Lange Stanis law P Radziszowski Department of Computer Science Rochester Institute of Technology Rochester NY 14623 arl9577spr csritedu and Xiaodong Xu Guangxi Academy of Sciences Nanning Guangxi 530007 China xxdmathssinacom Abstract In 1967 Erd733o ID: 57886
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1IntroductionGivenasimplegraphG,wewriteG!(a1;:::;ak)eandsaythatGarrows(a1;:::;ak)eifforeveryedgek-coloringofG,amonochro-maticKaiisforcedforsomecolori2f1;:::;kg.Likewise,forgraphsFandH,G!(F;H)eifforeveryedge2-coloringofG,amonochro-maticFisforcedintherstcolororamonochromaticHisforcedinthesecond.DeneFe(a1;:::;ak;p)tobethesetofallgraphsthatarrow(a1;:::;ak)anddonotcontainKp;theyareoftencalledFolkmangraphs.TheedgeFolkmannumberFe(a1;:::;ak;p)isthesmallestorderofagraphthatisamemberofFe(a1;:::;ak;p).In1970,Folkman[6]showedthatfork�maxfs;tg,Fe(s;t;k)exists.TherelatedproblemofvertexFolkmannumbers,whereverticesarecoloredinsteadofedges,ismorestudied[15,17]thanedgeFolkmannumbers,butwewillnotbediscussingthem.In1967,Erd}osandHajnal[5]askedthequestion:Doesthereex-istaK4-freegraphthatisnottheunionoftwotriangle-freegraphs?ThisquestionisequivalenttoaskingfortheexistenceofaK4-freegraphsuchthatinanyedge2-coloring,amonochromatictriangleisforced.AfterFolkmanprovedtheexistenceofsuchagraph,thequestionthenbecametondhowsmallthisgraphcouldbe,orusingtheabovenotation,whatthevalueofFe(3;3;4)is.Priortothispa-per,thebestknownboundsforthiscasewere19Fe(3;3;4)941[20,4].FolkmannumbersarerelatedtoRamseynumbersR(s;t),denedastheleastpositivensuchthatany2-coloringoftheedgesofKnyieldsamonochromaticKsintherstcolororamonochromaticKtinthesecond.Usingthearrowingoperator,itisclearthatR(s;t)isthesmallestnsuchthatKn!(s;t)e.TheknownvaluesandboundsforvarioustypesofRamseynumbersarecollectedandregularlyupdatedbythesecondauthor[19].Wewillbeusingstandardgraphtheorynotation:V(G)andE(G)forthevertexandedgesetsofgraphG,respectively.Acutisapartitionoftheverticesofagraphintotwosets,SV(G)and S=V(G)nS.Thesizeofacutisthenumberofedgesthatjointhetwosets,thatis,jffu;vg2E(G)ju2Sandv2 Sgj.MAX-CUTisawell-knownNP-hardcombinatorialoptimizationproblemwhichasksforthemaximumsizeofacutofagraph.2 upperbound.Lin[12]obtainedalowerboundon10in1972with-outthehelpofacomputer.All659graphson15verticeswit-nessingFe(3;3;5)=15[18]containK4,thusgivingthebound16Fe(3;3;4).In2007,twooftheauthorsofthispapergaveacomputer-freeproofof18Fe(3;3;4)andimprovedthelowerboundfurtherto19withthehelpofcomputations[20].ThelonghistoryofFe(3;3;4)isnotonlyinterestinginitselfbutalsogivesinsightintohowdiculttheproblemis.FindinggoodboundsonthesmallestorderofanyFolkmangraph(withxedparameters)seemstobedicult,andsomerelatedRamseygraphcoloringproblemsareNP-hardorlieevenhigherinthepolynomialhierarchy.Forexample,Burr[2]showedthatarrowing(3;3)eiscoNP-complete,andSchaefer[21]showedthatforgeneralgraphsF,G,andH,F!(G;H)isP2-complete.3ArrowingviaMAX-CUTBuildingo Spencer'sandothermethods,DudekandRodl[4]in2008showedhowtoconstructagraphHGfromagraphG,suchthatthemaximumsizeofacutofHGdetermineswhetherornotG!(3;3)e.TheyconstructthegraphHGasfollows.TheverticesofHGaretheedgesofG,sojV(HG)j=jE(G)j.Fore1;e22V(HG),ifedgesfe1;e2;e3gformatriangleinG,thenfe1;e2gisanedgeinHG.Lett4(G)denotethenumberoftrianglesingraphG.Clearly,jE(HG)j=3t4(G).LetMC(H)denotetheMAX-CUTvalueofgraphH.Theorem1(DudekandRodl[4]).G!(3;3)eifandonlyifMC(HG)2t4(G).ThereisaclearintuitionbehindTheorem1thatwewillnowdescribe.Anyedge2-coloringofGcorrespondstoabipartitionoftheverticesinHG.IfatrianglecoloredinGisnotmonochromatic,thenitsthreeedgeswhichareverticesinHGwillbeseparatedinthebipartition.Ifwetreatthisbipartitionasacut,thenthesizeofthecutwillcounteachtriangletwiceforthetwoedgesthatcrossit.4 stillshowarrowing.Weappliedmultiplestrategiesforremovingsetsofverticesandmostweresuccessful.Thisledtothefollowingtheorem:Theorem2.Fe(3;3;4)860.Proof.ForagraphGwithverticesZn,deneC=C(d;k)=fv2V(G)jv=idmodn;for0ikg.LetG=G941,d=2,k=81,andGCbethegraphinducedonV(G)nC(d;k).ThenGChas860vertices,73981edgesand542514triangles.Usingtheupperbound(1)andtheMATLABeigsfunction,weobtainMC(HGC)10849671085028=2t4(Gc):(2)Therefore,GC!