Topics in Chromatic Graph Theory

387 Pages·2015·4.927 MB·English

by  Lowell W. Beineke, Robin J. Wilson

TopicsinChromaticGraphTheory
Chromaticgraphtheoryisathrivingareathatusesvariousideasof‘colouring’(ofvertices,edges,etc.)to
exploreaspectsofgraphtheory.Ithaslinkswithotherareasofmathematics,includingtopology,algebra
andgeometry,andisincreasinglyusedinsuchareasascomputernetworks,wherecolouringalgorithms
formanimportantfeature.
Whileotherbookscoverportionsofthematerial,noothertitlehassuchawidescopeasthisone,in
whichacknowledgedinternationalexpertsinthefieldprovideabroadsurveyofthesubject.All15
chaptershavebeencarefullyedited,withuniformnotationandterminologyappliedthroughout.Bjarne
Toft(Odense,Denmark),widelyrecognizedforhissubstantialcontributionstothearea,actedas
academicconsultant.
Thebookservesasavaluablereferenceforresearchersandgraduatestudentsingraphtheoryand
combinatoricsandasausefulintroductiontothetopicformathematiciansinrelatedfields.
lowell w. beineke isSchreyProfessorofMathematicsatIndianaUniversity–PurdueUniversity
FortWayne(IPFW),wherehehasworkedsincereceivinghisPh.D.fromtheUniversityofMichigan
undertheguidanceofFrankHarary.Hisgraphtheoryinterestsincludetopologicalgraphtheory,line
graphs,tournaments,decompositionsandvulnerability.Hehaspublishedover100papersingraphtheory
andhasservedaseditoroftheCollegeMathematicsJournal.WithRobinWilsonhehasco-editedfive
booksinadditiontothethreeearliervolumesinthisseries.Recenthonoursincludeanawardinstitutedin
hisnamebytheCollegeofArtsandSciencesatIPFWandaCertificateofMeritoriousServicefromthe
MathematicalAssociationofAmerica.
robin j. wilson isEmeritusProfessorofPureMathematicsattheOpenUniversity,UK,and
EmeritusProfessorofGeometryatGreshamCollege,London.AftergraduatingfromOxford,hereceived
hisPh.D.innumbertheoryfromtheUniversityofPennsylvania.Hehaswrittenandco-editedmany
booksongraphtheoryandthehistoryofmathematics,includingIntroductiontoGraphTheory,Four
ColorsSufficeandCombinatorics:Ancient&Modern.Hiscombinatorialresearchinterestsformerly
includedgraphcolouringsandnowfocusonthehistoryofcombinatorics.Anenthusiasticpopularizerof
mathematics,hehaswontwoawardsforhisexpositorywritingfromtheMathematicalAssociationof
America.

ENCYCLOPEDIAOFMATHEMATICSANDITSAPPLICATIONS
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversity
Press.Foracompleteserieslistingvisitwww.cambridge.org/mathematics.
