HomiBhabhaNationalInstituteRecommendationsoftheVivaVoceBoardAsmemberso - PDF document

HomiBhabhaNationalInstituteRecommendationsoftheVivaVoceBoardAsmembersoftheVivaVoceBoard,werecommendthatthedissertationpreparedbyNarayananNentitled“Acyclic,K-intersectionedgecolouringsandOrientedcolour-ing.”maybeacceptedasfulllingthedissertationrequirementfortheDegreeofDoctorofPhilosophy. Date:Chairman:VArvind Date:Convener:CRSubramanian Date:Member:AjitADiwan Date:Member:MeenaMahajan Date:Member:VenkateshRamanFinalapprovalandacceptanceofthisdissertationiscontingentuponthecandidate'ssubmissionofthenalcopiesofthedissertationtoHBNI.IherebycertifythatIhavereadthisdissertationpreparedundermydirectionandrecommendthatitmaybeacceptedasfulllingthedissertationrequirement. Date:Guide:CRSubramanian DECLARATIONI,herebydeclarethattheinvestigationpresentedinthisthesishasbeencarriedoutbyme.Theworkisoriginalandtheworkhasnotbeensubmittedearlierasawholeorinpartforadegree/diplomaatthisoranyotherInstitutionorUniversity.NarayananN ...:pra;kx+:tya;aI+.dM...(tothenature) ACKNOWLEDGEMENTSIcannotexpressinwordshowmuchIamindebtedtomyparentsforthecontinuedsupporttheyhadgiventhroughoutmyeducationalcareerandmylife.IalsothankmyyoungerbrotherHari,formakingthelifesowonderfulandforalwaysbeingtheretosupportme.IthankmysupervisorCRSubramanianforintroducingmetotheeldofgraphcolouring,andalsoforsuggestingtheacyclicedgecolouringproblemstudiedinthethesis.HealsotaughtmesomeoftheprooftechniqueslikeLovaszLocalLemma.Ialsothankhimforhisguidanceandforhishelpinobtainingmanyoftheresultsreportedinthethesisandinpolishingthepresentationoftheresultsinthethesis.IthankProf.SopenaforintroducingtheorientedcolouringproblemandProf.BorowieckiforthegreattimeIhadvisitinghim.IthankmyteacherAmbatVijayakumarforintroducingmetographtheory.Ithankthemembersofmydoctoralcommittee,Arvind,Meena,VenkateshandRamanujamforthemanyconstructivesuggestionstheyhadgiven.IspeciallythankMeena,VenkateshandJamfortheirhelpthroughoutmystayatIMSc,forIalwaysusedtorushtothemwithanyproblemIfaceandtheyalwayshadsomesolution/suggestion.Vinu,or`achayan',AmitavaBhattacharyaor`atta'andSivaramakrishnanSivasubramaniamor`krishnan'aremybestfriends.TheywerealwaystheretohelpwhenIneededitmost.Rahulismycollaboratorandfriend.Hisamazingabilitytorememberanythingwasalifesaverformeinmanyoccasions.IwouldalwaysremembertheinnumerableoccasionswatchinghimplaychesseverydayontheFICS,withhisonlinecommentary,whichcanonlybeexpressedasawesome.IalsothankmycollaboratorsAnnaMariuszforthenicediscussionsessionsandforteachingmetheartofMushroompicking.IthankAravindandSomnathwhowereinstrumentalintheSaturdaylectureseriesoncombinatoricsweusedtoorganise.IalsothankJayalalandPiyushforbeingconnoisseursofmycooking.IthankallofmyfriendsincludingYogi,RahulJain,Vaibhav,Ved,Sunil,Sreekanth,Philip,Saptarshi,Raghu,Prakash,Suresh,KunalformakingthelifeinterestingatIMScandTIFR.Ithanktheadministration,especiallytheDirector,VishnuPrasadandourformerregistrarRamakrishnaManjaforthetimelyhelpinalladministrativeaffairs.Finally,Iexpressmysincerethankstotheanonymousreviewerswhosecommentshelpedtoimprovethepresentationofthethesis. AbstractInthisthesis,westudythreegraphcolouringproblems.Themainthemeofthisthesisistheacyclicedgecolouringproblem.Wealsostudytwootherproblems,namely,thek-intersectionedgecolouringandorientedvertexcolour-ing.Intheacyclicedgecolouringproblem,wearerequiredtondthemini-mumnumberofcolours0athatsufcestocolourtheedgesofagraphprop-erlysuchthattheunionofanytwocolourclassesformsaforest.Theacyclicedgecolouringconjecture(duetoFiamcik[Fia78]andalsoindependentlytoAlon,SudakovandZaks[ASZ01])statesthatitispossibletocolourtheedgesofanygraphGacyclicallywithatmost
(G)+2colours.Itisconsideredtobeadifcultproblemasverylittleisknownaboutexactortightestimatesofthisinvariantevenforhighlystructuredclassesofgraphs.Intherstpartofthisthesis,weobtainimprovedupperboundson0aforsomeclassesofgraphs.Wealsoshowthatcertainclassesofgraphssatisfytheacyclicedgecolouringconjecture.Someoftheseresultsareobtainedmakinguseofprobabilisticarguments,whiletheothersareprovedmakinguseofstructuralpropertiesoftheunderlyinggraphs.Thesecondpartdealswitharelatedproblemthatwecallthek-intersectionedgecolouring.Here,oneseekstondtheminimumnumberofcoloursthataresufcienttocolourtheedgessuchthatforanypairofadjacentvertices,thenumberofcommoncoloursreceivedontheedgesincidentonthemisatmostk.WeobtainanupperboundofO(
2=k)andshowthatthisboundisindeedtightforcompletegraphs.Inthethirdpart,welookattheorientedvertexcolouringofgraphs.Anorientedk-colouringofanorientedgraph#GisamappingC:V(#G)7![k]sothat(i)C(x)6=C(y)8(x;y)2A(#G)and(ii)C(x)=C(w)=)C(y)6=C(z)8(x;y);(z;w)2A(#G).Theorientedchromaticnumberoofanorientedgraph#Gisthesmallestksuchthatthereisanorientedk-colouring.Theori-entedchromaticnumberforanundirectedgraphGisthemaximumo(#G)overallorientations#GofG.Weobtainimprovedupperandlowerboundsonorientedchromaticnumberforcertainclassesofgraphsandproductsofgraphs. ContentsPreface1Howtoreadthisthesis.............................31Preliminaries41.1DenitionsandNotation.........................51.2GraphParametersandColourings....................8IAcyclicEdgeColouring112AcyclicEdgeColouring:RelatedResults122.1TheProblem...............................122.2RelatedWork...............................143Girthand0a(G)163.1TheProbabilisticMethod........................163.2Introduction...............................183.3Gettingbelowthebarrier—limitedimproperness...........193.4Ageneralrelationbetweengirthand0a.................243.5Remarks..................................273.6Anoteontheclaimed9
boundin[MR02]..............284Outerplanargraphs294.1Introduction...............................294.2Colouring.................................304.3Algorithmicaspects............................334.4Conclusions................................365Planarand3-foldgraphs375.1TheDischargingMethod.........................375.2Introduction...............................405.3Diggingoutsomestructure—thedischargingmethod.........425.4Extendingthepartialcolouring:reducibility...............47i PrefaceGraphtheoryisthestudyofgraphs.TheoriginsofgraphtheorydatesbacktoLeonhardEuler,whopublishedtherstpaperingraphtheoryin1736.Agraphisanabstractionofasetofentities(calledvertices)withabinaryrelationbetweenthem(callededges).Forexample,thesetofentitiesmaybeasetofcitiesornetworkhubs,persons,computerprocesses,ormobiletowersandsoon.Abinaryrelationbetweenapairofcitiesmaybethattheyareconnectedbyadirectroad.Inthecaseofmobiletowers,onemaysaythattwotowersarerelatediftherangeofareastheycoverintersect.Atypicalproblemwouldbetondoutwhetheritispossibletostartfromanycityandreachanyother.Inthecaseofmobilenetwork,givenasetoftowersandtheareastheycover,onemayneedtondouttheminimumnumberoffrequencybandsthatneedtobeallocatedsothatthesamefrequencyisnotusedonadjacenttowers.Thereispracticallyaninnitudeofsuchreallifeproblems.Graphtheorytriestoabstractoutsuchproblemsandtondasolutiontotheprobleminamathematicalsetting.Thismakesitimmenselyusefulindealingwithmanyproblemsfromdifferentbranchesofscienceorindustry.Someoftheareaswheregraphtheoryisgreatlyusedincludenetworkanalysis,VLSIdesign,molecularchemistry,DNAsequencing,molecularbiology,studyofevolutiongraphs,developmentofalgorithmsandstudyofneuralnetworks.Withtheadventofcomputerscience,graphtheorycametoprominenceandhasbecomeanessentialtool.Manyproblemsingraphtheoryareofanextremalnature.Forexample,onemayneedtondtheminimum(maximum)cardinalityofasetofverticessatisfyingcertainconditions.Forexample,acentralquestionistondoutthemaximumnumberofverticesinagraphsuchthattherearenoedgesbetweenanypair.Anothertypeofextremalproblemistopartitiontheverticesoredgessothatthepartitionsatisescertainproperties.Graphcolouringisabranchofgraphtheorywhichdealswithsuchpartitioningproblems.Forexample,supposethatwehaveaworldmapandwewouldliketocolourthecountriessothatiftwocountriesshareaboundaryline,thentheyneedtogetdif-ferentcolours.Wecantranslatethemaptoagraphbylettingcountriesberepresentedbyverticesandtwoverticesaremadeadjacentifandonlyifthecorrespondingcountriesshareaboundaryline.Thentheproblemofmapcolouringisequivalenttotheproblemofvertexcolouringitsgraph.Hencetheoriginalmapcolouringnowreducestovertexcolouringoftheassociatedgraph.1 Chapter0.PrefaceingMethodtoobtainsomenicestructuralpropertiesofgraphsofmaximumaveragedegree6.ThisworkwasdoneincollaborationwithAnnaFiedorwiczandMariuszHaushczakduringmyvisittotheUniversityofZielonaG¨ora,Poland.Theseresultsappearin[1].InChapter6,welookatafewotherclassesofgraphsandshowthattheysatisfytheacyclicedgecolouringconjecture.TheseincludetheHararygraphs,minimally2-connectedgraphsandfullysubdividedgraphs[3].Thesecondpart(Chapter7)dealswithanotheredgecolouringproblem,namely,thek-intersectionedgecolouring.Onceagain,wemakeuseofprobabilisticargumentsandtoolsliketheChernoffboundsandLov´aszLocalLemmatoobtainatightboundforthiscolouringnotion.ThisresultwaspresentedintheconferenceCID2007.Thisresultappearsin[7].Thenalpartdealswiththeorientedcolouringproblem.Thischaptercontainsrecentresultswehaveobtainedforsomeclassesofgraphs.TheseresultsarepresentedinChapter8.ThisisajointworkwithN.R.Aravind[2].WeconcludethethesiswithsomeremarksandopenproblemsinChapter9.HowtoreadthisthesisThe`pdf'versionofthisthesiscontainshyperlinkedtextandindexforeaseofnavigationandreference.Anyreference,gure,indexordenitionthatishyperlinkedappearsinasmallbox.Thereadercanleft-clickontheboxtonavigatedirectlytothepageinwhichitisrstdenedortothebibliographypages.The`preliminaries'chapteralsocontainstheimportanttermsthatwedeneintherightmarginofthetext.Thetermappearsontherightsideofthelineinwhichitisdened.Thereisaglossarypagefornotationsandshortdenitions.Thereadercangetaquickdenitionfromhereaswellasclickonthecorrespondingpagenumber(rstinthelist)togotothepagewhereitisrstdened.AnotherconventionusedistheuseoftwoQEDsymbols.Asquarelledblackstandsfortheendofargumentsforthemainproof.Anunlledsquaredenotestheendofargumentsfortheproofofanyclaimwithinamainproof.3 Chapter1.Preliminariesareadjacentiftheyshareacommonvertex.Thedegreeofavertexv,isthenumberofedgesincidentwithit.Wedenotethedegreedegreeofavertexbyd(v)(sometimesdG(v)tospecifythegraph).ThemaximumdegreeandminimumdegreeofagraphGaredenoted
