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: 57885
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1IntroductionGivenasimplegraphG,wewriteG!(a1;:::;ak)eandsaythatGarrows(a1;:::;ak)eifforeveryedgek-coloringofG,amonochromaticKaiisforcedforsomecolori2f1;:::;kg.Likewise,forgraphsFandH,G!(F;H)eifforeveryedge2-coloringofG,amonochromaticFisforcedintherstcolororamonochromaticHisforcedinthesecond.DeneFe(a1;:::;ak;p)tobethesetofallgraphsthatarrow(a1;:::;ak)eanddonotcontainKp;theyareoftencalledFolkmangraphs.TheedgeFolkmannumberFe(a1;:::;ak;p)isthesmallestorderofagraphthatisamemberofFe(a1;:::;ak;p).In1970,Folkman[6]showedthatfork�maxfs;tg,Fe(s;t;k)exists.TherelatedproblemofvertexFolkmannumbers,whereverticesarecoloredinsteadofedges,ismorestudied[16,18]thanedgeFolkmannumbers,butwewillnotbediscussingthem.Therefore,wewillskiptheuseofthesuperscriptewhendiscussingarrowing,asitisusuallyusedtodistinguishbetweenedgeandvertexcolorings.In1967,Erd}osandHajnal[5]askedthequestion:DoesthereexistaK4-freegraphthatisnottheunionoftwotriangle-freegraphs?ThisquestionisequivalenttoaskingfortheexistenceofaK4-freegraphsuchthatinanyedge2-coloring,amonochromatictriangleisforced.AfterFolkmanprovedtheexistenceofsuchagraph,thequestionthenbecametondhowsmallthisgraphcouldbe,orusingtheabovenotation,whatisthevalueofFe(3;3;4).Priortothispaper,thebestknownboundsforthiscasewere19Fe(3;3;4)941[21,4].FolkmannumbersarerelatedtoRamseynumbersR(s;t),whicharedenedastheleastpositivensuchthatany2-coloringoftheedgesofKnyieldsamonochromaticKsintherstcolororamonochromaticKtinthesecondcolor.Usingthearrowingoperator,itisclearthatR(s;t)isthesmallestnsuchthatKn!(s;t).TheknownvaluesandboundsforvarioustypesofRamseynumbersarecollectedandregularlyupdatedbythesecondauthor[20].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. smallestorderofanyFolkmangraph(withxedparameters)seemstobedicult,andsomerelatedRamseygraphcoloringproblemsareNP-hardorlieevenhigherinthepolynomialhierarchy.Forexample,Burr[2]showedthatarrowing(3;3)iscoNP-complete,andSchaefer[22]showedthatforgeneralgraphsF,G,andH,F!(G;H)isP2-complete.3ArrowingviaMAX-CUTBuildingo Spencer'sandothermethods,DudekandRodl[4]in2008showedhowtoconstructagraphHGfromagraphG,suchthatthemax-imumsizeofacutofHGdetermineswhetherornotG!(3;3).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)ifandonlyifMC(HG)2t4(G).ThereisaclearintuitionbehindTheorem1thatwewillnowdescribe.Anyedge2-coloringofGcorrespondstoabipartitionoftheverticesinHG.IfatrianglecoloredinGisnotmonochromatic,thenitsthreeedges,whichareverticesofHG,willbeseparatedinthebipartition.Ifwetreatthisbipartitionasacut,thenthesizeofthecutwillcounteachtriangletwiceforthetwoedgesthatcrossit.Sincethereisonlyonetriangleinagraphthatcontainstwogivenedges,thise ectivelycountsthenumberofnon-monochromatictriangles.Therefore,ifitispossibletondacutthathassizeequalto2t4(G),thensuchacutdenesanedgecoloringofGthathasnomonochromatictriangles.However,ifMC(HG)2t4(G),thenineachcoloring,allthreeedgesofsometriangleareinonepartandthus,G!