组合设计大集问题PPT课件

-,1,组合设计的大集与超大集已解决的和待解决的,康庆德河北师范大学数学研究所2009.7.29,Kirkmansschoolgirlproblem(T.P.Kirkman1847),SUNMONTUEWEDTHUFRISAT,大集问题的起源和背景,ThomasPenyngtonKirkman(英格兰教会的教区长),2020/5/15,-,3,a,50,31,01,41,51,00,10,11,20,40,61,30,60,21SUNMONTUEWEDTHUFRISAT,经典三元系的大集与超大集LSTS,LMTS,LDTS,LHTS,OLSTS,OLMTS,OLDTS.其它三元系的大集与超大集LT1,LT2,LT3,OLT1,OLT2,OLT3;LESTS,LEMTS,LEDTS.纯的有向三元系的大集与超大集LPMTS,LPDTS,OLPMTS,OLPDTS.可分解（几乎可分解）三元系的大集与超大集LKTS,LRMTS,LRDTS,OLKTS,OLRMTS,OLRDTS.LARMTS,LARDTS,OLARMTS,OLARDTS.图设计的大集与超大集路分解：P3-LGD,OP3-LGD,P3-OLGD,OP3-OLGD,P4-LGD,Pk-LGD.星(圈)分解：K1,3-LGD,K1,4-LGD,K1,k-LGD;C4-LGD.Hamilton圈(路)分解：LHCD,LHPD,LDHCD,LDHPD;LCS(v,v-1,).可分组设计的大集LGDD.拉丁方的大集LDILS,Golfdesign,.t-设计的大集LS(t,k,v),基本文献,C.J.Colbourn&J.H.Dinitz,TheCRCHandbookofCombinatorialDesigns,CRCPress(SecondEdition),2006.J.H.Dinitz&D.R.Stinson,ContemporaryDesignTheoryAcollectionofsurveys,Wiley,1992.Q.D.Kang,Onlargesetsofcombinatorialdesigns,AdvanceofMathematics,32(2003),269-284.,2020/5/15,-,6,A.经典三元系的大集与超大集LSTS,LMTS,LDTS,LHTS,OLSTS,OLMTS,OLDTS,OLHTS,LPMTS,LPDTS,OLPMTS,OLPDTS.,Sixtypesoftriplesandthecorrespondingtriplesystems,Theexistenceoftriplesystems,2020/5/15,-,9,经典三元系大集的存在谱,*AshortproofforLSTS(v)wasgivenbyL.Ji.*Lindner,Street,Colbourn,RosaandTeirlinckalsogavesomeresultsforLMTS(v).,2020/5/15,-,11,经典三元系超大集的存在谱,*遗留问题：,2020/5/15,-,12,LargeSetsofpureorintedtriplesystems,*遗留问题:,2020/5/15,-,13,OverlargeSetsofpureorintedtriplesystems,*遗留问题:,2020/5/15,-,14,B.其它三元系的大集与超大集LT1,LT2,LT3,OLT1,OLT2,OLT3,LESTS,LEMTS,LEDTS.,mixedtriplesystemsT1,T2,T3,ConclusionsforLTiandOLTi,*遗留问题:,Q.Kang,Z.Tian&L.Yuan,2003-2007,Aclassicaltripleconsistsofthreedistinctelements,butanextendedtripleisallowedtocontainrepeatedelements.,STS,MTS,DTS(LSTS,LMTS,LDTS)ESTS,EMTS,EDTS(LESTS,LEMTS,LEDTS).,Extendedtriplesystems,ExamplesofLESTS,ExamplesofLEMTS,ExamplesofLEDTS,AconstructionforLEDTS(7),ThereexistLEDTS(v)for,ThereexistLESTS(v)andLEMTS(v),2020/5/15,-,25,C.可分解三元系的大集与超大集LKTS,LRMTS,LRDTS,OLKTS,OLRMTS,OLRDTS,LARMTS,LARDTS,OLARMTS,OLARDTS.,2020/5/15,-,26,LKTS,LRMTS,LRDTS,LARDTS,LARMTS,RDTS,KTS,OLKTS,OLRDTS,ARDTS,OLARDTS,ARMTS,OLARMTS,RMTS,OLRMTS,DTS,MTS,STS,Theexistenceoftriplesystemswithresolvability,KTS(9)LKTS(9),KnownLKTS(v)andsmallorders405,LRMTS(12),OLARDTS(10),TriplingconstructionsforLKTS,ProductconstructionsforLKTS,TheexistenceforLKTS(v),Kirkman,Denniston,Schreiber,L.Wu,Y.Chang,G.Ge,L.Zhu,S.Zhang,J.Lei,L.Ji.before2005,Q.Kang&L.Yuan,2007,TriplingconstructionsforLRMTSandLRDTS,ProductconstructionsforLRMTSandLRDTS,TheexistenceofLRMTS(v)andLRDTS(v),Q.Kang,J.Lei,Z.Tian,L.Yuan,Y.Chang,J.Zhou,2000Kang&Tian,2006Kang&Yuan,2005Kang&Yuan,2002Kang&Tian,1996Kang&Lei,2020/5/15,-,38,D.图设计的大集与超大集P3-LGD,OP3-LGD,P3-OLGD,OP3-OLGD,P4-LGD,Pk-LGD.K1,3-LGD,K1,4-LGD,K1,k-LGD,C4-LGD.HCD,LHPD,LDHCD,LDHPD,LCS(v,v-1,).,LargesetsofP3-decompositions,LargesetsoforientedP3-decompositions,LargesetsofP3-decompositions,LargesetsoforientedP3-decompositions,Y.Zhang&Q.Kang,2006,Q.Kang&Y.Zhang,2002,2020/5/15,-,42,OverlargesetsoforientedP3-decompositions,OverlargesetsofP3-decompositions,Y.Liu&Q.Kang,2009,Y.Liu&Q.Kang,2008,ExamplesofLGDforcycle,pathandstar,2020/5/15,-,45,在完全图中：LargeSetsofHamiltoncycle(path)decompositions,*无遗留问题,ConstructionofLHCD2(v),在二分图中：LargeSetsofHamiltoncycle(path)decompositions,*无遗留问题,LargeSetsofdirectedHamiltoncycle(path)decompositions,*遗留问题:,1989Kang,2005Zhao&Kang,Tuscansquaresoforder6withorwithoutacross,withoutcross,withcross,Romansquare,Apairofrelasionaltuscansquaresoforder7,GoodtuscansquareT-TandT-1arerelational,问题:对于大于19的奇素数阶数是否存在atuscansquarewithacross？是否存在apairofrelationaltuscansquarewithacross？是否存在agoodtuscansquarewithacross？(阶数=3mod4),Largesetsofcyclesystems,问题:是否存在,2020/5/15,-,57,E.其它设计的大集与超大集Latinsquares,idempotentquasigroups,groupdivisibledesigns,golfdesigns,t-designs.,LargesetofidempotentLatinsquaresoforder5,Golfdesignoforder7(idempotentsymmetriclatinsquares),Overlargesetsofidempotentquasigroups,Largesetsofidempotentquasigroup,共轭不变子群大集超大集幂等拉丁方大集幂等拉丁方超大集单位元群n3,n6n3,n6幂等对称拉丁方大集幂等对称拉丁方超大集二阶子群n1mod2,n5n1mod2,n3LMTSOL