(G)and(G)(orsimply
and)respectively.Agraphissaidtoberegularif(G)=
(G).Verticesofdegreek,atleastk,andatmostkarerespectivelycalledak-vertex,k+-vertex,andk�-vertex.Agraphiscalledk-degenerate,ifeveryinducedsubgraphcontainsak�-vertex.Apathinagraphisasequenceofdistinctverticesu=v0;v1;:::;vk=vwhereforpatheveryik,(vi;vi+1)2E.Thelengthofapathisthenumberofedgesinit.Apathfromutoviscalledau�vpath.ApathonkverticesisdenotedbyPk.Acycleoflengthkisasequencev0;v1;:::;vk=v0ofverticessuchthatallbuttheendverticesaredistinctand(vi;vi+1)2Eforeveryi.WedenoteacycleonkverticesbyCk.Anedgethatjoinstwonon-consecutiveverticesofacycleiscalledachordofthecycle.WesaythatHisasubgraphofGifV(H)V(G)andE(H)E(G).Inaddition,ifsubgraphHcontainsalltheedges(u;v)2E(G)withu;v2V(H),thenHisaninducedsubgraphofG.WesaythatV(H)inducesHinG.AnedgeinducedsubgraphofagraphGisasubsetofedgesofGtogetherwithverticesthataretheendpointsofthegivenedges.WesaythatGisaconnectedgraph,ifforallpairsu;vofvertices,Gcontainsau�vconnectedpath.AmaximalconnectedinducedsubgraphisknownasaconnectedcomponentofG.Acut-vertexisavertexwhoseremovalincreasesthenumberofconnectedcomponentsofthegraph.ForagraphG,amaximalconnectedinducedsubgraphwithoutcut-verticesiscalledablockofG.A2-connectedgraphisaconnectedgraphonthreeormoreverticeswithoutanycut-vertex.Itiswell-knownthatina2-connectedgraph,foreverypairu;vofvertices,thereisacyclepassingthroughbothofthem.Agraphwithoutcyclesiscalledaforest.Further,ifitisconnected,thenitiscalledatree.forestApairofedgesareindependentiftheyarenotadjacent.Asetofpairwiseindependentedgesiscalledamatching.Ifasetofverticesarepairwisenon-adjacent,theyformanindependentset.Thelengthofashortestcycle,ifany,inagraphiscalleditsgirth,girthdenotedbyg(G).Ifalltheverticesinagrapharepairwiseadjacentitisacompletegraph.AcompletegraphofordernisdenotedKn.Ifaninducedsubgraphonkverticesiscomplete,itiscalledak-clique.cliqueThelinegraphL(G)ofagraphGisthegraphwithE(G)asitsvertexsetandapairofverticesinL(G)beingadjacentwheneverthecorrespondingedgesareadjacentinG.6 Chapter1.PreliminariesAgraphisembeddedintheplaneifitsverticesaremappedtodistinctpointsinR2andedgesaremappedtosimpleJordancurvesconnectingthecorrespondingpoints.Further,werequirethatnotwocurvesshareanypointsexceptpossiblyanend-point.Suchamappingiscalledaplanarembeddingofthegraph.Agraphisaplanargraphifitcanbeembeddedintheplane.Aplanargraphtogetherplanargraphwithanembeddingiscalledaplanegraph.GivenanyplanegraphG,noticethatR2nGisopen;itsconnectedregionsarethefacesofG.LetFbethesetoffacesinanyplanarembeddingofagraphG.Thewell-knownEuler'sformulastatesthefollowingfact.Fact1.1.1(Euler'sFormula).EveryconnectedplanegraphG=(V;E)satisesthefor-mulajVj�jEj+jFj=+2.UsingEuler'sformula,onecanshowthatanyplanargraphhasatmost3jVj�6edges.Wesaythatavertexliesonaface,ifitispartofanedgeformingtheboundaryoftheface.Agraphiscalledanouterplanargraphifithasaplanarembeddingsuchthateveryouterplanarvertexliesontheouter/unboundedface.AdirectedgraphGisapair(V;A)wheretheelementsofAareordered2-elementdirectedgraphsubsetsofV.ElementsofAarecalledarcs.Thearcs(x;y)andand(y;x)areoppositearcs.AnorientationofasimpleundirectedgraphGisadigraphobtainedbygivingeveryedgeofGoneofitstwopossibleorientations.Anorientedgraphisanorientationorientedgraphofsomesimpleundirectedgraph.Inshort,asimpledirectedgraphwithoutanypairofoppositearcsisanorientedgraph.Apairofverticesxandyinadirectedgrapharesaidtobeadjacentifatleastoneofthearcs(x;y),(y;x)belongstoA.LetG=(V;E)andH=(V0;E0)betwographs.WesaythatGandHareisomorphictoeachother,ifthereexistabijection':V7!V0suchthat(x;y)2Eifandonlyif('(x);'(y))2E0forallx;y2V.AhomomorphismfromagraphGtoagraphH,isamap :V(G)7!V(H)thatpre-homomorphismservestheedgerelations.Thatis,(x;y)2E(G)=)( (x); (y))2E(H).Analogously,isomorphismandhomomorphismcanbedenedbetweenapairoforientedgraphs.OperationsonGraphsOnecancombinetwoormoregraphstogetbiggergraphs.Wedenesomeoftheseoperationsbelow.7 Chapter1.PreliminariesSimilarly,aproperedger-colouringisamapC:E7![r]suchthatC(e)6=C(f)whenevereandfareadjacent.Thatis,aproperedger-colouringpartitionstheedgesetEintormatchings.TheminimumvalueofrforwhichGadmitsaproperedger-colouringiscalledthechromaticindexofGandisdenotedby0(G).Ingeneral,chromaticindexchromaticnumberandchromaticindexrespectivelydenotestheminimumnumberofcoloursrequiredtoproperlycolourtheverticesoredgesofagraph.Denition1.2.2.AproperedgecolouringofagraphGinwhichtheunionofanytwoacyclicedgecolouringcolourclassesformsaforest,iscalledanacyclicedgecolouringofG.Here,acolourclassstandsforthesetofalledgescolouredwithagivencolour.TheminimumnumberofcolourssufcienttocolourtheedgesofagraphacyclicallyisknownastheacyclicchromaticindexofG.Itisdenoted0a(G)ora0(G).Wefollow0atherstnotationinthisthesis.Acyclicvertexcolouringisananalogousnotionthatrequiresonetoproperlycolourtheverticesofagraphsuchthatthesubgraphinducedbyverticesofanytwocoloursisaforest.Itistobenotedthatanacyclicedgecolouringofagraphcorrespondstoanacyclicvertexcolouringofitslinegraphandviceversa.Withrespecttoaproperedgecolouring,wesaythatavertex(sayv)seesacolourx,ifanyoftheedgesincidentwithviscolouredx.Denition1.2.3.Given1k
,ak-intersectionedgecolouringofagraphGisak-intersec-tionedgecolouringproperedgecolouringofGinwhichthenumberofcommoncoloursseenbyanypairofadjacentverticesisatmostk.Inotherwords,ifweuseC(v)todenotethelistofcoloursseenbyavertexvinanedgecolouringC,thenforeveryedge(u;v),werequirethatjC(u)\C(v)jk.Theminimumnumberofcolourssufcientforsuchacolouringiscalledthek-intersectionchromaticindexandisdenotedby0k(G).0kDenition1.2.4.Anorientedr-colouringofanorientedgraph#GisamappingC:V7!orientedcolouring[r]suchthat(i)C(x)6=C(y)foranyarc(x;y)and(ii)C(x)=C(w)onlyifC(y)6=C(z),forallpairsofarcs(x;y)and(z;w).Lessformally,itmeansthatifthereisanarcfromaredvertextoabluevertexinthecolouredgraph,thentherecannotbeanarcfromabluevertextoaredvertex.9 Chapter1.PreliminariesForanorientedgraph#G,theminimumintegerrsuchthatthereisanorientedr-colouringof#Gisknownasitsorientedchromaticnumberandisdenotedbyo(#G).Theo(#G)orientedchromaticnumbero(G)ofanundirectedgraphGisdenedasthemaximumorientedchromaticnumberamongallpossibleorientations#GofG.Thatiso(G)=max#Go(#G)10 PartIAcyclicEdgeColouring Chapter2.AcyclicEdgeColouring:RelatedResultsknownfactthatforcompletegraphsofprimeorder0a=
+1.Ontheotherhand,forcompletegraphsofevenorder0a(K2n)
+2=2n+1.Thisisbecauseatmostonecolourclasscancontainnedges(aperfectmatching)andallotherscancontainatmostn�1edgeseach.Ifnot,theunionofanytwoperfectmatchingsdecomposesintocycles.Asimpleedgecountingargumentshowsthattheclaimholds.See[ASZ01]forfurtherdetails. Figure2.1:Anedgecolouringwhichisnotacyclic Figure2.2:Anacyclicedgecolouringofthesamegraph13 Chapter2.AcyclicEdgeColouring:RelatedResults2.2RelatedWorkThenotionsofacycliccolouringswereintroducedbyGr¨unbaumin[Gr¨u73].TheacyclicedgecolouringwasstudiedbyFiamcik(inthelate70's)whoalsoproposedaconjectureonthesame(whichwestatelater).Sinceany2-colouredcyclehasapairofedgesofthesamecolourjoinedbyanedge,itiseasytoseethatanydistance-2edgecolouring(whereanytwoedgesjoinedbyanotheredgearenotcolouredthesame)isacyclic.Sincethereisalwaysadistance-2colouringusingatmostO(
2)colours,itiseasytoseethat0a=O(
2).Burnstein[Bur79]provedthatanygraphofmaximumdegreeatmost4canbeacyclicallyvertexcolouredusingatmost5colours.Thisimpliesthatanysubcubicgraph(graphswithdegreeboundedby3)canbeacyclicallyedgecolouredusingatmost5colours.Thisisbecausethelinegraphofasubcubicgraphhasmaximumdegreeatmost4.Letusrecallthattheacyclicedgecolouringofagraphisanacyclicvertexcolouringofitslinegraph.In1991,Alon,McDiarmidandReed[AMR91]provedalinearupperboundof64
for0a,usingprobabilisticarguments.Thiswaslaterimprovedto16
usingessentiallythesamearguments,butwithatighteranalysisbyMolloyandReed[MR98].Theorem2.2.1.0a(G)16
foranygraphG.Fiamcikproposedthefollowingconjecture[Fia78]in1978.Itwasalsoindepen-dentlyproposedbyAlon,SudakovandZaks[ASZ01]in2001.Conjecture2.2.1.ForanygraphG,0a(G)
+2.Iftrue,thisconjecturehassomeinterestingconsequences.Ifitholdsfortheclassesofcompletegraphs,thentheperfect1-factorisationconjectureandperfectnear-1factori-sationconjecturearealsotrue.Precisely,theperfect1-factorisationconjecturestatesthattheedgesofK2n8n2canbeedgedecomposedinto2n�1perfectmatchingssothattheunionofanypairofthemformsahamiltoniancycle.Similarly,theperfectnear-1factorisationconjecturesaysthattheedgesofK2n+1hasadecompositioninto2n+1matchingsofsizensothattheunionofanypairformsahamiltonianpath.Theseconjecturesholdifandonlyif0a(K2n+1)=2n+1foralln.Forfurtherdetails,refer[ASZ01].14 Chapter2.AcyclicEdgeColouring:RelatedResultsAlon,etal.wereledtoproposetheaboveconjectureafterobservingthefollowingresultsbothappearinginthesamepaper.Theorem2.2.2.Foranygraphwithgirthg2000
