简介:LetMbeacompactorientable3-manifoldwithMconnected.IfV∪SWisaHeegaardsplittingofMwithdistanceatleast6,thenthe-stabilizationofV∪SWalongMisunstabilized.HenceMhasatleasttwounstabilizedHeegaardsplittingswithdifferentgenera.ThebasictoolisaresultondiskcomplexgivenbyMasurandSchleimer.
简介:InapaperbyZhangandChenetal.(see[11]),aconjecturewasmadeconcerningtheminimumnumberofcolorsXat(G)requiredinapropertotal-coloringofGsothatanytwoadjacentverticeshavedifferentcolorsets,wherethecolorsetofavertexvisthesetcomposedofthecolorofvandthecolorsincidenttov.WefindtheexactvaluesofXat(G)andthusverifytheconjecturewhenGisaGeneralizedHalingraphwithmaximumdegreeatleast6.AgeneralizedHalingraphisa2-connectedplanegraphGsuchthatremovingalltheedgesoftheboundaryoftheexteriorfaceofG(thedegreesoftheverticesintheboundaryofexteriorfaceofGareallthree)givesatree.
简介:LetP(G,λ)bethechromaticpolynomialofagraphG.TwographsGandHaresaidtobechromaticallyequivalent,denotedG~H,ifP(G,λ)=P(H,λ).Wewrite[G]={H|H~G}.If[G]={G},thenGissaidtobechromaticallyunique.Inthispaper,wefirstcharacterizecertaincomplete6-partitegraphswith6n+1verticesaccordingtothenumberof7-independentpartitionsofG.Usingtheseresults,weinvestigatethechromaticityofGwithcertainstarormatchingdeleted.Asaby-product,manynewfamiliesofchromaticallyuniquecomplete6-partitegraphswithcertainstarormatchingdeletedareobtained.
简介:GeneralizedSteiriertriplesystems,GS(2,3,n,g),areequivalenttomaximumconstantweightcodesoveranalphabetofsizeg+1withdistance3andweight3inwhicheachcodewordhaslengthn.ThenecessaryconditionsfortheexistenceofaGS(2,3,n,g)are(n-1)g≡0(mod2),n(n-1)g2≡0(mod6),andn≥g+2.Thesenecessaryconditionsareshowntobesufficientbyseveralauthorsfor2≤g≤11.Inthispaper,threenewresultsareobtained.First,itisshownthatforanygiveng,g≡0(mod6)andg≥12,ifthereexistsaGS(2.3.n.g)foralln,g+2≤n≤7g+13.thenthenecessaryconditionsarealsosufficient.Next,itisalsoshownthatforanygiveng,g≡3(mod6)andg≥15,ifthereexistsaGS(2,3,n,g)foralln,n≡1(mod2)andg+2≤n≤7g+6,thenthenecessaryconditionsarealsosufficient.Finally,asanapplication,itisprovedthatthenecessaryconditionsfortheexistenceofaGS(2,3,n,g)arealsosufficientforg=12,15.