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) SUN MON TUE WED THU FRI SAT

大集问题的起源和背景大集问题的起源和背景

Thomas Penyngton Kirkman ( 英格兰教会的教区长 )

<Lady’s and Gentleman’s Diary>

1 2 3 1 4 5 1 6 7 1 8 9 1 10 11 1 12 13 1 14 15

4 8 12 2 8 10 2 9 11 2 12 14 2 13 15 2 4 6 2 5 7

5 10 15 3 13 14 3 12 15 3 5 6 3 4 7 3 9 10 3 8 11

6 11 13 6 9 15 4 10 14 4 11 15 5 9 12 5 11 14 4 9 13

7 9 14 7 11 12 5 8 13 7 10 13 6 8 14 7 8 15 6 10 12

23:57:17 2

{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)

0 2 8 11 12 5 7 4 9 1 10 3 6

8 9 12 1 6 4 10 3 12 2 5 9 11 7 8

3 7 10 4 7 11 6 7 9 2 9 10 6 8 10 5 6 12 5 10 11

2 6 11 3 5 9 1 2 3 1 8 11 1 7 12 3 4 8 2 4 12

1 4 5 0 10 12 0 5 8 0 4 6 0 3 11 0 2 7 0 1 9

a b b b b b b b

a a a a a a

13 { , } Z mod 5) 13 (1 a bLKTS

7 2 { } ( Z ) mod Z15) 7( KT aS

323:57:17

• 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, Golf design,...

• t- 设计的大集 LSλ(t,k,v) …23:57:19 4

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),269-284.23:57:19 5

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.

23:57:19 6

y

zx

y

zx

y

zx1 2 3

1 2 3

[ , , ] [ , , ] [ , , ]

x y z x y z x y z

T T T

{ , , } , , ( , , )

S

x y z x y z x y z

teiner Mendelsohn Directed

y

z

x

y

z

x

y

z

x

Six types of triples and

the corresponding triple systems

23:57:19 7

(Kirkman 1( ) 1,3 8mo 4 6 )d . 7STS v v

( ) 0,1 mod 3, (Mendelsohn 1971) 6.MTS v v v

(Huan ( g ) &0 ,1 Me mod ndelsohn 1983) 3. DTS v v

The existence of triple systems

(Colbourn, Pull

e

( ) 0

ybl

,1 mod

ank & Rosa 1989

3.

)

HTS v v

23:57:19 8

( ) 0

(Bennett & Mendelsohn 1978

,1 mod 3

3,

6.

)

PMTS v v v v ，

( ) 0,1 mod 3, 3 (H. Shen 19 5). 9PDTS v v v

0 123,145,167,246,257,347,: .3568 8 0(7) {( \{ }, ) : }, mod 8,x xOLSTS x x x Z5 Z

23:57:19 9

7 7 0(9) {( { , }, ) : }, mod 7,x xLMTS a b x x Z5 Z

0

0 016 025 034 124 356

0 061 052 043 142 365

15 23 31 46 54 62

51 3

:

2 13 64 45 26

ab

ba

a a a a a a

b b b b b b

7 7 3 0(6) {( \{ }, ) : , }, mod 7,k k kx xOLPDTS x x k x Z5 Z I

10

20

30

1 2 3, 1 4 5, 2 1 6, 2 5 4, 3 4 1, 3 5 2, 5 3 6, 4 6 2, 6 4 3, 6 5 1;

1 2 4, 4 1 3, 2 1 5, 5 1 6, 5 2 3, 4 2 6, 3 6 1, 6 3 2, 3 5 4, 6 4 5;

1 5 2, 2 1 3, 1 6 4, 2 4 6, 6 2 5,

:

:

6 3 1 3 : ,

4 2, 3 5 6, 4 5 1, 5 4 3.

4 4 0(5,3) {( { }, ) : }, mod 4,x xLHTS a x x Z5 Z

0 10 ,201,30 ,1 2,2 1,

3 1,

123,132, 01, 02,

13, 23, 32,012,210,302,203,130,031.

: 03, a a a a

a a a a

a a a

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).

J. Lu, 1983; Luc Teirlinck, 1989 ( , )LSTS v

( , )LMTS v

( , )LDTS v

( , )LHTS v

6 | ( 1), | ( 2), ( , ) (7,1). v v v v

3 | ( 1), | ( 2), ( , ) (6,1). v v v v

3 | ( 1), | ( 2). v v v

3 | ( 1), | 4( 2), ( , ) (3,1). v v v v

Q. Kang, J. Lei & Y. Chang, 1993

Q. Kang & Y. Chang, 1991

Q. Kang & J. Lei, 1 996

23:57:20 10

经典三元系超大集的存在谱

* 遗留问题：

M. J. Sharry & A. P. Street, 1991 ( )OLSTS v

( )OLMTS v

( , )OLDTS v

( , )OLHTS v

1,3 mod 6, 3. v v

0,1 mod 3, 3, 6. v v v

0,1 mod 3 for =1, or 3 for =3.v v

0,1 mod 3, 4, =1,2,4. v v

L. Ji, 2007

Z. Tian & L. Ji, 2007

Z. Tian & M. Cheng, 2008

, for index >1. OLSTS OLMTS23:57:20 11

Large Sets of pure orinted triple systems

* 遗留问题 :

