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1、南京大学学报(自然科学)第37卷 第4期Vol.37,No.4JOURNALOFNANJINGUNIVERSITY2001年7月July,2001(NATURALSCIENCES)XVertexOpancyclicMultipartiteTournamentsZhouGoufei,ZhangKemin,XueGuohe(DepartmentofMathematics,NanjingUniversity,Nanjing,210093,China)Abstract:Ac2partitetournamentisanorientedgraphobtainedfromacomp
2、letec2partitegraph.Amultipartitetournamentisac2partitetournamentwithc≥2.Tbeingamultipartitetournament,we+-defineig(T)=max
3、d(x)-d(y)
4、overallpairsofverticesx,y∈V(T).WeprovethatifV1,V2,⋯,Vcarethepartitesetsofac2partite(c≥3)tournamentT,with
5、V1
6、≤
7、V2
8、≤⋯≤
9、V1
10、+1andig(T)≤1,thenTisvertex2pancycli
11、c.Keywords:multipartitetournaments,cycle,vertex2pancyclicityAc2partitetournamentisanorientedgraphobtainedfromacompletec2partitegraph.Amultipartitetournamentisac2partitetournamentwithc≥2.IfTisamultipartitetournament3andx∈V(T),wedenoteV(x)thepartitesettowhichxbelongsanddenotevT=mini{
12、Vi
13、}
14、,whereViarepartitesetsofT.Afactorinadigraphisaspanningcollectionofvertexdisjointcycles.AdigraphDispancyclicifitcontainscyclesoflengths3,4,⋯,
15、V(D)
16、andDisvertex2pancyclicifforeachw∈V(D)therearecyclesoflengths3,4,⋯,
17、V(D)
18、con2+-tainingw.ThelocalirregularityofadigraphDisdefinedasil(D)=max
19、d(
20、x)-d(x)
21、overallverticesx∈V(D)andthetheglobalirregularityisdefinedasig(D)=max+-
22、d(x)-d(y)
23、overallpairsofverticesx,y∈V(D).AdigraphDisstrongifforeachx,y∈V(D),thereisapathfromxtoy.AdigraphDisk2strongifD2XisstrongforallsetsofverticesX,
24、X
25、26、namentswithc≥4arepancyclic.Infact,[1]Yeoprovesthatwhenc≥5,allregularc2partitetournamentsarevertex2pancyclic.Ourmainresultsarebasedonthetechnicsof[1].Asforsurveysonmultipartitetournaments,see[2]and[1],33.1TerminologyandnotationsWeshallassumethatthereaderisfamiliarwiththestandardterminolo
27、gyongraphsandXFoundationitem:NSFC(19871041)Receiveddate:2000-11-15©1995-2006TsinghuaTongfangOpticalDiscCo.,Ltd.Allrightsreserved.·478·南京大学学报(自然科学)第37卷digraphsandreferthereaderto[3].LetD=(V,A)beadigraph.Ifxy∈A(D),thenwesaythatxdominatesyandyisdominatedbyx.Wealsodenotethisbyx→y.I
