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GRAPHTHEORY

WITHAPPLICATIONS

J.A.BondyandU.S.R.Murty

UniversityofWaterloo,

Ontario,Canada'

NORfH-HOLLAND

NewYork•Amsterdam•Oxford

®J.A.BondyandV.S.R.Muny1976

FirstpublishedinGreatBritain1976by

The·MacmillanPressLtd.

FirstpublishedintheU.S.A.1976by

ElseyierSciencePublishingCo.,Inc.

52VanderbiltAvenue,NewYork,N.Y10017

FifthPrinting,1982.

SoleDistributor

intheU.S.A:

ElsevierSciencePublishingCo.

.,Inc.

Library

ofCongressCataloginginPublicationData

Bondy,JohnAdrian.

Graphtheorywith,applications.

Bibliography:p.

Includes

index.

QA166.B671979511'.575-29826·

ISBN7 All formorbyanymeans,withoutpermission.

Printed

intheUnitedStatesofAmerica

·Toourparents

Preface

variety 'applications' gorithms shouldallbeattempted. arelisted. helpful appendixV. Many us

Chungphaisan

Preface

vii manuscriptandvaluablesuggestions, andtotheubiquitousG.O.M.forhis kindness andconstantencouragement. B. financialsupport.Finally,wewouldlike toexpressourappreciationtoJoan

Selwoodfor

artwork..

J.A.Bondy

U.S.R.Murty

Contents

Preface

1GRAPHSANDSUBGRAPHS

1.1GraphsandSimpleGraphs.

1.2

GraphIsomorphism

1.3

TheIncidenceandAdjacencyMatrices

1.4Subgraphs

1.5VertexDegrees_

1.6Pathsan"dConnection

1.7Cycles._

Applications

1.8The"ShortestPathProblem_

1.,9Sperner'sLemma.

2TREES

2.1Trees

2.2

CutEdgesandBonds..

2.3CutV'ertices.

2.4Cayley'sFormula.

Applications.

2.5TheCo"nnectorProblem

3CONNECTIVITY

3.1Connectivity.

3.2Block"s"_

4EULERTOURSAN-nHAMILTONCYCLES"

4.1EulerTours_

4.2HamiltonCycles.

Applications

4.3The",ChinesePostmanProblem

4.4TheTravellin,g'SalesDlanProblem

vi 1 4 7 8 10 12 14 15 21
25
27
31
32
,36" ' 42'
44.
47
51
53
62
65

Contents

5MATCHINGS

5.1Matchings

5.2

MatchingsandCoveringsinBipartiteGraphs

5.3PerfectMatchings.

Applications

5.4ThePersonnelAssignmentProblem'.

5.5

TheOptimalAssignmentProblem

. 6EDGECOLOURINGS

6.1EdgeChromaticNumber

6.2Vizing'sTheorem.

Applications

TheTimetablingProblem

7INDEPENDENTSETSANDCLIQUES

7.1IndependentSets.

7.2Ramsey's

7.3Turan'sTheorem.

Applications

7.4Schur'sTheorem.

7.5AGeometryProblem.

8VERTEXCOLOU'RINGS

8.1ChromaticNumber

8.2Brooks'Theorem.

8.3Haj6s'·.

8..4Chromatic

8.5GirthandChromaticNumber

Applications

8.6AStorageProblem

9PLANARGRAPHS

IX 70
72
76
80
86
91
93
96

·101

·103,

109

·112

·113

·117

·122

123
125
129
.131

·163

9.1 9.2 9.3 9.4 9.5' 9.6 9.7, 9.8

PlaneandPlanarGraphs.135

DualGraphs..139

Euler'sFormula.143

Bridges..145

Kuratowski's

Theorem.151

Nonhamiltonian

PlanarGraphs..160

Applications

APIa.narityAlgorithm.

x

10DIRECTEDGRAPHS

10.1DirectedGraphs.

10.2DirectedPaths

10.3DirectedCycles.

Applications

10.4AJobSequencingPr?blem.

10.5DesigninganEfficientC.omputerDrum

10.6MakingaRoadSystemOne-Way

10.7RankingtheParticipantsinaTournament.

11NETWORKS·

11.1Flows.

11.2 Cuts

11.3TheMax-FlowMin-CutTheorem

Applications

11.4Menger'sTheorems

11.5FeasibleFlows

12THECYCLESPACEANDBONDSPACE

12.1CirculationsandPotentialDifferences.

12.2

TheNumberofSpanningTrees.

Applications

12.3PerfectSquares.

AppendixIHintstoStarredExercises

AppendixIIISomeInterestingGra.phs.

AppendixIVUnsolvedProblems.

AppendixVSuggestionsforFurtherReading.

Glossary

ofSymbols·

IndexContents

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