( )LPMTS v

( )LPDTS v

0,1 mod 3, 4, 6,7v v v

J. Zhou, Y. Chang and L. Ji , 2008

0,1 mod 3, 4v v

J. Zhou, Y. Chang and L. Ji , 2006

, for ind >1. ex LPMTS LPDTS

23:57:20 12

Overlarge Sets of pure orinted triple systems

* 遗留问题 :

( )OLPMTS v

( )OLPDTS v

1,3 mod 6, 0, 4 mod 12 and 4v v v

L. Ji , 2006

0,1 mod 3 and 4v v

Y. Liu & Q. Kang, 2009

( ) for 6,10 mod 12;

, for index >1.

OLPMTS v v

OLPMTS OLPDTS

23:57:20 13

B.其它三元系的大集与超大集

LT1 , LT2 , LT3 ,

OLT1 , OLT2 , OLT3 , LESTS, LEMTS, LEDTS.

23:57:20 14

mixed triple systems—mixed triple systems—TT11, , TT22, , TT33

0 0

1 4 2 3

1(5), mod 5T

0 0

1 2 3 6

0 0

4 8 5 72 (9), mod 9T

1 2 4

0 6 0 5 0 33(7), mod 7T

3

3 3

3

A ( , ) contains blocks. An ( , ) ( ( , )) consists

of ( ) disjoint ( , ) ( ( , )). An ( , ) ( ( , ))

consists of ( ) di

2( 2)

2(

2

1sjoint ( , ) (

1( , ))

( -1) / 2

)

k s

s s

s

vT v LT v LT v

T v T v OLT v OLT v

T v T

v

v

v v

vv

, 1, 2,3 and 1,2.k s

23:57:21 15

( ) or ( ),

odd, odd

but 7 when ( , )

eve( ,

(3,

) 3

1 .

n,

)k

v

T v v

k

v

1 1

1

1 11

0 1 1

110

{[ , , ] ,[ , , ] ,[ , , ] :{

, , } },

, , where contains

(13) {({ , } , }

:

) :i

i

x y z y z x z x y x

LT a b

A

A

Z

y z

i i Z

i Z

0 12 46 70 59 38 23 57 69

81 04 001 890 134 458 367 903 156

782 680 791 035 026 924 025 470

ab a a a a a b b b

b b

5

2 2 2 2 2 2

2 2 2 2 2 2

2 on Z { }, two small sets:

[1,0,2] ,[2,0,4] ,[4,0,3] ,[ ,0,3] ,[ ,0,1] ,[0, ,2] , mod 5;

[1,0,4] ,[2,0,3] ,[1,0,3] ,[ ,0,2] ,[ ,0,4] ,[0, ,1] , mod 5

6, )

.

( 2LT

23:57:21 16

3 7 7(9) {({ , } , ) : , 1, 2}jiLT a b Z i Z j

6 1 0 23 35 12

46 04 25 0 6 1 4 3 5

54 42 30 65 31 52

3 2 4 0 6 1 602 562 613

145 015 241 403 534 051

506 436 163 264 320 210

ab ba a b a a a

a a a a a a

b b b b b b

b b b

2 10 03 3{[ , , ] :[ , , ] },z y x x y z

10 :

70 , , 1, 2.j ji i i Z j

23:57:21 17

1 4 4

0 1 401 1

{[1,2,3] ,[2,

(3) {( \{

3,1] ,[3,1, 2] }

}

,

, ) : }

, .x

xOLT Z x x

x Z

Z

x

Conclusions for LTConclusions for LTii and OLTand OLTii

3

3

3

3 3

| ( 2) a

(9)

( , 2 )

( , 2)

(7 ), (9 )

.

.

.

f

n

or

d 4

.

3

1 n n

LT

LT v

OLT v

OLT O

v

v

LT

v

n

3 3( , ) and ( , ) for odd .LT v OLT v

1 2

1 2

& ( ) or ( ),

except

odd, odd even, 3

| ( 2

( , ) (

) (

, )

( , )

, ) (5, 1)

odd( , )

.