109 J.M.BorweinandJ.D.VanderwerffConvexFunctions
110 M.-J.LaiandL.L.SchumakerSplineFunctionsonTriangulations
111 R.T.CurtisSymmetricGenerationofGroups
112 H.Salzmannetal.TheClassicalFields
113 S.PeszatandJ.ZabczykStochasticPartialDifferentialEquationswithLe´vyNoise
114 J.BeckCombinatorialGames
115 L.BarreiraandY.PesinNonuniformHyperbolicity
116 D.Z.ArovandH.DymJ-ContractiveMatrixValuedFunctionsandRelatedTopics
117 R.Glowinski,J.-L.LionsandJ.HeExactandApproximateControllability forDistributedParameter
Systems
118 A.A.BorovkovandK.A.BorovkovAsymptoticAnalysisofRandomWalks
119 M.DezaandM.DutourSikiric´GeometryofChemicalGraphs
120 T.NishiuraAbsoluteMeasurableSpaces
121 M.PrestPurity,SpectraandLocalisation
122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabischetal.AggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.TuckerTopicsinTopologicalGraphTheory
129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
131 H.MorimotoStochasticControlandMathematicalModeling
132 G.SchmidtRelationalMathematics
133 P.KornerupandD.W.MatulaFinitePrecisionNumberSystemsandArithmetic
134 Y.CramaandP.L.Hammer(eds.)BooleanModelsandMethodsinMathematics,ComputerScience,and
Engineering
135 V.Berthe´andM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
136 A.Krista´ly,V.D.RaˇdulescuandC.VargaVariationalPrinciplesinMathematicalPhysics,Geometry,and
Economics
137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications
138 B.CourcelleandJ.EngelfrietGraphStructureandMonadicSecond-OrderLogic
139 M.FiedlerMatricesandGraphsinGeometry
140 N.VakilRealAnalysisthroughModernInfinitesimals
141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation
142 Y.CramaandP.L.HammerBooleanFunctions
143 A.Arapostathis,V.S.BorkarandM.K.GhoshErgodicControlofDiffusionProcesses
144 N.Caspard,B.LeclercandB.MonjardetFiniteOrderedSets
145 D.Z.ArovandH.DymBitangentialDirectandInverseProblemsforSystemsofIntegralandDifferential
Equations
146 G.DassiosEllipsoidalHarmonics
147 L.W.BeinekeandR.J.Wilson(eds.)withO.R.OellermannTopicsinStructuralGraphTheory
148 L.Berlyand,A.G.KolpakovandA.NovikovIntroductiontotheNetworkApproximationMethod
forMaterialsModeling
149 M.BaakeandU.GrimmAperiodicOrderI:AMathematicalInvitation
150 J.Borweinetal.LatticeSumsThenandNow
151 R.SchneiderConvexBodies:TheBrunn–MinkowskiTheory(SecondEdition)
152 G.DaPratoandJ.ZabczykStochasticEquationsinInfiniteDimensions(SecondEdition)
153 D.Hofmann,G.J.SealandW.Tholen(eds.)MonoidalTopology
154 M.CabreraGarc´ıaandA´.Rodr´ıguezPalaciosNon–AssociativeNormedAlgebrasI:TheVidav–Palmerand
Gelfand–NaimarkTheorems
155 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables(SecondEdition)
156 L.W.BeinekeandR.J.Wilson(eds.)withB.ToftTopicsinChromaticGraphTheory
157 T.MoraSolvingPolynomialEquationSystemsIII:AlgebraicSolving
158 T.MoraSolvingPolynomialEquationSystemsIV:Buchberger’sTheoryandBeyond
Above:FrancisGuthrie,whoproposedthefour-colourproblem,andKennethAppeland
WolfgangHaken,whosolvedit.Below:GerhardRingel(right)andTedYoungs,whosolved
theHeawoodconjecture.(CourtesyofRobinWilson.)
Topics in Chromatic Graph Theory
Editedby
LOWELL W. BEINEKE
IndianaUniversity–PurdueUniversity
FortWayne
ROBIN J. WILSON
TheOpenUniversity
andtheLondonSchoolofEconomics
AcademicConsultant
BJARNE TOFT
UniversityofSouthernDenmark,Odense
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107033504
(cid:2)c CambridgeUniversityPress2015
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2015
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Topicsinchromaticgraphtheory/editedbyLowellW.Beineke,
IndianaUniversity-PurdueUniversity,FortWayne,RobinJ.Wilson,
TheOpenUniversityandtheLondonSchoolofEconomics;
academicconsultant,BjarneToft,UniversityofSouthernDenmark,Odense.
pages cm.–(Encyclopediaofmathematicsanditsapplications;156)
Includesbibliographicalreferences.