log
,0a(G)
+2.Theorem2.2.3.LetG2Gn;
bearandom
-regulargraphonnlabelledvertices.Thenasymptoticallyalmostsurely,0a(G)
+2.BothoftheaboveresultsareobtainedusingLov´aszLocalLemma(lll).Inthesamepaper,itisalsoshownthatforanygraphwithgirthatleastclog
,0a2
+2,wherecisasmallconstant.NesetrilandWormaldimprovedtheaboveupperboundforrandomregulargraphsto
+1in[NW05].Onefeatureoftheaboveresultsisthattheyarenotconstructiveanddonoteasilyyieldefcientalgorithms.Veryfewconstructiveresultsareknown.Someofthemarethefollowing.SanSkulrattanakulchai[Sku04]obtainedalineartimealgorithmtocolouranysub-cubicgraphusingatmost5colours.Subramanian[Sub06]proposedandanalysedagreedyheuristicwhichobtainsanacyclicedgecolouringinO(mn
2(log
)2)timeandusesO(
log
)coloursonanygraph.AlonandZaks[AZ02]obtainedahardnessresultthatevenforsubcubicgraphs,itisNP-completetodetermineexactlythevalueof0a.15 3Girthand0a(G)Inthischapter,weobtainanimprovedupperboundfortheacyclicchromaticindexassumingthegirthgtobeatleastsomeconstant.Thisimprovestheprevious16
boundmentionedbefore.Wemakeuseofprobabilisticarguments,especiallytheLov´aszLocalLemma(lll).Intherstpart,wegiveanintroductiontotheprobabilisticmethodandintroducelll.3.1TheProbabilisticMethodIncombinatorics,moderngraphtheoryandnumbertheory,theprobabilisticmethodhasattainedthestatusofoneofthestrongestandmostimportantoftools.Thisismorerelevantwhenoneisinterestedinshowingtheexistenceofacombinatorialobjectwithcertainproperties.Sincemanyofthegraphcolouringproblemsarepreciselyaboutexistence,theprobabilisticmethodhasbecomeaverypowerfultoolindealingwithcolouringproblems.Looselyspeaking,theideabehindtheprobabilisticmethodisthat,“toshowtheex-istenceofacombinatorialobjectwithgivenproperties,itisenoughifweshowthatarandomlychosenobject(fromasuitableprobabilitydistribution)hasthedesiredprop-ertieswithpositiveprobability”.Moreformally,theprobabilisticmethodsaysthefollowing.Aniteprobabilityspaceisaniteset togetherwithafunctionPr: ![0;1],suchthatPx2 Pr(x)=1.Subsetsof arecalledeventsandforanyA ,Pr(A)=Px2APr(x).Supposethattheelementsof areasetofcombinatorialobjectsandPisaproperty.IfPr(xdoesnotsatisfyP)1,thenitfollowsthatthereexistsanx2 such16 Chapter3.Girthand0a(G)thatxsatisesP.Itistobenotedthattheprobabilisticmethod,ingeneral,providesashortandsimple(althoughnotnecessarilyconstructive)proofofanexistentialstatement.Animportanttaskinapplyingtheprobabilisticmethodistodesignanappropriaterandomexperimentandcarryoutaprobabilisticanalysis.Wepresentsomeofthetoolsweuseandillustratetheapplicationofthisprincipleinthefollowing.Thersttoolwepresenthereistheunionboundontheprobability.Itstatesthattheprobabilityofadisjunctionofeventsisupperboundedbythesumoftheprobabilitiesoftheevents.i.e.,Pr(SiEi)PiPr(Ei).Aclassicexampleofhowthisfactcanbeeffectivelyusedisthefollowingproblem.SupposeweneedtondtheRamseynumberR(k),denedasthesmallestintegernsuchthatany2-colouringoftheedgesofthecompletegraphKnusingcolourssayredandblue,alwayscontainsamonochromaticcliqueofsizek.WeusetheunionboundtoshowalowerboundofR(k)2k=2.Itistobenotedthat,thebestknownconstructivelowerboundisexp�(1+o(1))log2k=loglogk
[FW81].Theorem.(Erd¨os)[AS00]R(k)2k=2forallk3.Proof:ColourtheedgesofKnredorblueindependentlyanduniformlyatrandom.LetESbetheeventthattheinducedcliqueonagivensetSofkverticesismonochromatic(i.e.,allofitsedgesarecolouredwithonecolour).Sinceeachofthe�k2
edgeshasaprob-ability1=2ofgettingcolouredagivencolour,andsincethecolouringsareindependent,wehavePr(ES)=22�(k2).Wenowupperboundtheprobabilitythatthereisatleastonemonochromatick-clique.Thereare�nk
distinctwaystoselectSandwecanusethesub-additivitypropertytoget,Pr([ES)XPr(ES)=nk21�(k2)Ifn2k=2,wecanshowthat�nk
21�(k2)1andhencewithpositiveprobabilitynoeventESoccurs.Thereforetherehastobeatleastonecolouringinwhichnomonochro-matick-cliqueispresent.Thiscompletestheproof. Now,weintroduceanothertoolfromprobabilitytheory,whichweuseinthisthe-sis.ItisalemmaknownasTheLov´aszLocalLemma(lll).Thelllisveryusefulinsituationswheretheeventsofinterestarenotnecessarilyindependent,butarealmost17 Chapter3.Girthand0a(G)independentinthesensethateacheventisdependentonlyoneventswhichhaveonlyalocalrelationshipwiththegivenevent.Thus,thetoolintroducedbyL´asloLov´aszensuresthatifthedependenciesareonlylocal,thenalltheeventsfailtoholdsimultane-ouslywithpositiveprobabilityprovidedtheconditionsofthelemmaaresatised.Thismeansthat,ifwedenetheeventsinsuchawaythatthefailureofallthemtogetherimpliestheexistenceofthedesiredcombinatorialobject,wecanapplylll.Formally,wehavethefollowinglemma.Lemma3.1.1(TheLovaszLocalLemma).LetA=fA1;:::;Angbeeventsinaprob-abilityspace andsupposethateacheventAiismutuallyindependentofalleventsinA�(fAigSDi),forsomeDiA.Ifthereexistx1;:::;xn2(0;1)suchthatPr(Ai)xiYAj2Di(1�xj);1inThenPr( A1^:::^ An)�0.Inatypicalapplicationoflll,wedesignarandomexperimentanddeneeventsAiinsuchawaythattheAiarebadeventswhoseabsenceensurestheexistenceofthedesiredcombinatorialobject.Thenweshowthattheseeventssatisfytherequirementsoflll.3.2IntroductionInthischapter,weproveimprovedupperboundsontheacyclicchromaticindex0a(G)forgraphswherethegirthgislowerboundedbyasmallconstant.Werecallthefactthatthebestupperboundknownfor0aforallgraphstodateis16
.Specically,weprovethefollowing.Theorem3.2.1.IfGisagraphsuchthatg(G)9,then0a(G)6
.Thatis,wecouldreducetheboundfrom16
to6
.Ifwerelaxthegirthtoaslightlyhighervalue,wecanfurtherimprovetheboundasthefollowingtheoremstates.Theorem3.2.2.IfGissuchthatg(G)220,then0a(G)4:52
.Weprovetheseresultsusingtheprobabilisticmethod.Theassumedlowerboundongirthisnotsurprisingandperhapsnaturalforasimplerandomexperiment,sinceforany18 Chapter3.Girthand0a(G)uniformlychosenrandomedgecolouring,evencyclesofshortlengtharemorelikelytobetwo-colouredthancyclesoflargelength.Wediscussthisindetailinthenextsectionwhenwetrytoformalisethisphenomenon.3.3Gettingbelowthebarrier—limitedimpropernessAswementionedinthepreviouschapter,16
isthebestupperboundknownfor0aandtheparticularrandomexperimentusedhastheinherentlimitationthatonecannothopetoimprovetheboundmuchmore.Wenoticedthatitwasmainlybecauseofthepropernessrequirement.TheinequalitiesintheapplicationoftheLocalLemmacorrespondingtothepropernessconstraintsarethemainbottleneckswhichrequire4e
colourstogothrough.Here,weattempttocrossthisbarrierwithaslightmodicationtotheeventspecicationbyrelaxingthepropernessrequirement,butinalimitedsense.Afterwards,wetakecareoftheimpropernesswithextracolours.Wedenethetypeofcolouringwearelookingforasfollows.Denition3.3.1.Forapositiveinteger,acolouringoftheedgesofagraphsuchthatthefollowingholdsiscalleda-improperacyclicedgecolouring.Here,theintegeristhemeasureofimproperness.iGivenanyvertexvandcolourx,atmostedgesincidenttovarecolouredx.iiTherearenoproperbichromaticcyclesiiiNocycleinthegraphismonochromaticallycoloured.Inthefollowingpartofthissectionweprovethattheminimumnumberofcoloursneededforsuchacolouringisupperboundedbyc
;1c3assumingasmalllowerboundforthegirth.Lemma3.3.1.ForagraphGwithgirthatleast9,thereexistsa2-improperacycliccolouringusingatmost3
colours.Lemma3.3.2.ForagraphGwithgirthatleast220,thereexistsa4-improperacycliccolouringusingatmost1:13
colours.19 Chapter3.Girthand0a(G)ofeventsofeachtypethatdependonagivenedge,andmultiplybythenumberofedgesinEtogetanupperboundonthenumberofeventsinuencingE.Thefollowingtwolemmaspresenttheestimates.TheproofofLemma3.3.3isstraightforward.Lemma3.3.4isalsonotdifcultandusesstandardarguments(see[AMR91]fordetails).Forthesakeofcompleteness,wegivetheargumentsforthesecondcaseofLemma3.3.4below.Lemma3.3.3.Theprobabilitiesofeventscanbeboundedasfollows:1.ForeachTypeIeventEe1;:::;e+1,wehavePr(Ee1;:::;e+1)=1 (c
).2.ForeachTypeIIeventEC;2kwherelengthofCis2k,Pr(EC;2k)1 (c
)2k�23.ForeachTypeIIIeventEC;`whereCisoflength`,Pr(EC;`)=1 (c
)`�1.Lemma3.3.4.Thefollowingholdsforanygivenedgee:1.Lessthan2
 !eventsofTypeIdependone.2.Lessthan
2k�2eventsofTypeIIdependone.3.Lessthan
`�2eventsofTypeIIIdependone.Wegiveargumentsforthecase2ofLemma3.3.4.Toseewhy
2k�2isinfactanupperboundontheeventsthatdependonagivenedge,wexanedgee.AneventofTypeIIdependoneonlyifeispartoftheassociatedcycleoflength2k.Hencewecanupperboundthenumberofeventsthatdependonewiththemaximumnumberofcyclesoflength2kthatcontaine.Toformsuchacycle,weneedapathoflength2k�2startingfromoneendpointofeandtheotherendpointshouldbejoinedtotheendpointofthepath.Thereareatmost
2k�2possibilitiesforsuchapathsinceateachvertextherearelessthan
choices.Othercasesfollowfromsimilararguments.InordertoapplytheLov´aszLocalLemma,letx0=1=(
),xk=1=(
)2k�2andy`=1=(
)`�1,bethevaluesassociatedwitheventsofTypesI,IIandIIIrespec-tively,where,, �1areconstantstobedeterminedlater.Recallthatweusegtodenotegirth.Weconcludethat,withpositiveprobabilitynoneoftheaboveeventsoccur,provided8kdg 2e;`g,thefollowinginequalitiesholdtrue.1 (c
)x0(1�x0)(+1)2
 !Ydg 2e(1�x)(+1)
2�2Yg(1�y)(+1)
�221 Chapter3.Girthand0a(G)Lemma3.4.2.Theprobabilitiesofeventsareasfollows:Foreach1.eventEf;gofTypeI,Pr(Ef;g)=p2 a
1�"=1 a
1+".2.eventEC;2kofTypeII,Pr(EC;2k)=(1�p)2ke�2k
".3.eventEC;2`ofTypeIII,Pr(EC;2`)2p`(1�p)` (a
1�")`�12a
1�" (a
)`.4.eventEC;2mofTypeIV,Pr(EC;2m)=p2m�a
1�"2
2 (a
1�")2m(a
1�")2 (a