& . 3

LT v LT v

OLT v OL

v v

v

T

v

vv

* * 遗留问题遗留问题 ::

Q. Kang, Z. Tian & L. Yuan, 2003-2007

23:57:21 18

system form of triple pairs triple's number system's number

( 1)

( 1)( 2) / 6

( 1)

( 1)( 2) /

,

, ,

, ,

, ,

,

, 3 2

3

( 1)

( 1)

xx

x

v

v v

v

x xy

xy yz zx

v

v

v

xxx

xxy

xyz

xxx

xxy

xyz

xxx

xxy

xx

xx xy yx

xy yz zx

xx

xx xy

ESTS

v v

v

vEMTS

ED xy

v

v v v

v

v v

TS y xy vyy v

x

, ,

,

( 1)

( 1)( 2),

xy yx xx

x

v v

v v

yx

x zy y xz yz v

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 ).Extended triple systems

23:57:21 19

Examples ofExamples of LESTS

4

{{0,0,0}}.

{{0,0,0},{1,1,0}}; {{1,1,1},{0,0,1}}.

{{0,0,0},{1,1,1},{2,2,2},{0,1,2}};

{{0,0,1},{1,1,2},{2,2,0}}; {{0,0,2},{1,1,0},

(1) :

(2) :

(3) :

(4) {

{2,2,1}}.

( , x

LESTS

LESTS

LESTS

LESTS Z

0

0

4

5 5

6 , 3 2

0

0

, ,0

,

,

{000,112,220,332,013}.

{000,113,221,334,442,014,02

)

3

: },

(5) {( ,

}.

) :

},

(6) {( , ) : , }, ,

x

x

s x s

x

s x

x Z

LESTS Z x Z

LE

x

x

STS Z s Z Z xx

A A

M

A

MM

A

0,0

1,0

2,0

{000,114,222,330,444,552,015,024,123,345},

{001,115,223,331,445,553,025,034,124},

{002,110,224,332,

440,554,035,125,134}.

M

M

M23:57:21 20

{ 0,0,0 }

{ 0,0,0 , 1,1,0 } { 1,1,1 , 0,0,1 }

{ 0,0,0 , 1,1,1 , 2,2,2 , 0,1,2 , 2,1,0 }

(1)

(2)

(3

: .

: ; .

: ;

; { 0,0,1 , 1,1,2 , 2,2,0 } { 0,0,2 , 1,1,0

)

(4

, 2,2,1 }.

LEMTS

LEMTS

LEMTS

LEMTS

4 4

5 5

6 ,

0

3 2

0

0

0

) {( , ) : },

(5) {( , ) : },

{000,112,220,332,013,310}.

{000,113,221,334,442,014,410,

(6) {( , )

023,320

,

,

,

:

}.

x

x

s x

x

x

Z x Z

LEMTS Z x Z

LEMTS Z s Z Z

x

x

x

A

M

A AA

0

, ,0

,0

1,0

2,0

{000,114,222,330,444,552,015,510,024,420,123,321,345,543},

{001,115,223

},

,331,445,553,025,520,034,430,124,421},

{002,110,224,332,440,554,035,530,125,521,134,431}

,

.

s x s x

M

M

M

M

M

Examples ofExamples of LEMTS

23:57:21 21

Examples of LEDTS

5 5 5 5

{(0,0,0)}

{(0,0,0), (1,0,1)} {(1,1,1), (0,1,0)}

{(0,0,1), (1,1,0)} {(1,0,0), (0,1,1)}

Nonexistence

(1)

(2)

(4) :

(5) {( , ) : } {( , )

: .

: ; ;

; .

, ( , ) : 1 4}

.

x k k

LEDTS

LEDTS

LEDTS

LEDTS Z x Z Z Z k A M

0

0 5

{131,212,343,424,104,203,302,401},

, .

{(0,0, ), (0, , ) : mod5},

{(0, , ), ( , ,0) : mod5},

( , ) : (3, 2), (1,4), (4,

x

k k k

k k k

k k

x x Z

k a b

k k a b

a b

A

M

1), (2,3), 1, 2,3, 4.k 23:57:21 22

A construction for LEDTS(7)

7

0

7

0

7 7

: 000, 113, 332, 226, 664, 445, 551, 436, 524, 165,

341, 253, 612, 104, 305, 201, 6

(7) {( , ) : } {

03,

( ,

402, 506;

: 300, 044, 43

), ( ,

3, 511, 15

) : 0 5}kx kLEDTS F x F F F k

A

A

M

0

5, 262, 636, 102, 213,

641, 650, 016, 352, 254, 314, 420, 456, 053;

: 411, 144, 505, 010, 242, 313, 616, 435, 546,

203, 640, 653, 026, 304, 251, 152,

362.

x

M

070 0, ; 3 , 3 , 0 5.kk k

kx x F k MA A M

23:57:21 23

There exist LEDTS(v) for0,1,2,3,4 mod 6 except 4;

5 mod 12; 11 mod 36.

v v

v v

Y. Liu & Q. Kang, 2009

(3 )

(3 ,3)

LEDTS v

LEDTS v ( ) 3, 6.LEDTS forv v v {

( : )

( , ) (

( )

).

n

LE

PECS g s

LEDTS g Ds s

LED

g

s

TS n s

TS

}( : )

( , ) (

( )

).

n

LEDTS gn s

PECS g s

LEDTS g s s

LEDTS g s

}

1.v There exist LESTS(v) and LEMTS(v)

23:57:21 24

C.可分解三元系的大集与超大集

LKTS, LRMTS, LRDTS, OLKTS, OLRMTS,

OLRDTS,LARMTS, LARDTS,

OLARMTS, OLARDTS.

