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Topics in Graph Theory
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Topics in Graph Theory
Graphs and Their Cartesian Product
Wilfried Imrich
Sandi Klavžar
Douglas F. Rall
AKPeters,Ltd.
Wellesley,Massachusetts
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Library ofCongressCataloging-in-PublicationData
Imrich,Wilfried,1941-
Topics in graph theory : graphs and their cartesian product / Wilfried
Imrich,SandiKlavžar,DouglasF.Rall.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-56881-429-2(alk. paper)
1. Graphtheory. I.Klavžar,Sandi,1962-II.Rall,DouglasF.III.Title.
QA166.I4692008
511’.5–dc22
2008020790
PrintedinIndia
1211100908 10987654321
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To our wives Gabi, Maja, and Naomi.
Without their love, patience, encouragement, support, and
understanding, the chances of this book being published
would have been infinitesimal at best.
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Contents
Preface xi
I Cartesian Products 1
1 TheCartesianProduct 3
1.1 Definitions,Fibers,andProjections . . . . . . . . . . . . . 3
1.2 ConnectednessandMoreExamples . . . . . . . . . . . . . 6
1.3 ProductsofSeveralFactors . . . . . . . . . . . . . . . . . . 8
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 HammingGraphsandHanoiGraphs 11
2.1 HypercubesandHammingGraphs . . . . . . . . . . . . . 11
2.2 HanoiGraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
II Classic Topics 17
3 Hamiltonian Graphs 19
3.1 ConditionsforHamiltonicity . . . . . . . . . . . . . . . . . 19
3.2 HamiltonianProducts . . . . . . . . . . . . . . . . . . . . . 20
3.3 HamiltonianPrisms . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Planarity and CrossingNumber 29
4.1 PlanarProducts . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 CrossingNumbersofProductsofCycles . . . . . . . . . . 31
4.3 MoreExactCrossingNumbers . . . . . . . . . . . . . . . . 35
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
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viii Contents
5 Connectivity 37
5.1 Vertex-Connectivity . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Edge-Connectivity . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Subgraphs 45
6.1 NontrivialSubgraphs . . . . . . . . . . . . . . . . . . . . . . 45
6.2 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . 47
6.3 BasicTrivialSubgraphs . . . . . . . . . . . . . . . . . . . . 49
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
III GraphicalInvariants 53
7 Independence 55
7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 LowerBounds . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.3 UpperBounds . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8 GraphColorings 63
8.1 VertexColorings . . . . . . . . . . . . . . . . . . . . . . . . . 64
8.2 GeneralizedColorings . . . . . . . . . . . . . . . . . . . . . 67
8.3 CircularColorings . . . . . . . . . . . . . . . . . . . . . . . . 69
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9 AdditionalTypesofColorings 73
9.1 ListColorings . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 L(2,1)-labelings . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.3 EdgeColorings . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
10Domination 81
10.1 DefinitionsandNotation . . . . . . . . . . . . . . . . . . . . 81
10.2 Boundsforγ inTermsofOtherInvariants . . . . . . . . 84
10.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
11DominationinCartesian Products 89
11.1 BinaryCodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
11.2 MultiplicativeResults . . . . . . . . . . . . . . . . . . . . . . 91
11.3 LowerBounds . . . . . . . . . . . . . . . . . . . . . . . . . . 92
11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
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Contents ix
IV MetricAspects 97
12Distance LemmaandWienerIndex 99
12.1 DistanceLemma . . . . . . . . . . . . . . . . . . . . . . . . . 99
12.2 ApplicationsoftheDistanceLemma . . . . . . . . . . . . 101
12.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
13ProductsandBoxes 105
13.1 Boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
13.2 MetricBoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
13.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
14Canonical Metric Representation 113
14.1 Representationα . . . . . . . . . . . . . . . . . . . . . . . . 113
14.2 PropertiesandApplicationsofα . . . . . . . . . . . . . . 117
14.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
V AlgebraicandAlgorithmicIssues 123
15PrimeFactorizations 125
15.1 UniquenessforConnectedGraphs. . . . . . . . . . . . . . 126
15.2 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 128
15.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
16Cancelation and Containment 133
16.1 CancelationProperties . . . . . . . . . . . . . . . . . . . . . 134
16.2 ContainmentProperties . . . . . . . . . . . . . . . . . . . . 136
16.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
17DistinguishingNumber 139
17.1 ExamplesandProductsofTwoFactors . . . . . . . . . . . 139
17.2 CartesianPowersofGraphs . . . . . . . . . . . . . . . . . . 142
17.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
18RecognitionAlgorithms 147
18.1 TheStructureTheorem. . . . . . . . . . . . . . . . . . . . . 147
18.2 APolynomialFactorizationAlgorithm . . . . . . . . . . . 149
18.3 RecognizingPartialCubes . . . . . . . . . . . . . . . . . . . 150
18.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
HintsandSolutionstoExercises 153
References 191
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