)2m.Lemma3.4.3.Thefollowingistrueforanygivenedgee:1.Lessthan2
eventsofTypeIdependone.2.Lessthan
eventsofTypeIIdependone.3.Lessthan2
`�1eventsofTypeIIIdependone,foreach`2.4.Lessthan
2m�2eventsofTypeIVdependone,foreachm2.ToapplyLov´aszLocalLemma,letx0=1=(
1+"),x1=1=(
1+2"),y`=(2a
1�")=(
)`andzm=(a
1�")2=((
)2m)bethevaluesassociatedwitheventsofTypeI,II,IIIandIV,wherelengthsofcyclesIIIandIVare2`and2m,respectively.Here;; ;�1arerealvaluestobedeterminedlater.Weconcludethatwithpositiveprobabilitynoneoftheaboveeventsoccur,provided8k;`;mdg 2e,wehave1 a
1+"x0(1�x0)4
(1�x1)2
Ydg 2e(1�y)4
�1Ydg 2e(1�z)2
2�2e�2k
"x1(1�x0)4k
(1�x1)2k
Ydg 2e(1�y)4k
�1Ydg 2e(1�z)2k
2�22a
1�" (a
)`y`(1�x0)4`
(1�x1)2`
Ydg 2e(1�y)4`
�1Ydg 2e(1�z)2`
2�2(a
1�")2 (a
)2mzm(1�x0)4m
(1�x1)2m
Ydg 2e(1�y)4m
�1Ydg 2e(1�z)2m
2�2Setting== ==1000anda=4000andusingthefactthat(1�1 z)z1 48z2wehave,26 Chapter3.Girthand0a(G)maticascomparedtolongcycles.Similarly,whenwetrytokillbichromaticcyclesinapropercolouringbyrandomlyrecolouringsomeoftheedgeswithasetofnewcolours,shortcycleshaveahighprobabilityofsurvival.Itwouldbeinterestingtondawayofhandlingshortcyclesusingprobabilisticarguments.3.6Anoteontheclaimed9
boundin[MR02]Theproofof0a(G)9
givenin[MR02]isbasedonapplyingaspecialisedversionofLov´aszLocalLemma,tothefollowingrandomexperiment:chooseacolourforeachedgeindependentlyanduniformlyatrandom,fromasetCofa
coloursforsomea�1.Itiseasytoseethattherequirementsofthelocallemmaarenotmetintheproofgiven.Wegivebelowanargumentexplainingwhytheproofisnoteasilyrectiableevenifweignoretheacyclicityrequirementsandonlywanttoensureproperness.Morepre-cisely,weshowthatanyproof,whichisbasedonapplyingLocalLemmaontherandomexperimentstatedaboveandwhichassumesauniformvalueforalltheconstants(as-sociatedwithvariousevents),requiresthata4e.Itisnaturaltoassumethattheconstantsareuniformlythesameunlessonewantstolookatproofswhichmakeuseofthestructureofthegraphunderconsideration.Withrespecttotherandomexperiment,consideranunfavourableeventthatapairofincidentedgese;freceivethesamecolour.DenoteitbyEe;f.Clearly,Pr(Ee;f)=1 a
andthenumberofothereventswhichmayinuenceagiveneventisatmost4
.Letx0betheuniformconstantchosenforallevents.Applyingthelocallemma,weseethatnoneofthesebadeventshold,if1 a
x0(1�x0)4
.Writex0as1 
.Itfollowsthattheinequalityofthelocallemmaholdsonlyif1 a1 (1�1 
)4
1 e�4=.Letf()=1 e�4=.Tondtheextrema,wehavef0()=�1 2e�4=+4 3e�4==0,whichyields=4.Sincef0()=�4 2�1 
f(),wegetf00()=�4 2�1 
f0()+�1 2�8 3
f().Sincef00()=�1 64e0,themaxi-mumvalueoff()namely1 4eisachievedat=4.Henceweneedtohave1 a1 4eorequivalently,a4e.28 Chapter4.Outerplanargraphs Figure4.1:Outerplanargraphwithvertexuofdegree2Byinductivehypothesis,Gnucanbeacyclicallyedgecolouredusingcoloursfrom[
(Gnu)+1][
+1].LetC(v)andC(w)betherespectivesetsofcoloursusedonedgesincidentatvandwintheacycliccolouringofGnu.Wehavethefollowingcases.Case1:[
(Gnu)=
�1]Usinginductivehypothesis,Gnucanbeacyclicallyedgecolouredusingcoloursfrom[
].Since
+1=2C(v)[C(w),wecanapplytheExtensionLemmaandobtainanacyclic[
+1]-colouringofG.Hence,fortherestoftheproof,wemayassumethat
(Gnu)=
.Case2:[(v;w)=2E]Inthiscase,weconsiderthegraphH=(Gnu)[(v;w).Wehave
(H)
andHis2-connectedandouterplanar.Byinductivehypothesis,wehaveanacyclic(
+1)-colouringofHfromwhichwegetanacyclic(
+1)-colouringofGnu.Inthiscolouring,thecolourusedon(v;w)ismissingfromC(v)[C(w)andhencewecanapplytheExtensionLemmaandobtainanacycliccolouringofGwith[
+1].Henceforth,weassumethat
(Gnu)=
andalsothat(v;w)2E.Case3:[dv=3]Since(v;w)2E,C(v)\C(w)6=;.ThenjC(v)[C(w)jjC(v)j+jC(w)j�12+(
�1)�1
+1.ByapplyingtheExtensionLemma,wegetanacycliccolouringofGwith[
+1].Case4:[dv=4]ByLemma4.2.2,thereexistsavertexxsuchthatuandxarebothdegree2verticesinGhavingvasacommonneighbour.Sincetherecanbenochord32 Chapter4.OuterplanargraphsmaximumdegreeofB.Sincem=PBmB,thistakesatotalofO(mlog
)=O(nlog
)time.Now,sincearticulationverticesaresharedbymorethanoneblock,weneedtoper-mutethecolouringsofedgesincidentatarticulationverticessoastoremovepotentialconictsamongedgesincidentatarticulationvertices.Forthispurpose,werstrootthetreeBC(G)atanarticulationvertexaofGandorderthearticulationverticesandblocksofGbasedonthepreorder-traversalorderofBC(G).ForeacharticulationvertexaofGconsideredinthisorder,letB0betheparentofaandB1;:::;Bkbeitschildren.Whenwecometoprocessa,wedistributetheremainingcoloursof[
+1]notusedontheedgesfromB0whichareincidentata,totheremainingedges(fromotherblocksB1;:::Bk)incidentata.ForeachblockBconsideredinthisorder,wepermutecolourclassesofedgesinBsoastomatchthecoloursusedonedgesincidentata(theparentofB)withthosedistributedbya.ThistakescareofconictsateacharticulationvertexaofG.ItiseasytoseethatthiscanbeachievedinO(n)timewithsuitabledatastructures.Henceitsufcestoshowthat2-connectedouterplanargraphscanbeacyclicallyedgecolouredinO(nlog
)time.Nowonwards,weassumethatGisa2-connectedouterplanargraph.WeassumetheadjacencylistrepresentationforstoringG.Thetwooccurrencesofeachedge(i;j)(oneinAdj[i]andtheotherinAdj[j])arelinkedtoeachother.Thesetofcoloursusedsofaronedgesincidentatavertexuarestoredinaheightbalancedbinarysearchtree(BST)Col(u)orderedbythecolourvalues.Inaddition,weassumetwoqueuesQ3andQ4whereQ3isaqueueonthoseverticesofdegree2havinganeighbourofdegree3.Q4isaqueueonthoseverticesofdegree2havinganeighbourvofdegree4suchthatvhasaneighbourofdegree2.Alldegreesarewithrespecttothegraphbeingconsideredinthecurrentrecursiveinvocation.Alldatastructuresmentionedbeforeareassumedtobegloballyavailableineachrecursivecall.4.3.1CorrectnessandComplexitySinceBlockColOPisessentiallytheproofofTheorem4.1.1statedasanalgorithm,thecorrectnessfollowsimmediately.Sowefocusonthecomplexityofthealgorithm.Byaddingtheedge(v;w)toBnuwheneverrequired,weensurethattheinputgraphtoeachrecursivecallisalways2-connected.Also,sinceeachrecursivecallworksona34 Chapter4.Outerplanargraphsgraphwithonevertexlessthanitsparentcall,thereareatmostnrecursivecalls.OnecanbuildQ3andQ4initiallyonceintherstinvocationofBlockColOPinO(n)timebyscanningtheadjacencylists.Afterthis,foreachrecursivecall,weonlyneedtoupdateQ3andQ4anddonotneedtocomputethemfromscratch.ItiseasytocheckthatthisupdatecanbedoneinO(1)time.Hence,totaltimerequiredinallrecursivecallsforStep4isO(n).AfteruhasbeenfoundinStep4,checkingeachoftheifconditionsinSteps5,10and14canbedoneinO(1)timeperrecursivecall.Step7involvesremovingthecolouroftheedge(v;w)fromeachofCol(v)andCol(w)if(v;w)isnotpartofE(B)andhasbeenexplicitlyaddedtoB0tomakeit2-connected.ThiscanbedoneinO(log
)timeperrecursivecall.WenowneedtoestimatethetimerequiredforanapplicationoftheExtensionLemma.Recallfromitsproofthatweneedtondacolourc=2�Col(v)[Col(w)
andalsoacolourc0=2�Col(w)[fcg
.Forj=1;2;:::,wekeepndingthej-thsmallestcolourwhichisnotinCol(w)untilwendonewhichisnotalsoinCol(v).SincethereexistssuchacolourandsincejCol(v)j3,wedon'tneedtogobeyondj=4.Foreachj,thej-thsmallestcolourwhichisabsentfromCol(w)canbefoundinO(log
)timebymaintainingthesizeofeachsubtreeatitsrootintheBSTassociatedwithCol(w).Similarly,onecanndc0also.Thus,thetotaltimerequiredforallapplicationsofEx-tensionLemmaisO(nlog
)sincethereareatmostnrecursivecalls.Step16issimilartoapplyingtoExtensionLemmaandthisalsorequiresthesametimeonthewhole.Thus,theoveralltimerequiredbyBlockColOP(B)isO(nlog
)intheworstcase.Hence,anarbitraryouterplanargraphcanbeacyclicallyedgecolouredusingatmost
+1coloursinO(nlog
)time. AlgorithmBlockColOP(B) 1:ifBisasingleedge(u;v)then2:colour(u;v)with1andRETURN.3:endif4:FindavertexuhavingexactlytwoneighboursvandwinBsuchthateither(i)degreeofvinBisatmost3or(ii)degreeofvinBisexactly4andvhasaneighbourxhavingdegree2inB.5:if
(Bnu)
(B)orif(v;w)=2E(B)then35 Chapter5.Planarand3-foldgraphstherecannotbeanyminimalcounterexampleandthusthehypothesishastobetrue.ThedischargingmethodhascometoprominencesinceitsinitialuseintheproofofthePlanarFourColourTheorembyAppelandHacken[AH76].5.1.1WhyDischarging?Inthefollowingandformostpracticalpurposes,weconsiderchargesasrealnumbers.Thedischargingmethodessentiallyconsistsoftwophases.Intheinitialphase,weassignchargestographcomponentslikevertices,edges,faces,cornersetc.Thisphaseisknownasthechargingphase.Wethenndthetotalchargebyusingsomeknownrelationshipinvolvingthesecomponents.Forexample,inthecaseofplanargraphs,onemaymakeuseofEuler'sformulatondthesumofcharges.OtherthanEuler'sformula,onemayalsousesomeotherrelationbetweenthenumberofedgesandvertices(aswedoinourcase)oranyotherrelationshipthatworksbest.Inthesecondphase,weredistributethechargesaccordingtoasetofcarefullyfor-mulatedrulesdependingonwhatweintendtoshow.Thisprocessiscalleddischargingduetoitssimilaritytothedischargingofelectricalchargesinaphysicalsystem.Thusthemethodisknownasthedischargingmethod.Afterthedischargingstep,welookatthedistributionofcharges.Basedonthisandusingsomestructuralpropertiesofthegraphs,wearriveatsomeconclusions.5.1.2Example:TwosimpleapplicationsofthedischargingmethodWegivetwosimpleexamplestodemonstratethetechniqueofthedischargingmethod.Therstexampleisawell-knownfactanditspurposeistoillustratethepowerfulideabehindthedischargingprocedure.Supposeweneedtoprovethewellknownfactthateveryplanargraphcontainsavertexofdegreeatmostve.Itsufcestoproveitformaximalplanargraphs.GivenanymaximalplanargraphG,wegivechargestotheverticesandfacesofthegraphGasfollows.Toeachvertexv,assignachargeofd(v)�6.ToeachfacefofG,weassignacharge2jfj�6.Here,jfjdenotesthenumberofedgesinf.NowthetotalchargebecomesXv2V(d(v)�6)+Xf2F(2jfj�6)=2jEj�6jVj+4jEj�6jFj=6(jEj�jVj�jFj)=�1238 Chapter5.Planarand3-foldgraphsWemaygeneralisetheaboveresultbyusingtheconceptofmaximumaveragedegree.Theorem5.2.5.LetGbeagraphwithmad(G)6.Then0a(G)2