(3;3).Abenetofconvertingtheproblemofarrowing(3;3)toMAX-CUTisthatthelatteriswell-knownandhasbeenstudiedextensivelyincomputerscienceandmathematics(seeforexample[3]).ThedecisionproblemMAX-CUT(H;k)askswhetherornotMC(H)k.ItisknownthatMAX-CUTisNP-hardandthisdecisionproblemwasoneofKarp's21NP-completeproblems[13].Inourcase,G!(3;3)ifandonlyifMAX-CUT(HG;2t4(G))doesn'thold.SinceMAX-CUTisNP-hard,anattemptisoftenmadetoapproximateit,suchasintheapproachespresentedinthenexttwosections. 3.2Goemans-WilliamsonMethodTheGoemans-WilliamsonMAX-CUTapproximationalgorithm[9]isawell-known,polynomial-timealgorithmthatrelaxestheproblemtoasemi-deniteprogram(SDP).ItinvolvestherstuseofSDPincombinatorialapproximationandhassinceinspiredavarietyofothersuccessfulalgo-rithms(seeforexample[12,8]).Thisrandomizedalgorithmreturnsacutwithexpectedsizeatleast0.87856oftheoptimalvalue.However,inourcase,allthatisneededisafeasiblesolutiontotheSDP,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):DeneonepartofthecutasS=fijyi=1g.Sinceinourcaseallgraphsareweightless,wewillusewi;j=(1iffi;jg2E(H);0otherwise:Next,theintegerprogram(3)isrelaxedbyextendingtheproblemtohigherdimensions.Eachyi2f�1;1gisnowreplacedwithavectorontheunitspherevi2Rn,asfollows:Maximize1 2Xiwi;j(1�vivj)(4)subjectto:kvik=1foralli2V(H):IfwedeneamatrixYwiththeentriesyi;j=vivj,thatis,theGrammatrixofv1;:::;vn,thenyi;i=1andYispositivesemi-denite.Therefore,(4)isasemideniteprogram.3.3SomeCasesofArrowingUsingtheGoemans-Williamsonapproach,wetestedawidevarietyofgraphsforarrowingbyndingupperboundsonMAX-CUT.Thesegraphsincluded 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"isthesolutionof(4)fromSDPLR-MCandSBmethod.G786isthegraphofTheorem3.oneoftheMAX-CUTmethodswouldbeabletoprovearrowing.Indeed,wewereabletodosowithL(785;53).NoticethatalloftheupperboundsforMC(HL(785;53))are857220,thesameas2t4(L(785;53)).OurgoalwasthentoslightlymodifyL(785;53)sothatthisvaluebecomessmaller.LetG786denotethegraphL(785;53)withoneadditionalvertexconnectedtothefollowing60vertices:{0,1,3,4,6,7,9,10,12,13,15,16,18,19,21,22,24,25,27,28,30,31,33,34,36,37,39,40,42,43,45,46,48,49,51,52,54,55,57,58,60,61,63,66,69,201,204,207,210,213,216,219,222,225,416,419,422,630,642,645}G786isstillK4-free,has61290edges,andhas428881triangles.TheupperboundcomputedfromtheSDPsolversforMC(HG786)is857753.Wedidnotndanicedescriptionforthevectorsofthissolution.SoftwareimplementingSpeeDPbyGrippoetal.[10],analgorithmdesignedtosolvelargeMAX-CUTSDPrelaxations,wasusedbyRinaldi(oneoftheauthorsof[10])toanalyzethisgraph.Hewasabletoobtainthebounds857742MC(HG786)857750,whichagreeswith,andimprovesoverourupperboundcomputation.Since2t4(G786)=857762,wehavebothfromourtestsandhisSpeeDPtestthatG786!(3;3),andthefollowingmainresult.Theorem3.Fe(3;3;4)786:WenotethatndingalowerboundonMAX-CUT,suchasthe857742MC(HG786)boundfromSpeeDP,followsfromndinganactualcutofacertainsize.Thismethodmaybeuseful,asndingacutofsize2t4(G)showsthatG6!(3;3). 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