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1、AbstractIn1966,inordertosettleRingel’sconjecture,Rosaintroducestheconceptofgraphlabellings:Agraphlabellingisamappingfromthevertexsetofagraphtoasetofintegers.Byvariousconstraintswehavemanytypesofgraphlabellings.Clearly,graphlabellingisanimportantbranchofgraphtheorysinceitca
2、nbeappliedtoawideofscientificareas.However,therearemanynewproblemsyieldinginattackingfamousconjectures,suchasGracefulTreeConjecture,StronglyGracefulTreeConjecture,Odd-gracefulTreeConjecture,FelicitousTreeConjectureandsoon.Basedontheknownconclusionsandresultsongraphlabelling
3、s,myresearchingmainlyfocusonthefollowingaspects:ChapterOnedistributesasimpleintroductiontothedevelopmentofgraphtheoryandgraphlabellings.Basicterminologyandnotationofgraphtheoryaredefined,andthedefinitions,conjecturesandsomeresultsofgraphlabellingsaregiven.ChapterTwoworksmain
4、lyongracefullabellingofgraphs.Welistsomeresultsincurrentresearchinggracefullabellingofgraphs,andtrytoshowasummaryofstudyinggracefullabellingofgraphs.Weinvestigateparticulargracefullabellingsinordertofindsomeregularitiesofgracefullabellingsthatcanbeusedtoconstructlargescaleo
5、fgracefulgraphs,anddotheoperationofmovingedgeprincipleforattackingGracefulTreeConjecture.InChapterThree,we,firstofall,introducethestudyonodd-gracefullabelingsofgraphs.BasedonthestudyofGnanajothi,forsolvedtheconjectureproposedbyBarrientos:Everybipartitegraphisodd-graceful,an
6、dobtainedsomeresults.Wesolvetheodd-gracefulnessofseveralparticularclassesofbipartitegraphs.iTheFourthChapteristostudygraphfelicitouslabellingsonwhichafewofresultshasbeenobtained.Weprovethatsomeparticularbipartitegraphsadmitfelicitouslabellings,andfurthermore,byoutexperienc
7、esonworkingthistopic,weproposeaconjecture:Everybipartitegraphadmitsfelicitouslabellings.Keywords:gracefullabelling;odd-gracefullabelling;felicitoulabelling;bipar-titegraph;matchingverticesii摘要1966年,为了解决Ringel’sconjecture,Rosa等人提出了图的标号的概念,所谓图的标号是指:一个图的顶点标号是图的顶点集到整数集的映射,而根据对
8、边标号的不同要求,产生了各种各样的图的标号.图的标号是图论中十分重要的研究课题之一,它们在众多的科学领域有着广泛的应用，许多研究者在此方面作了大量的工作,但图标号仍有很多问题没有