+29.Theproof,whichfollowseasilyfromTheorem5.2.3isgivenbelow.Proof:Bytheverydenitionofmaximumaveragedegree,wenoticethatitisahereditaryproperty.Thatis,ifthemadofagraphisstrictlylessthan6,thenallsubgraphsalsohaveitasanupperbound.Wenoticethatmad(G)6impliesthatthetotaldegreeofGlessthan6n,andhencethenumberofedgesisstrictlylessthan3nandhenceisatmost3n�1.SincethispropertyalsoholdsforallsubgraphsofG,theconditionsofTheorem5.2.3aresatisedandasbefore,theresultfollowsimmediately. Thetheorems5.2.1and5.2.3areprovedbymakinguseofthedischargingmethod.Weobtainasetofreduciblecongurationsforacyclicedgecolouringthecorrespondingclassesofgraphs.Wethenshowthatthissetofcongurationsareunavoidableintheclassbyusingthedischargingmethod.5.3Diggingoutsomestructure—thedischargingmethodWemakeuseofthestructureenforcedonthegraphsbythehereditarypropertythatthenumberofedgesislinearlybounded.Weusethedischargingmethodtonetunethestructuretosuitourneeds.Thatis,toenableustoextendapartialacycliccolouring.Thefollowinglemmasareinstrumental.Lemma5.3.1.LetGbeagraphsuchthatjE(G)j2jV(G)j�1and(G)2.ThenGcontainsatleastoneofthefollowingcongurations:Conf1:A2-vertexadjacenttoa5�-vertex.Conf2:A3-vertexadjacenttoatleasttwo5�-vertices.Conf3:A6-vertexadjacenttoatleastve3�-vertices.Conf4:A7-vertexadjacenttoseven3�-vertices.Conf5:Avertexvsuchthatatleastd(v)�3ofitsneighboursare3�-verticeswithoneofthemhavingdegreetwo.42 Chapter5.Planarand3-foldgraphsLemma5.3.2.LetGbeanygraphsuchthatjE(G)j3jV(G)j�1and(G)3.ThenatleastoneofthefollowingcongurationsoccursinG:Conf1:Avertexofdegree3havinga11�neighbour.Conf2:Avertexofdegree4havingatleasttwo11�neighbours.Conf3:Avertexofdegree5havingatleastthree11�neighbours.Conf4:Avertexv,12d(v)14suchthatjN5�(v)jd(v)�2.Conf5:Avertexofdegree15havingatleastfourteen5�neighbours.Conf6:Avertexvsuchthat16d(v)17,andallofitsneighboursare5�vertices.Conf7:AvertexvsuchthatjN5�(v)jd(v)�5,oneoftheneighboursbeingofdegreethree.Intheremainingpartofthissection,weprovetheabovetwolemmas.5.3.1Proofoflemma5.3.1Proof:LetG,beanygraphsatisfyingthehypothesis.Wehavem2n�1and(G)2.Weusethedischargingmethodinthefollowing.ChargingPhaseInthisphaseweassigntoeachvertexv,acharge�(v)=d(v)�4.WethenobtainthetotalchargeXv2V�(v)=Xvd(v)�4=Xvd(v)�4n=2m�4n�2usingthefactthatm2n�1.DischargingPhaseAfterthecharging,weproceedtolettheverticesdischargeaccordingtothefollowingrule.DischargingRule:Eachvertex,ifitsdegreeisatleastsix,sendsachargeof+1toeachdegreetwoneighbourandachargeof+1 2toeachofitsdegreethreeneighbours.43 Chapter5.Planarand3-foldgraphsLetthenewchargeofeachvertexvbe�0(v).Sincethetotalchargeinthesystemisconserved,Xv�0(v)=Xv�(v)=�2(5.1)WenowproveLemma5.3.1bywayofcontradiction;i.e.,weshowthatifanygraphsatisfyingthehypothesisdoesnotsatisfyLemma5.3.1,thenPv�0(v)�0contradictingEquation(5.1)ofconservationofcharges.Wenowanalysethepossiblevaluesof�0(v)ifLemma5.3.1doesnothold.InthecaseConf1doesnotoccurinthegraph,foreachvertexofdegreetwo,bothofitsneighboursareofdegreeatleastsixandthusgivesachargeof+1eachtov.Therefore�0(v)=�2+1
jN6+(v)j=�2+2=0.Whend(v)=3andConf2doesnotholdinG,wehave�0(v)=�1+1 2
jN6+(v)j�1+1 2
2=0.Verticeswithdegree4anddegree5donotparticipateinthedischargingandthusretaintheirchargeswhicharenon-negative.Foradegreesixvertexv,�0(v)=2�1
jN2(v)j�1 2
jN3(v)j.Wehavetwocases.Intherstcase,weassumethatjN2(v)j=.Inthiscase,sinceConf3isprohibited,jN3(v)j4andhence�0(v)=2�1 2
jN3(v)j2�1 2
4=0.OtherwisethevertexvhasadegreetwoneighbourandtheabsenceofConf5assuresthatthereareatmosttwo3�neighbours.Thisimpliesthat�0(v)2�2=0.Similarly,whend(v)=7,wehave�0(v)=3�1
jN2(v)j�1 2
jN3(v)j.Firstweassumethatvhasaneighbourofdegreetwo.Then,becauseConf5doesnotoccur,weseethatvcanhaveatmostd(v)�4=7�4=3neighboursofdegreeatmostthree.Therefore�0(v)3�3=0.Intheothercasewherevdoesnothaveadegreetwoneighbour,wehave�0(v)=3�1 2
jN3(v)j3�1 2
6=0,sinceConf4doesnotoccur.Finally,consideranyvertexvofdegreeatleasteight.ThenifvhasadegreetwoneighbourandConf5isnotallowed,thevertexvcanhaveatmostd(v)�4neighboursofdegreeatmostthreeandtherefore�0(v)0.Forthecasewhenvdoesnothaveaneighbourofdegreetwo,wehave�0(v)=d(v)�4�1 2
jN3(v)jd(v)�4�1 2
d(v)=1 2
d(v)�40.44 Chapter5.Planarand3-foldgraphsSupposed(v)=3andConf1isnotpresentinthegraph.Then,�0(v)=�3+1
jN12+(v)j=�3+3=0.Ifd(v)=4,then�0(v)=�2+2 3
jN12+(v)j�2+2 3
3=0,becauseConf2isnotpresent.Assumethatd(v)=5andConf3doesnotoccur.Thenwehave,�0(v)=�1+1 3
jN12+(v)j�1+1 3
3=0.Forverticesvsuchthat6d(v)11,thechargeremainintactandthus�0(v)=�(v)0.Nowsupposethat12d(v)14.Wehave�0(v)=d(v)�6�1
jN3(v)j�2 3
jN4(v)j�1 3
jN5(v)jbythedischargingrule.Ifweassumethatthevertexvhasaneighbourofdegree3,then,sinceConf7doesnotoccurinthegraph,thenumberof5�neighboursthatthevertexvcanhaveisatmostd(v)�6andhence�0(v)0.Otherwise,weobservethat�0(v)d(v)�6�2 3
jN5�(v)jd(v)�6�2 3
(d(v)�3)0,sinceConf4isabsent.Whend(v)=15,�0(v)=9�jN3(v)j�2 3
jN4(v)j�1 3
jN5(v)j.Asinthepreviouscase,ifweassumethatthevertexvisadjacenttoa3-vertex,then,sinceConf7isprohibited,wecanseethatvcanhaveatmostd(v)�6neighboursofdegreeatmost5implyingthat�0(v)0.Ontheotherhand,whenvisnotadjacenttoa3-vertexwehave�0(v)9�2 3
jN5�(v)j9�2 3
130,becauseConf5doesnothold.Forthecases16d(v)17,wenoticethat�0(v)=d(v)�6�jN3(v)j�2 3
jN4(v)j�1 3
jN5(v)j.Ifvisadjacenttoa3-vertex,thentheabsenceofConf7impliesthatvcanbeadjacenttoatmostd(v)�6verticesofdegreeatmost5andhence�0(v)0.Ifvdoesnothavea3-vertexneighbour,�0(v)d(v)�6�2 3
jN5�(v)jd(v)�6�2 3
(d(v)�1)0,sinceConf6doesnotoccur.Finally,whend(v)18,�0(v)=d(v)�6�jN3(v)j�2 3
jN4(v)j�1 3
jN5(v)j.Supposingthatvisadjacenttoa3-vertexandConf7doesnotoccur,weseethatthevertexvcanbeadjacenttoatmostd(v)�6verticesofdegreeatmost5onceagainimplying�0(v)0.Again,ifvisnotadjacenttoa3-vertex,then�0(v)d(v)�6�2 3
jN5�(v)jd(v)�6�d(v)
2 30.46 Chapter5.Planarand3-foldgraphsSincethecharge�0(:)isnon-negativeforeveryvertex,weobtainacontradictionwithequation(5.2). 5.4Extendingthepartialcolouring:reducibilityInthesequelwefrequentlyusethefollowingnotations.LetCdenoteanacyclicedgecolouringofagraphGusingkcolours,forsomeintegerk.WedenotebyC(v)thesetofcoloursassignedbyCtotheedgesincidenttothevertexv.ForanysubsetWV(G)wedeneC(W)=Sw2WC(w).Thecolourofanedge(u;v)inthecolouringCisdenotedbyC(uv).Letu;vbetwodistinctverticesofG.ThesetofneighbourswofthevertexvinGforwhichC(vw)2C(u)isdenotedWG(vu).NoticethattheorderofvanduisimportanthereandthatthesetWG(vu)couldbeempty.Recallthatak-colouringisacolouringwherethecoloursarefrom[k].ThenextlemmaisamodicationoftheExtensionLemma(Lemma4.2.3)fromthepreviouschapter.ThegraphobtainedfromGbydeletingtheedge(u;v)isdenotedG�uv.Lemma5.4.1.LetGbeagraph,(u;v)2E(G)andletCbeanacyclick-colouringofG�uv.IfjC(v)[C(u)[C(WG�uv(vu))jk,thenthecolouringCcanbeextendedtoGwithoutusingadditionalcolours.Proof:Itisenoughtocolourtheedge(u;v)withanycolourfromtheset[k]�(C(u)[C(v)[C(WG�uv(vu))),toobtainanacyclick-colouringofG. 5.4.1ProofofTheorem5.2.1Proof:WeprovethetheorembyshowingthatalltheveunavoidablecongurationsmentionedinLemma5.3.1arereducible.Thisimpliesthatthereisnominimumcoun-terexampletothetheoremsinceatleastoneofthecongurationsinpresent.SupposethatHisaminimumcounterexampletothetheorem(onewiththenumberofedgesbeingaminimum).Letk=
(H)+6.47 Chapter5.Planarand3-foldgraphsRecallingtheargumentsgivenintheintroduction,wemayassumewithoutlossofgeneralitythatHis2-connected.Otherwise,wecanobtainanacyclick-colouringofeachofitsblocksandcombinethem(byrenamingsomecoloursifneeded)togetanacyclick-colouringoftheentiregraph.IfHis2-connected,(H)2,whichweassumefortherestoftheproof.AccordingtoLemma5.4.1itsufcesifweshowthatthereexistsanedge(u;v)forwhichjC(v)[C(u)[C(WH�uv(vu))jk.Weshowtheexistenceofsuchanedgeinthefollowing.Weconsideranumberofcases,dependingonwhichoftheveunavoid-ablecongurationsoccurinH.IneachcasewepointoutsuchanedgeandweapplyLemma5.4.1provingthereducibility.ByLemma5.3.1thegraphHcontainsatleastoneofthevegivencongurations.Conguration1:WhenConf1ispresent,Hcontainsa2-vertexxadjacenttosome5�-vertexy.Letzbetheremainingneighbourofx.Moreover,letH0=H�xz.SinceHwasaminimumcounterexample,0a(H0)k.LetCbeanyacyclick-colouringofH0andletC(x)=a.Wehavethefollowingtwocases.Case1:IfjC(z)\fagj=0thenjC(z)[fagj
(H).Therefore,applyingLemma5.4.1,Hhasanacyclick-colouring,contradictingtheassumption.Case2:TheothercaseiswhenjC(z)\fagj=1.UsingthefactthatdH(y)5weseethatjC(y)j5.ThereforejC(z)[C(y)j
(H)+3.Again,usingthegen-eralisedextensionlemmaweextendthecolouringCtoanacyclick-colouringofH,acontradiction.Conguration2:Thismeansthatthereissome3-vertexxinHadjacenttotwo5�-vertices,sayz1andz2.Considerytobethethirdneighbourofx.LetH0=H�xz1.SinceH0haslessedgesthanH,0a(H0)kduetotheminimalityofH.LetCbeanacyclick-colouringofH0.Wehavethefollowingcases.Case1:jC(z1)\C(x)j1.Here,wehavejC(x)[C(z1)[C(WH0(xz1))j
(H)+4.Usingextensionlemma,weextendthecolouringCtoanacyclick-colouringofH,acontradiction.Case2:jC(z1)\C(x)j=2.Subcase2.1WhenC(xy)62C(z2),wehavejC(z1)[C(z2)[C(x)j
(H)+3.Thereforewerecolour(inH0)theedge(x;z2)withacolour62C(z1)[C(z2)[C(x),obtaininganacyclick-colouringC0ofH0reducingittothepreviouscase.Subcase2.2C(xy)2C(z2).48 Chapter5.Planarand3-foldgraphsCase2.2.aIfC(xz2)62C(y),thenjC(x)[C(y)[C(z1)j