23:57:21 25

LKTS

LRMTSLRDTS

LARDTS LARMTS

RDTS

KTS

OLKTS

OLRDTS

ARDTS

OLARDTS

ARMTS

OLARMTS

RMTS

OLRMTS

DTS MTS

STS

23:57:21 26

The existence of triple systems with resolvability

(Ray-chandhur

( )

i

3 mod

& Wilson, 1971)

6KTS v v

( )

(Bermond & Soteau, 19

0 mod 3 and 6

)

79

RMTS v v v

( )

(Bermond & Soteau, 19

0 mod 3 and 6

)

79

RDTS v v v

A ( ) 1

(Bennett & Soteau, 1981)

mod 3 and 4 RMTS v v v

A ( ) 1

(Bennett & Soteau, 1981)

mod 3 and 4 RDTS v v v

23:57:21 27

KTS(9) LKTS(9) 7,A B Z

2 4 1

5 6 3

0 A B

124

214

124

356

0ABAB

502

46

223

2

13 2

A

B

A

B

115

1

01

3

451

6

2A

B B

A

340

51

133

3

62 3

A

B

A

B

23:57:21 28

Known LKTS(v) and small orders ≤ 405 (Kirkman 1850)

1 (Sylv

9,

3, 27, 81, 2

(9)

43,(3 ) ester 1893)

(Denniston 1974) (15)

k

LKTS

kLKTS

LKTS

1 (Schreiber 1976, Wu 1990)

17,35, 1 (Denniston 1979, L. Wu 1990)

15,

33

, 9(3 1

9, 297,

1)

(3

)

k

k

LKTS

LKTS m

k

m k

5, 25,43, 1 (Denniston 1979)

51, 105, 153, 315,

45, 75, 129, 135, 225, 387, 405,

(3 )

(

2 (Y. Chang, G. 3 ) 41 Ge

k

k

LKTS m

LK

m k

TS k

369,

201,

& L. Wu, 19

(201)

(3 91

99)

(Y. Chang & G. Ge, 1999)

1 (G. Ge 2000)

7,13, 1 (

) 2

(

73

3 J

,

.)

k

k

LKTS

LKTS

LKTS m

k

m k

Zhou & Y. Chang, 2009)

21, 39, 63, 117, 1 89. 23:57:21 29

LRMTS(12)5 2 5 2({ , } ( ), ) : , }ijA B Z Z i Z j Z

3 24 42 3 1 0 1 0

04 3 1 0 2 40 2 14 3

12 1 130 0 2 2 1

0 0 3 24 42 1 1 3

2 40 1 14 04 3 3 2 2

4 03 2 0 2 2 30 2 0 0 01

123

4 103 1 0 2 0 4

0 4 431 4 132 34 3 1 10 1 4 43 423 4 32 34 3 21

A A A A AB A A A A BA A

B B B B B B B B B

4 23 32 4 0 1 1 0

03 4 1 13 2 30 2 0 4

1

0 0 1 4 23 32 1 4

2 30 1 2 03 4 4 13 2

3 04 2 0 2 40 20 022 1 140 02 20 1 3 104 1 0 2 21

0 3 4 24 3 42 4 3 3

3

124 1 3 142 4 3 3 4 1 10 1 3 4 014 3

A A A A AB A A A A BA A

B B B B B B B B B

00

01

23:57:21 30

5 5{( \{ }, )(4 : , j=1,2,3}) jxx xOLARDTS Z5 ZA

10

2 j0 5x 0

30

{123,214,341,432},

{312,421,134,243}, (mod 5),

{231,142,413,324},

j x x

Z5

A

A A A

A

51

5{( \{ }, ) : }(4) xxOLARM xTS Z5 ZA

241 547 845 673 374 652 238 872

356 698 276 942 816 483 496 614

789 132 319 518 259 917 715 953

(9)OLRMTS

23:57:21 31

55 (7)={({ , } , ): , 1,2,3}k

xLARDTS a b Z x Z k 10

1 32 4 0 14 2 3 4 30

243 04 31 02 013 120 421: ab a a a ba b a

b b b

20

42 23 1 4 10 2 0 3 0

1 3 40 043 012 3 2 314 241: a a ba ab a b a

b b b

30

4 2 4 3 2 1 30 20 01

13 023 140 04 12 431 324: a a b b a a a b a

b b b

OLARDTS(10)

