1. Graphtheory.2. Hypergraphs.I.Title.
QA166.V6492009
511’.5–dc22
2008047206
PublishedbyNova SciencePublishers,Inc. <New York
To Julian, Olesea and Georgeta for unlimited love and support
The Essence of Mathematics is in its generalizations,
The Beauty of Mathematics is in its ideas,
The Power of Mathematics is in its absolute truth...
Contents
Preface xi
I GRAPHS 1
1 BasicDefinitionsandConcepts 5
1.1. Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2. GraphModeling Applications . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3. GraphRepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4. Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5. BasicGraphClasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6. BasicGraphOperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7. BasicSubgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.8. Separation andConnectivity . . . . . . . . . . . . . . . . . . . . . . . . . 34
2 TreesandBipartiteGraphs 39
2.1. TreesandCycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2. TreesandDistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3. MinimumSpanningTree . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4. BipartiteGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 ChordalGraphs 51
3.1. Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2. Separators andSimplicialVertices . . . . . . . . . . . . . . . . . . . . . . 52
3.3. Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4. Distances inChordalGraphs . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5. Quasi-triangulated Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 PlanarGraphs 67
4.1. PlaneandPlanarGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2. Euler’sFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3. K andK ArenotPlanarGraphs . . . . . . . . . . . . . . . . . . . . . . 71
5 3,3
4.4. Kuratowski’sTheoremandPlanarityTesting . . . . . . . . . . . . . . . . . 73
4.5. PlaneTriangulations andDualGraphs . . . . . . . . . . . . . . . . . . . . 76
vii
viii Contents
5 GraphColoring 79
5.1. Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2. DefinitionsandExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3. StructureofColorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4. ChromaticPolynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.5. ColoringChordalGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.6. ColoringPlanarGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.7. PerfectGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.8. EdgeColoringandVizing’sTheorem . . . . . . . . . . . . . . . . . . . . 112
5.9. UpperChromaticIndex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Traversals andFlows 123
6.1. EulerianGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2. Hamiltonian Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3. NetworkFlows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
II HYPERGRAPHS 131
7 BasicHypergraphConcepts 135
7.1. PreliminaryDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2. Incidence andDuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.3. BasicHypergraph Classes . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4. BasicHypergraph Operations . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.5. Subhypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.6. Conformality andHellyProperty . . . . . . . . . . . . . . . . . . . . . . . 154
8 Hypertrees andChordalHypergraphs 161
8.1. Hypertrees andChordalConformalHypergraphs . . . . . . . . . . . . . . 161
8.2. AlgorithmsonHypertrees . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.3. CyclomaticNumberofaHypergraph . . . . . . . . . . . . . . . . . . . . . 174
9 SomeOtherRemarkableHypergraphClasses 181
9.1. BalancedHypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.2. Interval Hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.3. NormalHypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
9.4. PlanarHypergraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
10 HypergraphColoring 193
10.1. BasicKindsofClassicHypergraph Coloring . . . . . . . . . . . . . . . . . 193
10.2. GreedyAlgorithmfortheLowerChromaticNumber . . . . . . . . . . . . 197
10.3. BasicDefinitionsofMixedHypergraph Coloring . . . . . . . . . . . . . . 201
10.4. GreedyAlgorithmfortheUpperChromaticNumber . . . . . . . . . . . . 207
10.5. Splitting-Contraction Algorithm . . . . . . . . . . . . . . . . . . . . . . . 213
10.6. Uncolorability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
10.7. UniqueColorability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227