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组合设计的大集与超大集 已解决的和待解决的

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. Kirkman’s schoolgirl problem ( T. P. Kirkman 1847 ). 大集问题的起源和背景. SUN MON TUE WED THU FRI SAT. Thomas Penyngton Kirkman ( 英格兰教会的教区长 )

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组合设计的大集与超大集 已解决的和待解决的

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  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

  2. Kirkman’s schoolgirl problem(T. P. Kirkman1847) 大集问题的起源和背景 SUN MON TUE WED THU FRI SAT Thomas Penyngton Kirkman (英格兰教会的教区长) <Lady’s and Gentleman’s Diary>

  3. {a,50,31},{01,41,51},{00,10,11},{20,40,61},{30,60,21} SUN MON TUE WED THU FRI SAT (1850 Sylvester , Cayley 1974 Denniston)

  4. 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：K 1,3-LGD, K 1,4-LGD, K 1,k-LGD ;C4-LGD. Hamilton圈(路)分解：LHCD, LHPD, LDHCD, LDHPD ; LCS(v,v-1,λ) . • 可分组设计的大集LGDD. • 拉丁方的大集LDILS, Golfdesign,... • t-设计的大集 LSλ(t,k,v) …

  5. 基 本 文 献 • C. J. Colbourn & J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press (Second Edition), 2006. • J. H. Dinitz & D. R. Stinson, Contemporary Design Theory – A collection of surveys, Wiley, 1992. • Q. D. Kang, On large sets of combinatorial designs, Advance of Mathematics, 32(2003),269-284.

  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.

  7. Six types of triples and the corresponding triple systems

  8. The existence of triple systems

  9. 经典三元系大集的存在谱 *A short proof for LSTS(v) was given by L. Ji. *Lindner, Street, Colbourn, Rosa and Teirlinck also gave some results for LMTS(v).

  10. 经典三元系超大集的存在谱 * 遗留问题：

  11. Large Sets ofpureorinted triple systems *遗留问题:

  12. Overlarge Sets ofpureorinted triple systems * 遗留问题:

  13. B.其它三元系的大集与超大集LT1 , LT2 , LT3 ,OLT1 , OLT2 , OLT3 ,LESTS, LEMTS, LEDTS.

  14. mixed triple systems—T1, T2, T3

  15. Conclusions for LTi and OLTi Q.Kang, Z.Tian & L.Yuan, 2003-2007 * 遗留问题:

  16. Extended triple systems A classical triple consists of three distinct elements, but an extended triple is allowed to contain repeated elements. STS, MTS, DTS (LSTS, LMTS, LDTS ) ⇒ ESTS, EMTS, EDTS (LESTS, LEMTS, LEDTS ).

  17. Examples ofLESTS

  18. Examples ofLEMTS

  19. Examples ofLEDTS

  20. A construction forLEDTS(7)

  21. There exist LESTS(v)and LEMTS(v) There exist LEDTS(v)for

  22. C.可分解三元系的大集与超大集LKTS, LRMTS, LRDTS, OLKTS, OLRMTS, OLRDTS,LARMTS, LARDTS, OLARMTS, OLARDTS.

  23. OLKTS STS KTS OLRDTS OLRMTS LKTS RDTS RMTS LRDTS LRMTS DTS MTS LARMTS LARDTS ARDTS ARMTS OLARDTS OLARMTS

  24. The existence of triple systemswith resolvability

  25. KTS(9) LKTS(9)

  26. Known LKTS(v) and small orders≤405

  27. LRMTS(12)

  28. OLARDTS(10)

  29. Tripling constructions for LKTS Product constructions for LKTS

  30. The existence for LKTS(v) Kirkman, Denniston, Schreiber, L.Wu, Y. Chang, G. Ge, L. Zhu, S. Zhang, J. Lei, L.Ji. … before 2005 Q. Kang & L. Yuan, 2007

  31. Tripling constructions for LRMTS and LRDTS Product constructions for LRMTS and LRDTS

  32. The existence of LRMTS (v) and LRDTS(v) Q. Kang, J. Lei, Z. Tian, L. Yuan, Y. Chang, J. Zhou, …

  33. 1996 Kang & Lei 2000 Kang & Tian 2005 Kang & Yuan 2006 Kang & Yuan 2002 Kang & Tian

  34. D. 图设计的大集与超大集P3-LGD, OP3-LGD, P3-OLGD, OP3-OLGD, P4-LGD , Pk-LGD.K 1,3-LGD, K 1,4-LGD, K 1,k-LGD , C4-LGD. HCD, LHPD, LDHCD, LDHPD , LCS(v,v-1,λ) .

  35. Large sets of P3-decompositions

  36. Large sets of oriented P3-decompositions

  37. Large sets of P3-decompositions Q. Kang & Y. Zhang, 2002 Large sets of oriented P3-decompositions Y. Zhang & Q. Kang, 2006

  38. Overlarge sets ofP3-decompositions Y. Liu & Q. Kang, 2008 Overlarge sets oforientedP3-decompositions Y. Liu & Q. Kang, 2009

  39. Examples of LGDforcycle, path and star

  40. 在完全图中：Large Sets ofHamilton cycle (path) decompositions * 无遗留问题

  41. Construction of LHCD2(v)

  42. 在二分图中：Large Sets ofHamilton cycle (path) decompositions * 无遗留问题

  43. Large Sets ofdirectedHamilton cycle (path) decompositions *遗留问题: 1989 Kang 2005 Zhao & Kang

  44. Tuscan squares of order 6 with or without a cross without cross Roman square with cross

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