719 045 570 609 810 406 078 013 902 320

842 627 916 518 259 231 152 954 614 417

563 893 348 724 736 987 439 268 375 865

197 450 705 960 081 064 807 130 029 203

428 762 691 185 925 312 521 549 146 741

356 389 834 247 673 798 943 826 537 658

971 504 057 096 108 640 780 301 290 032

284 276 169 851 592 123 215 495 461 174

635 938 483 472 367 879 394 682 753 586

11 11 10{( \{ }, ) : , 1, 2,3}, 1j j jx xZ x x Z j x

110 :

210 :

310 :

23:57:22 32

Tripling constructions for LKTS

( )

( )

LKTS v

TKTS v (Denn(3 iston, 1979)) LKTS v

( )

( )

LKTS v

TRISQ v (L. Zhu &S. Zhang, 200) 0( )3LKTS v

( ) 3 mod 6TRISQ v v ( ) ( ) 3LKTS v LKTS vProduct constructions

for LKTS

(S. Zhang, L. Zhu, 2003)

(

)

)

(LKTS

LR u

v( ) LKTS uv

3

(3

( )

( )

)

LR u

RPIC

L v

S

S

v

KT

(L. Ji & J. Lei, 2004)23:57:22 33

The existence for LKTS(v)

2 1{1,7,11,13,17,35,43,67,91,123} {2 25 1: , 0},p qm p q 1 1

3 5 (2 7 1) (2 13 1 ) ji

u wsra b

i j

v m

unknown ( ) : 57,69,87,93,111,123,141,147,159,LKTS v v

, , 1, , , 0, 2 for 1 1.i ja r s b u w a u w b m ,Kirkman, Denniston, Schreiber, L.Wu, Y. Chang, G. Ge, L. Zhu, S. Zhang, J. Lei, L.Ji. … before 2005

prime powers 7 mod 12, 1, , 1.i i jq u w r s Q. Kang & L. Yuan, 2007

1 1

3 (2 1) (4 1)ji

u wsr

ii j

v q

23:57:22 34

Tripling constructions for LRMTS and LRDTS

( )

( )

LRMTS v

TRIQ v (Chang, 2001( ) )3 LRMTS v

( )

( )

LRDTS v

DTRIQ v (Zhou & Chan( g, 200) 3)3LRDTS v

( ) & ( ) 3 | and 2 mod 4TRIQ v DTRIQ v v v Product constructions for LRMTS

and LRDTS

(Chang & Zhou, 2003)

( ) LRMTS uv

)

( )

( )

(

TRI

LRMT

u

u

LR

S

v

Q

)

( )

( )

(

DTR

LRDTS

u

u

LR

Q

v

I ( ) LRDTS uv

23:57:22 35

The existence of LRMTS (v) and LRDTS(v)

4 6 2, {2 ,2 ,7 ,13 ,5 : 12 };r r r r rq qv r

2r+1

, {1,4,5,7,11,13,17,23,25,35,37,41,43,47,53,

55,57,61,65,67,

3

91,123} {2 25 1: , 0}.

k

s

v m

r s

m

(2 7 1) , (2 13 1) , 2 mod 4, 0; n nv v v v nv

, {35,38,46,47,48,51,56,60};

, {0,1,2,3,4,6,7,8,9,14,16,18

6 3

12 ,20,22,24,28 ;12 ,32}

v s

v

s

tt

3 (2 , 1)k nv q prime power 7 mod 12, 0, 1.q n k

unknown ( ) & ( ):

18,30,42,54,57,66,78,87,90,93,

LRMTS v LRDTS v

v

Q. Kang, J. Lei, Z. Tian, L. Yuan, Y. Chang, J. Zhou, …

23:57:22 36

2000 Kang & Tian 1 4 1: 1nv M n 2 2 1: 7, 31,127, 0nM q qv n

3 2 1: prime power 7 mod 12, 0nM q q nv

1 2(3 ) (3 )Mv M

1 3(3 ) (3 ), 0t tM M tv

4 2 13 1: 0nv M n

2006 Kang & Yuan

2005 Kang & Yuan

There exist & for

1

(

0,4 ,7 ,13 ,25 ,25 4

)

( 1)

) (n n n n n

OLARMTS v OLARDTS v

v n

There exist & f( ), ( ) ( ) orOLKTS v OLRMTS v OLRDTS v

2002 Kang & Tian 23:57:22 37

There exist & fo( ( ) r)LARMTS v LARDTS v

4 , 2(7 1), 2(31 1), 2(127 ,) 11 .n n n n nv 1996 Kang & Lei (7Th )ere exists an , but no (7). LARLARDTS MTS

( ) & ( ):

10,13,

unknow

19,22,25,28,31,34,3 ,

n

7

LARMTS v LARDTS v

v

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) .