(3;3)e.2Noneofthemethodsusedallowedfor82ormoreverticestoberemovedwithouttheupperboundonMCbecominglargerthan2t4.3.2Goemans-WilliamsonMethodTheGoemans-WilliamsonMAX-CUTapproximationalgorithm[8]isawell-known,polynomial-timealgorithmthatrelaxestheprob-lemtoasemideniteprogram(SDP).ItinvolvestherstuseofSDPincombinatorialapproximationandhassinceinspiredavarietyofothersuccessfulalgorithms(seeforexample[13]).Thisrandomizedalgorithmreturnsacutwithexpectedsizeatleast0.87856oftheoptimalvalue.However,inourcase,allthatisneededisthesolu-tiontotheSDP,asitgivesanupperboundonMC(H).AbriefdescriptionoftheGoemans-Williamsonrelaxationfollows.TherststepinrelaxingMAX-CUTistorepresenttheproblemasaquadraticintegerprogram.GivenagraphHwithV(H)=f1;:::;ngandnonnegativeweightswi;jforeachpairofverticesfi;jg,wecanwriteMC(H)asthefollowingobjectivefunction:Maximize1 2Xiwi;j(1�yiyj)(3)subjectto:yi2f�1;1gforalli2V(H):6 G 2t4(G) min SDP L(127;5) 19558 20181 20181 L(457;6) 347320 358204 358204 L(761;3) 694032 731858 731858 L(785;53) 857220 857220 857220 G786 857762 857843 857753 Table2:PotentialFe(3;3;4)graphsGandupperboundsonMC(HG),where\min"isthebound(1)and\SDP"istheso-lutionof(4)fromSDPLR-MC,SDPLR,andSBmethod.G786isthegraphofTheorem3.andSDPupperbounds.MultipleSDPsolversthatweredesignedtohandlelarge-scaleSDPandMAX-CUTproblemswereused.Specif-ically,wemadeuseoftwoversionsofSDPLRbySamuelBurer[1],bothusinglow-rankfactorization.SDPLR-MCisaversionofthesoft-warespecicallyfortheMAX-CUTrelaxation.TheregularsoftwareSDPLRismeantforanySDP.SBmethodbyChristophHelmberg[10]implementsaspectralbundlemethodandwasalsoappliedsuccess-fullyinourexperiments.Table2liststheresults.Inallcases,allthreesolversgavethesameresult.NotethatalthoughnoneofthecomputedupperboundsoftheL(n;s)graphsimplyarrowing(3;3)e,allSDPboundsmatchthoseoftheminimumeigenvaluebound.Thisisdistinctfromotherfamiliesofgraphs,includingthosein[4],astheSDPboundisusuallytighter.Thus,thesegraphsweregivenfurtherconsideration.L(127;5)wasgivenparticularattention,asitisthesamegraphasG127,whereV(G127)=Z127andE(G127)=ffx;ygjx�y 3mod127g(thatis,thegraphG(127;3)asdenedintheprevioussection).IthasbeenconjecturedbyExoothatG127!(3;3)e.Healsosuggestedthatsubgraphsinducedonlessthan100verticesofG127mayaswell.FormoreinformationonG127see[20].NumerousattemptsweremadeatmodifyingthesegraphsinhopesthatoneoftheMAX-CUTmethodswouldbeabletoprovearrowing.Indeed,wewereabletodosowithL(785;53).NoticethatalloftheupperboundsforMC(HL(785;53))are857220,thesameas2t4(L(785;53)).OurgoalwasthentoslightlymodifyL(785;53)8 5AcknowledgmentsThethirdauthorissupportedbytheGuangxiNaturalScienceFoun-dation(2011GXNSFA018142).WewouldliketothankG.RinaldiandL.GrippofortheirenthusiasticaidinthecomputationofMAX-CUTboundswiththeirSpeeDPalgorithm[9].References[1]SamuelBurerandRenatoD.C.Monteiro.Anonlinearpro-grammingalgorithmforsolvingsemideniteprogramsvialow-rankfactorization.MathematicalProgramming(SeriesB),95(2):329{357,February2003.Softwareavailableathttp://dollar.biz.uiowa.edu/~sburer.[2]StefanA.Burr.1976.ResultmentionedinbookbyM.GareyandD.Johnson.ComputersandIntractability:AGuidetotheTheoryofNP-Completeness,1979.W.H.FreemanandCompany.[3]ClaytonW.Commander.MaximumCutProblem,MAX-CUT.InChristodoulosFloudasandPanosPardalos,editors,Encyclo-pediaofOptimization,pages1991{1999.Springer,secondedi-tion,2009.[4]AndrzejDudekandVojtechRodl.OntheFolkmanNumberf(2;3;4).ExperimentalMathematics,17(1):63{67,2008.[5]PaulErd}osandAndrasHajnal.Researchproblem2{5.JournalofCombinatorialTheory,2:104,1967.[6]JonFolkman.Graphswithmonochromaticcompletesubgraphsineveryedgecoloring.SIAMJournalonAppliedMathematics,18(1):19{24,January1970.[7]PeterFranklandVojtechRodl.Largetriangle-freesubgraphsingraphswithoutK4.GraphsandCombinatorics,2:135{144,1986.10 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