(H)+3.Thuswerecolour,theedge(x;y)inHwithacolour62C(x)[C(y)[C(z1)toobtainanacyclick-colouringC0ofH0andweareintherstcase.Case2.2.bIfC(xz2)2C(y),thenjC(x)[C(z1)[C(z2)[C(y)j
(H)+5andusingLemma5.4.1,weextendthecolouringCtoanacyclick-colouringofH,obtainingacontradiction.Conguration3:TheoccurrenceofConf3impliesthatthereisa6-vertexxadjacenttove3�-vertices.Letanytwoofthembez;z1.Letybetheremainingneighbourofx.AlsoletH0=H�xz.DuetotheminimalityofH,wehave0a(H0)k.LetCbeanyacyclick-colouringofH0.Case1:jC(x)\C(z)j1orC(xy)62C(z).ItfollowsjC(x)[C(z)[C(WH0(xz))j
(H)+5andhencebyLemma5.4.1weextendthecolouringCtoanacyclick-colouringofH,acontradiction.Case2:jC(x)\C(z)j=2andC(xy)2C(z).AssumewithoutlossofgeneralitythatC(xz1)2C(z).Subcase2.1IfjC(y)\C(x)j=1,werecolourtheedge(x;y)(inH0)withacolour62C(x)[C(y)toobtainanacyclick-colouringC0anditreducestothepreviouscase.Subcase2.2IfjC(y)\C(x)j2,thenjC(x)[C(z)[C(WH0(xz))j
(H)+5andbyLemma5.4.1weextendthecolouringCtoanacyclick-colouringofH,againacontradiction.Conguration4:IfHhasa7-vertexxadjacenttoseven3�-vertices,thenletzbeoneofitsneighboursandletH0=H�xz.SinceH0haslessedgesthanH,0a(H0)k.LetCbeanacyclick-colouringofH0.WeobservethatjC(x)[C(z)[C(WH0(xz))j10.RecallthatWisthesetofneighbourswofxinH0forwhichC(xw)2C(z).There-fore,accordingtoLemma5.4.1andsince
(H)7,Hhasanacyclick-colouring,acontradiction.Conguration5:Ifnoneofthecases1-4occurs,thentheremustbeavertexxinHsuchthatatleastdH(x)�3ofitsneighboursare3�-verticesandoneofthem,sayz,isofdegree2.LetusconsiderthegraphH0=H�xz.SinceH0haslessedgesthanH,wehave0a(H0)k.LetCbeanacyclick-colouringofH0.LetC(z)=fag,C1=fC(xy):y2NH(x)&dH(y)�3g,andC2=fC(xy):y2NH(x)�fzg&dH(y)3g.49 Chapter5.Planarand3-foldgraphsCase1:Ifa62C1,thenjC(x)[C(z)[C(w)j
(H)+1,wherewistheneighbourofxinH0satisfyingC(xw)=a.Therefore,fromLemma5.4.1,Hhasanacyclick-colouring,acontradiction.Case2:Ontheotherhandifa2C1,letybetheneighbourofzinH0.Thereisacolour62C1[C(y)suchthatwecanrecolour(inthegraphH0)theedge(z;y)with,obtaininginthiswayanacyclick-colouringC0ofH0anditreducestothepreviouscase. 5.4.2ProofofTheorem5.2.3Proof:Followingtheideaofthepreviousproof,letusassumethatHisaminimalcounterexampletoTheorem5.2.3.Forbrevity,inthefollowingofthesectionkstandsfor2
(H)+29.WeknowfromLemma5.4.1thatitisenoughifweshowthatthereexistssomeedge(u;v)forwhichjC(v)[C(u)[C(WH�uv(vu))jk.Claim5.4.1.Anygraphwith2cannotbeaminimumcounterexampletothelemma.Proof:LetHbeaminimumcounterexamplewith(H)2.Letebeanedgeincidenttoaminimumdegreevertex.BytheminimalityofH,wehave0a(H�e)k.ItiseasytoseethattherequirementofLemma5.4.1ismetfortheedgeeandwecanextendanyk-acycliccolouringofH�etoH. Hencewemayassumewithoutlossofgeneralitythat(H)3.Therefore,applyingLemma5.3.2,Hcontainsatleastoneofthe7congurations.Weconsiderdifferentcasesdependingonwhichofthe7unavoidablecongurationsoccurinH.AsinthepreviousproofweshowtheexistenceofasuitableedgeinordertoapplyLemma5.4.1thusprovingthereducibilitybywayofcontradiction.Conguration1:Thereexistsa3-vertexx,adjacenttoa11�-vertexyinH.Letusassumethatzisanotherneighbourofx(inH)andletH0=H�xz.FromthefactthatHistheminimalcounterexampleweseethatH0hasanacyclick-colouring,sayC.Moreover,sincedH(y)11,wehavejC(z)[C(x)[C(WH(xz))j2
(H)+8.AccordingtoLemma5.4.1,itfollowsthatHhasanacyclick-colouring,acontradiction.50 Chapter5.Planarand3-foldgraphsConguration2:IfHcontainsa4-vertexxadjacenttoatleasttwo11�-verticessayy1;y2,letzbeanyotherneighbourofx(inH)andletH0=H�xz.SinceHisaminimalcounterexample,H0hasanacyclick-colouring.Moreover,becausedH(y1);dH(y2)11,wehavejC(z)[C(x)[C(WH0(xz))j2
(H)+18.FromLemma5.4.1itfollowsthatHhasanacyclick-colouring,acontradiction.Conguration3:IfHhasa5-vertexxadjacenttoatleastthree11�-verticesy1;y2;y3,thenletzbeanyotherneighbourofx.LetH0=H�xz.Asbefore,H0hasanacyclick-colouringC.WealsohavedH(y1),dH(y2);dH(y3)11and
(H)5implyingjC(z)[C(x)[C(WH0(xz))j2
(H)+28.Asinthepreviouscase,itfollowsthatHhasanacyclick-colouring,acontradiction.Conguration4:IfHhasavertexxofdegree12or13or14,adjacenttoatleastdH(x)�2verticesofdegreeatmost5thenletzbeanyofthemandletH0=H�xz.FromthefactthatHistheminimalcounterexamplewehavethatH0hasanacyclick-colouringC.Fromthefactthatall,exceptatmosttwo,neighboursofxinHareofdegreeatmost5wehavejC(z)[C(x)[C(WH0(xz))j2
(H)+19.AccordingtoLemma5.4.1itfollowsthatHhasanacyclick-colouring,acontradiction.Conguration5:Ifthereisa15-vertexxadjacenttoatleastfourteen5�-vertices,thenletzbeanyofthemandletH0=H�xz.AsbeforeH0hasanacyclick-colouringC.WenoticethatjC(z)[C(x)[C(WH0(xz))j
(H)+252
(H)+10,because
(H)15.ItfollowsfromLemma5.4.1thatHhasanacyclick-colouring,acontradiction.Conguration6:IfHcontainsavertexxofdegreeeither16or17,suchthatallitsneighboursare5�-vertices,thenletzbeanyofthisneighboursandletH0=H�xz.FromthefactthatHistheminimalcounterexamplewehavethatH0hasanacyclick-colouringC.Moreover,sinceallneighboursofxinHareofdegreeatmost5wehavejC(z)[C(x)[C(WH0(xz))j322
(H).AccordingtoLemma5.4.1itfollowsthatHhasanacyclick-colouring,acontradiction.Conguration7:IfHhasavertexx,havingatleastdH(x)�5neighboursofdegreeatmost5,oneofwhichsayz,isofdegree3,thenletH0=H�xz.FromthefactthatHistheminimalcounterexample,H0hasanacyclick-colouringC.LetC1=fC(xy):y2NH(x)anddH(y)�5g,C2=fC(xy):y2NH(x);y6=zanddH(y)5g.IfC(z)\C1=;,thenjC(z)[C1[C2[C(WH0(xz))j
(H)+7.FromLemma5.4.151 Chapter6.SomemoregraphssatisfyingAEC6.2.1HararyGraphsProof:(Theorem6.1.1)ConsidertheHararygraphG=Hn;2onnvertices0;1;2;:::;n�1.Wecolourthegraphintwophases.Intheinitialphase,wecolourthegraphsothatitisproperand`almostacyclic'andthenrectifythecolouringinthenextphasebyrecolouringafewedges.Initially,wecolourthegraphGasfollows.FirstPhase:Firstofall,weassumethatn�6toavoidsometechnicalities.Forsmallervaluesofn,wecaneasilyverifythattheresultistrue.Wealsonotethatalltheadditionsandsubtractionsinthefollowingarewithrespecttomodulonarithmeticunlessstatedotherwise.1.Alltheedgesoftheform(i;i+1)arecolouredwiththecolourimod3+1,0in�1.2.Theedges(i;i+2)arecolouredwith4ifeitheri0mod4ori1mod43.Theedges(i;i+2)arecolouredwith5ifeitheri2mod4ori3mod44.Ifn1mod3,thenrecolourtheedge(n�1;0)withcolour2.5.Ifn0mod4,recolourtheedges(n�2;0)and(1;3)withcolour6.6.Ifn1mod4,recolourtheedge(1;3)withcolour6.7.Ifn2mod4,recolourtheedges(0;2)and(1;3)withcolour6.8.Ifn3mod4,recolourtheedge(n�2;0)withcolour6.Thisformstherstphaseofthecolouring.AsexempliedinFigure6.1,theHararygraphHn;2consistsofahamiltoniancycleCn=(0;1;2;:::;n�1;0)onnverticesandeveryotheredgeisachordofthiscyclebetweenverticesthatareatdistance2fromeachotherinthiscycle.Fromnowon,theedgesofthehamiltoniancycleCnarecalledhamiltonianedgesandotheredgesaschordedges.Wecallthecyclesformedbychordedgesaloneaschordcycles.Weobservethefollowingfact.55 Chapter6.SomemoregraphssatisfyingAEC Figure6.4:Thecolouringwhenn2mod4andn3mod4resp.abnormalpair.Similarlycolour6isnoteligibleasitismissingfromvertexn�1.Colour4iseligibleforbothpairsleadingtothe2-4bipathfn�5;n�3;n�2;0;n�1;1;2gterminatingat2.Whenn2mod3,theonlyabnormalpairisf(n�1;0);(1;2)gcoloured2.Noticethatsincewerecolour(n�2;n�1)by6(Step10)wheneveritiscoloured1,therearenoabnormalpairscoloured1.Asbefore,5isnoteligible.Colour4isnoteligibleasitisabsentatvertex2.Colour6iseligible(sincewerecolour(n�2;n�1)),leadingtothe2-6bipathf3;1;2;0;n�1;n�2gterminatingat3.Sincecolour6appearsonbothCnandchordcyclesinthisparticularcase,toseethattherearenobicolouredcyclesinvolvingthecolour6,wealsonoticethatthe6-4bipathf3;1;n�1;n�2;0;2gterminatesat2,the6-3bipathf1;3;2;0;:::gterminatesat1andnally,the1-6bipathf2;0;1;3;4gterminatesat2.Thereisno6-5bipath.Asbefore,thisestablishesthatabnormalpairsorexceptionaledgesarenotpartofbichromaticcycles.Case4:n3mod4Inthisnalcase,theonlyexceptionaledgesare(n�2;0)coloured6and(n�1;0)61 Chapter6.SomemoregraphssatisfyingAECcoloured2(whenn1mod3).Sincewerecolouronlyoneedgewith6,itcannotbepartofanybicolouredcycle.Whenn1mod3,theabnormalpairsaref(n�3;n�2);(n�1;0)gandf(n�1;0);(1;2)gallcoloured2.Colour5ismissingfromvertex0,andisnoteligibleforeitherpair.Colour4isnoteligibleforthepairf(n�3;n�2);(n�1;0)gasitismissingfromn�2.Itiseligibleforthepairf(n�1;0);(1;2)gleadingtothe2-4bipathf3;1;2;0;n�1;n�3;n�2gthatterminatesat3.Whenn2mod3,theabnormalpairsaref(n�2;n�1);(0;1)gcoloured1andf(n�1;0);(1;2)gcoloured2.Sincecolour4ismissingfromn�2andcolour5ismissingfrom0,neitheriseligibleforthepairf(n�2;n�1);(0;1)g.Thecaseforf(n�1;0);(1;2)gfollowsfromtheargumentsabove(forn1mod3)asthecolouringofchordedgesremainsthesame.Thisprovestheclaimforthisnalcaseandestablishesthatabnormalpairsandex-ceptionaledgesarenotpartofbichromaticcycles. ThuswehaveapropercolouringCinwhichthehamiltoniancycleCnandthechordcyclesareproperlycolouredwithatleast3colours.Wehavealsoprovedthattherearenobichromaticcyclesthatusestheabnormalpairsorexceptionaledges.Thus,weonlyneedtoshowthattherearenobicolouredcyclesthatusebothhamiltonianedgesandnormalchordedges.Thiswedointhefollowing.Claim6.2.3.ThepropercolouringCisalsoacyclic.Proof.Inthefollowing,weshowthattherearenobicolouredcyclesthatusesbothhamiltonianedgesandchordedges.Sincetherearenobichromaticcyclesthatusescolour6(sinceexceptionaledgescannotbepartofthem),weonlyneedtoconsidertheedgescoloured1-5fortherestoftheargument.Sincecolours1;2;3appearsonlyonhamiltonianedgesand4;5onlyonchordedges,itsufcesifweshowthattherearenobichromaticcyclesthatuseshamiltonianandchordedgesalternately.Wealsomakeuseofthefactthattherearenobicolouredcyclesthatusesabnormalpairsorexceptionaledges.Supposethatthereissome2-colouredcycleformedbychordedgesandhamiltonianedges.Weknowthatnoneofitsedgescanbeanexceptionaledge.Wealsonoticethatanyhamiltonianedge(j;j+1)whichisnotexceptionaliscolouredwithjmod3+1.Startfromsomevertexi(ofthiscycle)sothattheedge(i;i+1)isahamiltonianedge62 Chapter6.SomemoregraphssatisfyingAECCase2:xexistsandC(ux)=.Thereareatmost