23:57:22 38

Large sets of P3-decompositions

3 3

00

10 0

3

0, 21, 1 , 0 2, 102

{( { , }, ) : 0,1, }

:

:

;

0, 12, 10, 0 2, 20.

- (5)

jx

j jx

ab b a b a

ba b

Z a b j xP LGD

xa

Z

b a

3 4- (4) {( , ) : 0 3}jP LGD Z j

0 1 2 3

0 1 2 1 2 0 2 3 0 3 0 2

0 2 3 1 0 3 2 0 1 3 2 1

0 3 1 1 3 2 2 1 3 3 1 0

23:57:22 39

。 。 。x y z

Large sets of oriented P3-decompositions

1 2 33 3 3 P P P

x y z x y z x y z

33 4

1

20

4

0 0, 3 , 0 , 2 , 12, 31, 1 2, 23 ,

10 , 13, 2 3, 21 , 302, 201, 032;

1, 2 , 03, 3 2, 1 3, 20 , 01, 21,

3 1, 0 3, 12 ,

- (6) {( { , }, ) : , 1, 2}

:

:

jx

ab ba a b b a a a a a

a b

P LGD Z a b x Z j

b b

ba ab a a a a b b

b b b

31 , 230, 132, 102.b3

3 5

10

20

5

1, 4, 0 , 0 , 10, 20, 30, 01 , 02 , 03 , 04 ,

32, 41, 3 2, 31 , 12 , 24 , 340, 213, 143, 423;

0 , 4 , 0

1, 1 4, 2 3, 3 2, 4

- (7) {( { , }, ) : ,

:

:

1

, 2}jx

ab ab a b b a a a a a a a

P LGD Z a b x Z

a

b b b b b b

ab ba a a a

j

a

1, 14, 23, 2 3, 1 4,

3 0, 10 , 021, 432, 341, 120, 130, 031, 240, 042.

a b b b b

b b23:57:22 40

| 2( 2)

3 (for even )

0,1 mod 4 (for odd )

v

v

v

3- ( )P LGD v {Large sets of P3-

decompositions

13 | ( - 2) , mod 2,- ( ) 3v vP LG v vD

Large sets of oriented P3-decompositions

33 | 2( 2) and ( , ) (3,1- ) )(P LGD v v v

Y. Zhang & Q. Kang, 2006

Q. Kang & Y. Zhang, 2002

23:57:22 41

42

Overlarge sets of oriented P3-decompositions

Overlarge sets of P3-decompositions

3 ( - )P OLGD v { 5, 0,1 mod 4 (for 1)

3 (for 2)

v v

v

13 1 , - 1 mod 2, 3) ( vP OLGD v v

33 - ) ( P OLGD v { 5 (for 1)

3 (for 2)

v

v

Y. Liu & Q. Kang, 2009

Y. Liu & Q. Kang, 2008

23:57:22

Examples of LGD for cycle, path and star

4 2

4 2

{(1234), (1243), (1324)}- (4) :

- no

.

ne(5) :

C LGD

C LGD

[1,1, 2,3], [2, 2, -3, -1], [3,3, 2, -1];

[1,1,3, 2], [2, 2, -1, -3], [3,3, -1, 2];

[1, 2,1,3], [2, -3, 2, -1], [3, -1,3, 2];

[1, 2, -1, -2],[2,3, -2, -3], [1,3, -1, -3];

[1,1,1, -3], [2, 2, 2,1], [3,3,3, -2].

4 4 7- (7 on ) C LGD Z

{4

4 4

- ( )

-

consists of blocks

( )

;

co

v( -1) / 8

ns ( - 2)(ists - -3) )of .(/

C GD v

C LG

v

v vD v C GD v

4 4 5 on - (6) { }C L ZGD [ ,1, 2],[ , -1, -2],[1,2, -1, -2];

[ , 2, -2],[ ,1, -2],[1,1,1,2];

[ ,1, -1],[ , -1,2],[2,2,2, -1].

{

23:57:22 43

1,3- (6)K LGD

23:57:22 44

320 540 420 510 310

541 310 510 321 421

321 542 421 520 320

531 532 430 310 432

431 543 410 542 420

542 530 541 520 531

540 510 210 320 430

521 531 410 210 431

432 420 521 532 210

532 430 431 530 321

421 432 543 540 410

543 520 530 5

0 1

4

2 3 4 5

1 521

0

0

0

0

0

0

0

0124 1435 2031 3254 4051

1450 2134 3015 4025 5324

2045 3215 4103 5243 0531

3401 4152 50

5301 0231

24 0321 1

1254 241

354

4352 5031 0

5 3405

5412 0132 1524 2034 3

123 140

50

51

4

2 24

M

N

A

B

C

D

E

4 - (6)P LGD

0 mod 6,

0 2 for { }

0 5 for , , ,

,

, { }

x x

x

x A B

M N

C D E

4 - (4)P LGD0123 0132 0213 0231 0312 0321

1302 1203 2301 2103 3201 3102

1 for odd pri- ( me) power ; k kP LGD kq q

Conclusion

1,3

1,3 3

- ( )

