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1、-------AbstractIn1966,inordertosettleRingel’sconjecture,Rosaintroducestheconceptofgraphlabellings:Agraphlabellingisamappingfromthevertexsetofagraphtoasetofintegers.Byvariousconstraintswehavemanytypesofgraphlabellings.Clearly,graphlabellingisanimportantbranchofgraphtheorysinceitcanbeappliedtoawideofs
2、cientificareas.However,therearemanynewproblemsyieldinginattackingfamousconjectures,suchasGracefulTreeConjecture,StronglyGracefulTreeConjecture,Odd-gracefulTreeConjecture,FelicitousTreeConjectureandsoon.Basedontheknownconclusionsandresultsongraphlabellings,myresearchingmainlyfocusonthefollowingaspects
3、:ChapterOnedistributesasimpleintroductiontothedevelopmentofgraphtheoryandgraphlabellings.Basicterminologyandnotationofgraphtheoryaredefined,andthedefinitions,conjecturesandsomeresultsofgraphlabellingsaregiven.ChapterTwoworksmainlyongracefullabellingofgraphs.Welistsomeresultsincurrentresearchinggracefu
4、llabellingofgraphs,andtrytoshowasummaryofstudyinggracefullabellingofgraphs.Weinvestigateparticulargracefullabellingsinordertofindsomeregularitiesofgracefullabellingsthatcanbeusedtoconstructlargescaleofgracefulgraphs,anddotheoperationofmovingedgeprincipleforattackingGracefulTreeConjecture.InChapterThr
5、ee,we,firstofall,introducethestudyonodd-gracefullabelingsofgraphs.BasedonthestudyofGnanajothi,forsolvedtheconjectureproposedbyBarrientos:Everybipartitegraphisodd-graceful,andobtainedsomeresults.Wesolvetheodd-gracefulnessofseveralparticularclassesofbipartitegraphs.i-----------TheFourthChapteristostudy
6、graphfelicitouslabellingsonwhichafewofresultshasbeenobtained.Weprovethatsomeparticularbipartitegraphsadmitfelicitouslabellings,andfurthermore,byoutexperiencesonworkingthistopic,weproposeaconjecture:Everybipartitegraphadmitsfelicitouslabellings.Keywords:gracefullabelling;odd-gracefullabelling;felicit
7、oulabelling;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分析课题之一,它们在众多的科学领域有着广泛的应用，