coloursseenbythevertexw(thisistrueforanyvertex).Thusthereisatleastonecolourfree(notused)atw.Since(u;v)isuncoloured,theedge(v;w)canberecolouredwithsomeavailablecolour differentfrom.Thisreducestheproblemtooneofthepreviouscases( isnotseenbyuorC(ux)6= ).ThusHcannotbeacounterexampletothetheoremleadingtoacontradiction.Thetheoremfollows. 6.3.1FullySubdividedGraphsProof:(Theorem6.1.3)Weprovideanalgorithmicproofofthetheorem.Ouralgorithmproceedsiterativelybyextendingapartial(possiblyempty)colouring.Ateachstep,itextendsthepartialcolouringtooneormoreedgestillthegraphisentirelycoloured.Weuseamaximumof
+1coloursforthecolouring.Wexanorderingoftheedges(arbitrary)andcolourtheminthatorder.Wenoticethatineveryfullysubdividedgraph,everyedgejoinsavertexofdegreeatmost
andavertexofdegreetwointroducedinthesubdivision.Wecalltheverticesintroducedinthesubdivisionasnewvertices.Theremainingverticesarereferredtoasoldvertices.Atstepi,weneedtocolouredgeei=(ui;vi)whereuiisanoldvertexandviisanewvertex.Noticethattheuncolourededgeeihasatmost
�1edgesadjacenttoitatuiandexactlyoneatvi.Therearetwocases.Supposethatthetwosetsofcoloursseenbytheendpointsofeidonothaveanycommoncolour.Inthiscase,thereareatmost
coloursusedatuiandvitogetherandwehaveatleastonecolouravailableforcolouringei.Sincetherearenocommoncoloursattheendpointsofei,therecannotbeanybichromaticcyclepassingthroughit.Thiscompletestherstcaseandweareabletoextendthepartialcolouring.Supposethatthereisacommoncolourcusedatbothendpointsofei.Wenoticethatthereareatmost
�1coloursusedateitheroftheendpointsofeitogetherandwehaveatleast2coloursavailableforcolouringei.Letfbetheedgeincidenttouithatiscolouredc.Thentheonlypossiblebicolouredcyclethrougheiusescandthecolouroftheotheruniqueedgeadjacenttof.Hencewehaveatleastonecolouravailabletoextendthepartialcolouringtoincludeei.66 Chapter6.SomemoregraphssatisfyingAECThusweareabletoinductivelyextendthepartialacyclicedgecolouringtothewholegraphusingatmost
+1colours. 67 PartIIk-intersectionEdgeColouring Chapter7.kintersectionedgecolouring.be
,thek-iecbecomestheusualproperedgecolouring.Adistance-2edgecolouringisaproperedgecolouringwhereedgescolouredthesameareseparatedbyapathonatleast2edges.TheminimumnumberofcoloursthatsufcesforsuchacolouringofGisthedistance-2chromaticindexofG.Ifaproperedgecolouringisalsoadistance-2edgecolouring,itisa1-iec.Thatis,thenumberofcommoncoloursseenbyadjacentverticesisexactly1.Itiseasytoseethattherearegraphsthatrequire (
2)coloursinanydistance-2edgecolouring.Sobyrestrictingktobe1,weknowthatthek-iecbecomesequivalenttoadistance2edgecolouringandthusrequires (
2)colours.Nowconsiderthecasewhenk=
�1andgraphsareregular.Thismeansthatforeach(u;v)2E,thek-iecCissuchthatjC(u)\C(v)jk�1orequivalently,C(u)6=C(v).Inotherwords,eachpairofadjacentverticesseeadifferentsetofcolours.ThistypeofcolouringiscalledanadjacentvertexdistinguishingedgecolouringorAVDcolouringforshort.Wesawthatweneed (
2)coloursintheworst-caseforadistance-2edgecolour-ing,while
+1isanupperboundforaproperedgecolouring.Theconceptofk-intersectionedgecolouringsimultaneouslygeneralisesalltheabovenotionsbyallowingthemaximumnumberofcommoncolourstobeboundedbysomek,whichliesbetween1and
,inclusiveofboth.Ourstudyismotivatedbytheinteresttoknowwhathappenstothechromaticindexwhenthemaximumnumberofcommoncoloursvariesfrom1to
.ItfollowsfromthedenitionthatforanyproperedgecolouringofagraphG,weneedatleast
colours.V.G.Vizingin[Viz64]showedthatthereisaproper
+1edgecolouringforanygraph.Hisproofisconstructiveandimmediatelyprovidesadeterministicpolynomialtimealgorithmtoobtainsuchacolouring.WemakeuseofVizing'sresultaswellastwotoolsfromprobabilisticmethod,namelytheLov´aszLocalLemma(SymmetricandGeneralforms)andtheChernoffbound.See[AS00,MR02,EL75]forfurtherdetailsontheprobabilistictools.Specically,weprovethefollowing.Theorem7.1.1.Letfk(
)=maxG:
(G)=
f0k(G)gthenfk(
)=(
2 k).70 Chapter7.kintersectionedgecolouring.colouringC0.Foranyedgee=(u;v)2E(G),letse=jC0(u)\C0(v)j�1.Letestandforthenumberofcommoncoloursi(otherthanthecolourofe)suchthatedgeseu;iandev;igetthesamecolourinthenewcolouringC.A“bad”eventisthatforsomeedgee=(u;v),theverticesuandvhaveatleastk+1coloursincommoninthenewcolouringC,orequivalentlyek.Absenceofeverysucheventimpliesthatwehavethedesiredcolouring.WehaveExp(e)=se .LetB(n;p)denotethesumofnindependentlyandidenticallydistributedindicatorvariableseachhavingexpectationp.Bythewell-knownChernoffbound(see[AS00,MR95]),wehavePr(B(n;p)(1+)np)e (1+)1+npforany�0:Notethate=B(se;1=).Further,eisstochasticallydominatedbyB(
;1=).Wenowset=d2
ke.Forthesakeofsimplicity,weignoretheceilings(withoutaffectingthecorrectnessofthearguments)andtreat=2
k.Hence,wehavePr(ek)Pr(B(
;1=)k)e�(0:38)k 2:Usingtheassumptionthatk20log
,wegetPr(ek)e�(3:8)log

�3:8:Thus,weobtainanupperboundontheprobabilityofhavingmorethankcoloursincommonbetweentheendpointsofanyxededge.Sincetwosucheventsaredependentonlyiftheyshareanedge,eacheventismutuallyindependentofallbutatmost2
(
�1)otherevents.Now,toapplythesymmetricformofLov´aszLocalLemma,weneedtoverifythate
�3:8(2
(
�1)+1)1whichholdsif2e
2
3:81()2e
1:81()(2e)5=9
whichistrueforevery
3.72 Chapter7.kintersectionedgecolouring.Hence,theresultfollowsforthecasek20log
byxing=d2
ke.Sinceanynumberstrictlylessthanbutsufcientlycloseto2sufces,amoretightanalysisshowsthatwecandoawaywiththeadditivesub-linearterm.Weskipthedetails. Nowwelookatthecasewhenk20log
wheretheaboveargumentsfailbecausetheuniformupperboundontheprobabilitiesPr(ek)isnotsufcientlysmalltoapplythesymmetricversionofLocalLemma.Proof:(Lemma7.1.2:Part2)Weassumethat
6,sinceotherwisekisatmost5andevenadistance-2colouringsufces,sincethenumberofcoloursusedwouldbeatmost2
(
�1)+122
2=kfortherangek
5.ForprovingPart2,weusethemostgeneralformofLov´aszLocalLemmaasgivenbelow.Lemma7.2.2(TheLovaszLocalLemma(generalform)).LetA=fA1;:::;Angbeeventsinaprobabilityspace suchthateacheventAiismutuallyindependentofalleventsinA�(fAig[Di),forsomeDiA.Alsosupposethatthereexistx1;:::;xn2(0;1)suchthatPr(Ai)xiYAj2Di(1�xj);1inThenPr( A1^:::^ An)�0.Considerthefollowingrandomexperiment.Coloureachedgeuniformlyandinde-pendentlyatrandomwithoneofthec=22
2=kcolours.Wedenethefollowingtwotypesofbadevents.TypeIApairofincidentedgese;freceivethesamecolour.DenotethiseventbyEe;f.TypeIIForanedgee=(u;v),andsetsS1E(u)nfeg,S2E(v)nfeg,withjS1j=jS2j=k,theseedgesareproperlycolouredwithasetofkcolours.WedenotethiseventbyEe;S1;S2.SupposethattherandomcolouringCissuchthatnoneoftheaboveeventshold.ThenCisproper,andfurther,notwoadjacentverticessharemorethankcoloursincommon.Weshowthatthishappenswithpositiveprobability.Inordertoapplylll,observethattheprobabilitiesofeachtypeofeventcanbeupperboundedasinthefollowinglemma.73 Chapter7.kintersectionedgecolouring.ToverifythatthecolouringCsatisestherequiredproperties,itissufcienttoshowthatthefollowinginequalitieshold:1 c2 c1�2 c22
1�k!2k ck22
2k k!(k�1)!(7.1)andk! ckk!2k ck1�2 c2k2
1�k!2k ck2k2
2k k!(k�1)!:(7.2)Simplifyingandtakingroots,weseethatboth,(1)and(2),followfrom121�2 c4
1�k!2k ck4
2k k!(k�1)!;(7.3)Substitutingc=22
2=k,(3)isequivalentto121�k 11
24
1�k!k 11
2k!4
2k k!(k�1)!;(7.4)andfurtherto1214k 11
24 (k�1)!(k 11)k;(7.5)where,1=1�k 11
211
2 kand2=1�k!kk (11)k
2k(11)k
2k (k!kk):Byusingtheassumptions
6andk
,onecanverifythat1;21=4andalsothatthesumoftheexponentsof1and2in(5)isatmost1=2.Thisestablishesinequality(5).Itfollowsthatthereexistak-intersectionedgecolouringofGusing22
2=kcolours. Wecanalsoshowthatif1klog
,then0k(G)13
2 k.Theprooffollowsfromessentiallythesameargumentmakinguseofthefactthatkisatmostlog
.Hencetheupperboundcanbeimprovedto13
2 kforallgraphsirrespectiveofthevalueofk.75 8OrientedColouring8.1IntroductionTheconceptoforientedcolouringwasintroducedbyBrunoCourcellein[Cou94].Sincethen,manyresearchershaveworkedontheproblemduetoitsimportanceinmobilecommunicationandVLSIdesign.Evenforsimpleclassesofgraphslike2-dimensionalgridgraphsorplanargraphs,wedonotknowtightboundsonthevalueofthisparameter.Inthenextsection,westartwithafewimportantdenitionsandthenproceedtopresenttheresultswehaveobtained.8.1.1DenitionsandResultsAnorientedgraph#G=(V;A)isanorientationoftheedgesofasimpleundirectedgraphG=(V;E).Thatis,#Gdoesnotcontainloopsoroppositearcs.Anorientedk-colouringofanorientedgraphisapartitionofitsvertexsetintoklabelledsubsetssuchthatnotwoadjacentverticesbelongtothesamesubset,andallthearcsbetweenapairofsubsetshavethesameorientation.Precisely,anorientedk-colouringofanorientedgraph#GisamappingC:V7![k]suchthat(i)C(x)6=C(y)foranyarc(x;y)2A(#G)and(ii)C(x)=C(w)onlyifC(y)6=C(z)forallarcs(x;y)and(z;w)inA(#G).NoticethatC(x)inthischapterstandsforthecolourofthevertexxw.r.t.thecolouringC.Theorientedchromaticnumberofanorientedgraph#Gisthesmallestk2Nthatadmitsanorientedvertexk-colouringof#Gandisdenotedbyo(#G).Onecanalsoviewanorientedk-colouringasahomomorphismfrom#Gtoasuitableorientedgraphonkvertices.Ahomomorphismfromadirectedgraph#Gtoadirectedgraph#Hisa78 