-

for 6,7,13;

for 8,14,20,6 5) ;(

K LGD v

K LGD v tv

v

1,4 2

1,4 4

for 5,13;

for 6,10,

- ( )

4 3. - ( )

K LGD v

K

v

tLGD v v

4

4 2

4 4

for 4,6,7;

for 4,5;

for 6, 4

- ( )

- ( )

3- ( )

;

P LGD v

C LGD v

C

v

LGD

v

v tv

23:57:22 45

1, 2- ( 1 for 3 ; )kK LGD k k

在完全图中： Large Sets of Hamilton cycle (path) decompositions

* 无遗留问题

( )LHCD v

2 ( )LHCD v

( )LHPD v

2 ( )LHPD v

3odd v

4 even v

3odd v

2even v

. . , 1998D E Bryant

. . , 1998D E Bryant

. , . , 2005H Zhao Q Kang

. , . , 2005H Zhao Q Kang

23:57:22 46

*2 1 1 ( ) {( { }, ) : ( )},v G vLHCD v Z Sym Z

0 4 1 3 2

0 1 4 2 3

1 0 2 4 3

4 0 3 1 2

4 3 0 2 1

Construction of Construction of LHCDLHCD22(v)(v)

2[ ]

2

1

(2 1,2where ) .

v

i

G i i

for 6 :v ，

*5

(14)(23)

{(1),(1243),(13)(24),(12),(13),

(14),(24),(34),(123),(132),(124),(13

4)

(

}

)G

G

Sym Z

23:57:22 47

在二分图中： Large Sets of Hamilton cycle (path) decompositions

* 无遗留问题 H. Zhao & Q.Kang, 2006

, -cyclen nK H 2| (( 1)!) ,n

, 1 -pathn nK H *, -cyclen nK H

*, 1 -pathn nK H

*, -pathn nK H

, -pathn nK H

2| (( 1)!)n

2| (2 1)(( 1)!) , (2 1) |n n n

2 1,2 1 \ -cyclet tK F H 2 1,2 \ -patht tK f H

even 2 for any

odd 3 for even

n

n

23:57:22 48

Large Sets ofLarge Sets of directed Hamilton cycle (path) decompositions

* 遗留问题 :

( ) for odd 3

( ) for even 2

LDHCD v v

LDHPD v v

( 1) and ( ) for

odd composite and prime 7,11,13,17,19.

LDHCD v LDHPD v

v v

1989 Kang

2005 Zhao & Kang

( 1) and ( )?

for odd prime 23

LDHCD p LDHPD p

p

23:57:23 49

Tuscan squares of order 6

with or without a cross

1 2 3 4 5 6

2 4 6 1 3 5

3 6 2 5 1 4

4 1 5 2 6 3

5 3 1 6 4 2

6 5 4 3 2 1 1 2 3 4 5 6

2 1 6 5 4 3

3 1 5 2 6 4

4 2 5 3 6 1

5 1 4 6 3 2

6 2 4 1 3 5

2 3 4 5 6

2 4 3 6 5

3 2 6 5 4

4

1

1

1

1

1

6 2 5 3

5 3 6 4 2

6 3 5 2 4 1

without without crosscross

with with crosscross

Roman Roman squaresquare

23:57:23 50

A tuscan square of order with a cross v

1 2 3 4 5

1 3 2 5 4

2 1 5 4 3

3

0

0

0

0

0

5 1 4 2

4 2 5 3 1

5 2 4 1 3 0

:

A consists of directed Hamilton cycles

A contai

(

ns dis

1)

joint ( 1) ( 1! ( )-1)