ISBN978-1-107-03350-4(Hardback)
1. Graphcoloring–Dataprocessing. 2. Graphtheory–Dataprocessing.
I. Beineke,LowellW.,editor. II. Wilson,RobinJ.,editor.
QA166.247.T672015
511(cid:3).56–dc23 2014035297
ISBN978-1-107-03350-4Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Contents
ForewordbyBjarneToft pagexiii
Preface xv
Preliminaries 1
LOWELLW.BEINEKEandROBINJ.WILSON
1. Graphtheory 1
2. Graphcolourings 9
1 Colouringgraphsonsurfaces 13
BOJANMOHAR
1. Introduction 13
2. Planargraphsare4-colourableand5-choosable 14
3. Heawood’sformula 18
4. Colouringwithfewcolours 20
5. Gro¨tzsch’stheoremanditsgeneralizations 23
6. Colouring–flowduality 25
7. Theacyclicchromaticnumber 29
8. Degeneratecolourings 30
9. Thestarchromaticnumber 31
10. Summary 32
2 Brooks’stheorem 36
MICHAELSTIEBITZandBJARNETOFT
1. Introduction 36
2. ProofsofBrooks’stheorem 37
3. Criticalgraphswithfewedges 41
4. Boundingχ by(cid:3)andω 45
5. Graphswithχ closeto(cid:3) 48
6. Notes 50
viii Contents
3 Chromaticpolynomials 56
BILLJACKSON
1. Introduction 56
2. Definitionsandelementaryproperties 57
3. Logconcavityandotherinequalities 59
4. Chromaticroots 60
5. Relatedpolynomials 64
4 Hadwiger’sconjecture 73
KEN-ICHIKAWARABAYASHI
1. Introduction 73
2. Completegraphminors:earlyresults 74
3. Contraction-criticalgraphs 75
4. Algorithmicaspectsoftheweakconjecture 79
5. Algorithmicaspectsofthestrongconjecture 81
6. Theoddconjecture 82
7. IndependentsetsandHadwiger’sconjecture 85
8. Othervariantsoftheconjecture 86
9. Openproblems 89
5 Edge-colourings 94
JESSICAMCDONALD
1. Introduction 94
2. ElementarysetsandKempechanges 96
3. Tashkinovtreesandupperbounds 97
4. TowardstheGoldberg–Seymourconjecture 101
5. Extremegraphs 103
6. Theclassificationproblemandcriticalgraphs 105
7. Thedichotomyofedge-colouring 108
8. Finalthoughts 109
6 List-colourings 114
MICHAELSTIEBITZandMARGITVOIGT
1. Introduction 114
2. Orientationsandlist-colourings 118
3. Planargraphs 121
4. Precolouringextensions 128
5. Notes 129
7 Perfectgraphs 137
NICOLASTROTIGNON
1. Introduction 137
Contents ix
2. Lova´sz’sperfectgraphtheorem 139
3. Basicgraphs 141
4. Decompositions 142
5. Thestrategyoftheproof 146
6. BookfromtheProof 148
7. Recognizingperfectgraphs 151
8. Bergetrigraphs 152
9. Evenpairs:ashorterproofoftheSPGT 154
10. Colouringperfectgraphs 155
8 Geometricgraphs 161
ALEXANDERSOIFER
1. Thechromaticnumberoftheplane 161
2. Thepolychromaticnumber:lowerbounds 162
3. ThedeBruijn–Erdo˝sreductiontofinitesets 165
4. Thepolychromaticnumber:upperbounds 167
5. Thecontinuumof6-colourings 169
6. Specialcircumstances 171
7. Spaceexplorations 172
8. Rationalspaces 173
9. Oneoddgraph 175
10. Influenceofsettheoryaxioms 175
11. Predictingthefuture 177
9 Integerflowsandorientations 181
HONGJIANLAI,RONGLUOandCUN-QUANZHANG
1. Introduction 181
2. Basicproperties 183
3. 4-flows 185
4. 3-flows 185
5. 5-flows 187
6. Boundedorientationsandcircularflows 188
7. Moduloorientationsand(2+1/t)-flows 190
8. Contractibleconfigurations 191
9. Relatedproblems 194
10 Colouringrandomgraphs 199
ROSSJ.KANGandCOLINMCDIARMID
1. Introduction 199
2. Denserandomgraphs 202
3. Sparserandomgraphs 208
4. Randomregulargraphs 214