Chapter8.OrientedColouringmappingthatpreservesthearcs.Thatis,:V(#G)7!V(#H)isahomomorphismif((u);(v))2A(#H)foreveryarc(u;v)inA(#G).Hencewenotethato(#G)isthesmallestorderofanorientedgraph#Hsuchthatthereisahomomorphismfrom#Gto#H.TheorientedchromaticnumberofanundirectedgraphGdenotedo(G),isthemaximumo(#G)overoverallorientations#GofG.Thework[Sop01]ofSopenacontainsanumberofinterestingresultsonthisproblem.Anautomorphismofanorientedgraph#GisabijectionfromV(#G)toitselfthatpreservesedges,non-edgesanddirectionsoftheedges.Ifanautomorphismdoesnotmapanyvertextoitself,wecallitanon-xingautomorphism.RecallthatK2denotesanedgeandPkdenotesanundirectedpathonkvertices.AlsorecallthedenitionsofCartesianproductandStrongproductofgraphsdenedinChapter1.ForanygraphG,letGnandGndenoterespectivelythecartesianandstrongprod-uctsofGwithitselfntimes.ThegraphHd=K2discalledthehypercubeofdimensiond.InotherwordsHdisthecartesianproductofdedges.Thed-dimensionalhypergrid(mesh)denotedMdisthecartesianproductofdpaths.ThegraphMm;n=PmPniscalledanmngrid.WecallthegraphSm;n=PmPnanmnstrong-grid.Atournamentisanorientationofanundirectedcompletegraph.Letnbeaprimenumberoftheform4k+3.Letc1;c2;:::;cdbethenon-zeroquadraticresiduesofn.Itisknownthatd=n�1 2.Deneadirectedgraph#Tn=T(n;c1;:::;cd)overV=f0;1;:::;n�1gasfollows.Foreveryx;y2V;x6=y,(x;y)isanarcify=x+ciforsomei2[d].Itiswell-knownthat#TnisatournamentandiscalledthePaleytournament.AgraphGisarctransitiveifforanytwoarcse;finG,thereexistsanautomorphismmappingetof.Inotherwordsanarc-transitivegraphisagraphsuchthatanytwoarcsareequivalentundersomeelementofitsautomorphismgroup.Itisawell-knownfact[Fri70]thataPaleytournamentisarctransitive.Weobtainthefollowingresultsono(G)whenGisaproductofundirectedgraphsororientedgraphs.Wealsoproposeaconjecture.Theorem8.1.1.Let#Gbeanorientedgraphand#TbeaPaleytournamentsuchthato(#G)=j#Tj.Let#PkbeanyorientationofPk.Assumethatthereisahomomorphism:V(#G)7!V(#T).Theno(#G#Pk)=o(#G);8k2.Corollary8.1.1.1.Fortheorientedproduct#Hdofdorientededges,wehaveo(#Hd)=3.79 Chapter8.OrientedColouringConsiderthedotted2-pathsx-a-yandx-b-y.Wehavethefollowingtwocases.Case1:The2-pathsx-a-yandx-b-yareidenticallyoriented.WecanseefromFigure8.2thatforeverypossibleorientationofa2-path,thereareatleast2verticespandqsuchthattheears0-p-1and0-q-1areidenticallyoriented.Nowwecolouraandbsuitablybylookingattheorientationsofthearcsbetweenaandbandbetweenpandq.Case2:Theorientationsofx-a-yandx-b-yarenotthesame.Nowwenoticethatwehaveatleast2possiblecolours(sayfr;sg)satisfyingtheorientationofx-a-yaswellasadisjointsetof2-colours(sayft;ug)forx-b-y.Nowitiseasytocheckthatbetweenfr;sgandft;ug,thereisatleastonearcwhichsatisestheorientationof(a;b).Hencewecaninductivelyextendthecolouringto(#S2;n).ThelowerboundisexplicitfromtheorientedgraphdepictedinFigure8.3whichrequires8coloursinanyorientedcolouring. Figure8.3:2x5graphrequiring8colours.8.2.3.1ThesecondresultWenowprovethat10S3;n67.Herewehaveahugegapbetweentheupperandlowerbounds.Onceagain,wemaptheverticesto#T67toshowtheupperbound.Thelowerboundfollowsfromthefactthat,manyorientationsofS3;5(e.g.Figure8.4)requiresatleast10colours.Let#GbeanyorientationofS3;n.Asbefore,weconstructahomomorphismfrom#GtoaPaleytournament,namely#T67.Werstrecallsomedenitions.Anorientationvectorofsizemisasequence=(1;:::;m)inf0;1gm.GivenasequenceX=(x1;:::;xm)ofverticesinanorientedgraph#G,an-successorofXisavertexysuchthatforeachi,(xi;y)2A(#G)ifi=1and(y;xi)2A(#G)ifi=0.WesaythatanorientedgraphhasthepropertyP(m;k)ifforanysequenceXofmdistinctverticesin83 Chapter8.OrientedColouring Figure8.4:AnorientationofS3;5thatrequires10colours.thegraphandforanyorientationvectorinf0;1gm,thereareatleastk-successorsofX.NotethatthepropertyP(m;k)impliesthepropertyP(n;k)forallnm.Paleytournamentsaregoodcandidatesforsatisfyingsuchpropertiesandinparticular,weshallusethefollowingfact:Fact8.2.1.#T67satisesP(4;1)[Esp07]aswellasP(2;2)[BKN+99].WenowmapS3;nto#T67usinginductiononn.Basecase:n=1.Inthiscase,#Gisjustanorientedpathon3verticesanditiseasytomapthe3verticesof#Gto3distinctverticesin#T67usingtheP(2;2)property.Inductionstep:Assumethatthesubgraphinducedbythevertices(i;j):i2f0;1;2g;j2f0;1;:::;n�2garemappedto#T67.Now(0;n�1)hasexactly2neigh-bourswhichhavealreadybeenmapped(coloured)todistinctverticesin#T67andsince#T67hasthepropertyP(2;2),wecanextendthemappingsothat(0;n�1)ismappedtoavertexin#T67thatisdifferentfromtheimageof(2;n�2).Wenowseethat(1;n�1)hasfourneighboursthathavealreadybeenmappedtofourdistinctverticesin#T67andusingthepropertyP(4;1)of#T67,weextendthemappingtothevertex(1;n�1).Wecannowmap(2;n�1)to#T67aswellsinceithasthreedistinctcolouredneighboursandwemakeuseofthepropertyP(3;1)of#T67.Thiscompletestheproofofpart(ii)ofTheorem8.1.3. 84 A;nea;k+:sMa;Za;ya;ea;.cCe+. a;d:pa:=+ea;[a;a;TRa;~yad;ZRa;kMÁ.sa;vRa;~yal+.ea;.ca;nMaZa;a;~:Maya;~yana;a;~tya;nDaO;;va.sa:ÁÁItisbyaskingquestionsthatonendsthehid-dentruth.Scienceistheeyeofall.Apersonwithoutitisreallyblind.9RemarksandOpenproblemsInthischapter,wepresentsomeobservationsonthecolouringproblemsanditscurrentstatus.Wealsonotethetechnicaldifcultieswefacedindealingwiththeseproblemsandofthesewhatweexpecttobeeasyandwhatweexpecttobedifcult.Weconcludethesectionwithalistofafewopenproblems.9.1RemarksonAcyclicEdgeColouringWeareinterestedinnding0a,i.e.,theminimumnumberofcoloursneededtoacycli-callyedgecolouragivengraphG.Ithasbeenknownforsometimethatalinearnumberofcoloursissufcient.TheacyclicedgecolouringconjectureduetoFiamcik[Fia78](alsomentionedin[ASZ01])predictsthat
+2coloursaresufcientforanygraphofmaximumdegree
.Theconjecturehasbeenshowntobetrueforgraphshavingahighgirthaswellasforalmostallregulargraphs.Weshowthattheconjectureistrueforsomeotherclassesofgraphs.Weprovideatightboundfortheclassofouterplanargraphs,hararygraphsHn;2andtheclassoffullysubdividedgraphsinthiswork.Weimprovetheupperboundforgraphswithgirthatleast9usingprobabilisticarguments.Wethenprovideimprovedupperboundsfortheclassesofplanargraphs,three-foldgraphs,triangle-freeplanargraphsandtwo-foldgraphsmakinguseofthedischargingmethod.Wehadshown(earlier)thattheconjectureholdstrueforpartial2-trees,andcartesianproductsofpathsandcycles.Thiswork,donejointlywithmycolleagueRahulMuthuformsapartofhisPh.D.thesisandtheinterestedreadercanreferto[MNS06]and[MNS]forfurtherdetails.85 Chapter9.RemarksandOpenproblemsThefollowingtablegivesacollectionofclassesofgraphstogetherwithknownupperboundson0aforitsmembers.Therstcolumnmentionsthegraphclass,thesecondliststhebestknownupperbound,thirdcolumnstatesifitisknowntosatisfytheacyclicedgecolouringconjectureornot,andthelastcolumnindicatesifthemembersoftheclasscontainatmostalinearnumberofedges(intermsofitsorder). Class Upperbound0a Conjecture? Sparse? Outerplanar
+1 True Yes. Partial2-tree
+1 True Yes,jEj2jVj Partialtorus
+1 True Yes Kp,paprime.
+1 True No g2000
log

+2 True ? RandomRegular a.a.s.
+1 a.a.s.True. Yes. 2-degenerate
+1 True Yes PlanarGraphs 2
+29 Open Yes 3-foldgraphs 2
+29 Open Yes triangle-freeplanar
+6 Open Yes 2-foldgraph
+6 Open Yes Kn
+O(
2=3) Open No G:jEjcn 16
Open Yes Garbitrary 16
Open No WecanseefromTable9.1thatalltheclassesthatareknowntosatisfytheconjecture,exceptfortheclassofcompletegraphsofprimeorder,arereasonablysparse.Wemakeaspecialmentionofthea.a.sresultonrandomgraphswhichwasprovedusingLov´aszLocalLemma.Theproofmakesuseofatypicalpropertyofrandomregulargraphsthat,foragivendegreed,anytwosmallcycles(oflengthlessthansomeconstantdepend-ingond),inarandomd-regulargraphareseparatedbyalongpath(againaconstantdependingond).Wefeelthatitisreasonabletobelieveintheconjectureforthecaseswhenthenumberofedgesislinearinthenumberofvertices.Wealsoexpectthatitmightbepossibletoprovea(1+o(1))
boundforthegeneralgraphs.86 GlossaryNotationDescriptionPageList(u;v)Edge(arc)betweenuandvinanundirected(di-rected)graph5E(v)Thesetofedgesincidentwithavertexv5G2HCartesianproductofGandH8GHStrongproductofGandH8KnCompletegraphonnvertices6C(uv)Thecolouroftheedge(u;v)inthecolouringC470a(G)Smallestintegerksuchthatthereisaproperedgek-colouringthatisacyclic9C(v)SetofcoloursseenbyvintheedgecolouringC90k(G)k-intersectionchromaticindex9o(G)OrientedchromaticnumberofG10d(v)ordG(v)ThedegreeofvinG6k-intersectionedgecolouringAproperedgecolouringofagraphinwhichthenumberofcommoncoloursseenbyanypairofadjacentverticesisatmostk9k-vertexAvertexofdegreek6k+-vertexAvertexofdegreeatleastk6k�-vertexAvertexofdegreeatmostk6x-ybipathAbichromaticpathoflength4thatusescoloursxandy59acyclicedgecolouringAproperedgecolouringwheretheunionofanytwocolourclassesformanacyclicsubgraph9adjacentverticesuandvareadjacentif(u;v)2E(G)5adjacentedgesEdgeseandfareadjacentiftheyshareacommonvertex590 GlossaryNotationDescriptionPageListblockAmaximalconnectedsubgraphwithoutcut-vertices6cartesianproductGHsothatV(GH)=V(G)V(H)and([u1;u2];[v1;v2])2E(GH)ifeitheru1=v1and(u2;v2)2E(H)oru2=v2and(u1;v1)2E(G)8chordAnedgethatjoinstwonon-consecutiveverticesofacycle6,53connectedgraphAgraphinwhichthereisapathbetweenanypairofvertices6cut-vertexAvertexwhoseremovalincreasesthenumberofconnectedcomponentsofthegraph6cycleAsequencev0;v1;:::;vk=v0ofverticessuchthatallexcepttheendverticesaredistinctand(vi;vi+1)2Eforeachi6degreeNumberofedgesincidentwithavertex6directedgraphApair(V;A)wheretheelementsofAare2-elementorderedsubsetsofV7edgeAnelementofEinG=(V;E)5edger-colouringAmapC:E7![r]suchthatC(e)6=C(f)when-evereandfareadjacent9forestAgraphhavingnocyclesisaforest6graph(directedgraph)AnorderedpairofsetsG=(V;E)whereEisacollectionof2-element(ordered)subsetsofV5homomorphismAmappingbetweentheverticesoftwographsthatpreservestheedge(arc)relations791