v

v

DHCD v

LDHCD v DHCD v

*( 1) {( { }, ) : ( )} v vLDHCD v Z Sym Z

23:57:23 51

A pair of relasional tuscan squares of order 7

(143652) (152436)1 8 2 3

(13)(245) (1632)4 5 6 13

(154)(36) (134625) (142653)7 14 9 10 11 1

B B B B

B B B B

B B B B B B

2

2

5 3 6 1 2 4

3 2 5 4 1 6

4 5 1 6 3 2

6 2 1 4 3 5

5 6 4 2 1 3

0

0

0

0

0

02 6 5 4 3 1

01 5 2 3 4 6

T

1

6 4 3 2 5 1

6 5 4 2 1 3

1 2 3 4 5 6

5 2 3 6 1 4

2 4 1 6 3 5

0

0

0

0

0

03 1 5 4 6 2

04 1 2 6 5 3

T

1

2

TT

T

23:57:23 52

Good tuscan square T -----T and T -1 are relational

1 5 7 6 2 4 3

1 2 3 4 5 6 7

2 6 1 7 3 5 4

4 1 6 5 3 7 2

3 1 2 6 4 7 5

6 4 2 5 3

0

0

0

0

0

0

0

7 1

7 4 5 2 1 3 6

5 1 4 7 06 3 2

T

4 2 5 1 3

4 1 2 3 5

5 2 4 3 1

3 4 1 5 2

2 1 3 5 4

1

0

0

4 5 2

0

0

0

03

T

6order 8order

23:57:23 53

问题: 对于大于 19 的奇素数阶数 是否存在 a tuscan square with a cross ？ 是否存在 a pair of relational tuscan square with a cross ？ 是否存在 a good tuscan square with a cross ？ ( 阶数 =3 mod 4)

There is a tuscan square of order with a cross

and

There is a pair of relational tuscan squares of order

( 1) ( )

(

LDHCD v LDHPD v

LD CD

v

v

H

and

There is a good tuscan square of order

1)

( )

( 1) ( )and

v LDHPD v

LDHCD v LDHPD

v

v

23:57:23 54

( ,0, 2,1,4),

( ,0,1,2,3),

( ,3,0,4,1),

( , 4,0,3,2),

( ,1,3,4,2),

( 0,1,3 4,2).

6

,

v

( ,0,5,4,3,2),

( ,0,5,3,4,1),

( ,3,0,1,2,5),

( , 4,0,2,1,5),

( ,1, 4,0,3,2),

( ,3,1,5,2,4),

( 0,1,3

7

,5,4,2).

v

( ,0,1, 2,3, 4,5),

( ,0, 2,1, 4,3,6),

( , 4,0,5, 2,6,3),

( ,3,0,6,1,5, 4),

( , 2,3,0,6,1,5),

( ,1, 4,0,5, 2,6),

( ,1,3,5,6, 4, 2),

( 0,1,3,5,6, 4, 2)

8

.

v

*-1 1( , -1,2) {( { }, ) : ( )}v G vLCS v v Z Sym Z

2[ ]

2

1

(2 1,2 )

v

i

G i i

23:57:23 55

Large sets of cycle systems

问题 : 是否存在

1

2

2 | ( 1), 2 | ( 1), | ( ).k

i

k v v v v i

( , 1, ) 2 | , 2 | ( 1), | ( 2)!LCS v v v v v

( , , ) LCS v k

( , 1, ) | ( 2)! Kang 1989 LM v v v

1

2

( , , ) | ( 1), | ( ).k

i

LM v k k v v v i

( , 1, 2) for 0,1,2 mod 4 LCS v v v

( , 1, 2) for 3 mod 4, 15 ?LCS v v v v

for 7,11 Zhao & ng Kav

23:57:23 56

E. 其它设计的大集与超大集

Latin squares, idempotent quasigroups,

group divisible designs, golf designs, t-designs.

23:57:23 57

Large set of idempotent Latin squares of order 5

0 2 4 1 3

2 1 3 4 0

4 3 2 0 1

1 4 0 3 2

3 0 1 2 4

0 3 1 4 2

4 1 0 2 3

3 4 2 1 0

2 0 4 3 1

1 2 3 0 4

0 4 3 2 1

3 1 4 0 2

1 0 2 4 3

4 2 1 3 0

2 3 0 1 423:57:23 58

Golf design of order 7(idempotent symmetric latin squares)

2 4 5 1 6 3

6 4 5 3 0

0 3 1 5

6 2 1

0 2

4

3 6 1 5 4 2

5 0 6 2 4

4 1 0 3

2 6 5

3 0

1

4

5 3

6 2 1

2 0 6 5

3 6 4 0 1

1 5 0 4 3 2

5

3 4

2 6 5

6 3 0 1

1 0 6 4 2

4 2 1 0 5 3

0

1

2

3

6 1 4 3 2 5

0 5 2 4 3

6 5 3 4

0 1

6

1 0

4 5 6

3 2 5 0

2

4

5

2

6 1

04 3 1 6

5 3 4 2 1 0 6

23:57:23 59

Overlarge sets ofidempotent quasigroups

0 0 0 0 0 0

0

0

0

1 4 2 5 3

5 2 1 3 4

4 5 3 1 2

2 3 5 4 10

0 3 1 4 2 5

0 3 4 5 2

4 2 5 0 3

5 0 3 2 4

1

1 1 1 1 1 1

1

1

13 5 2 4 0

2 1 4 0 3 5

0 3 5 1 4

5 1 4 3 0

4 0 3 5 1

1 5

2

2

2 2 2 2 2 2

2

2 0 4 3

3 4 2 1 0 5

3

3

3

3 3 3 3

0 5 4 2 1

2 1 5 0 4

1 4 2 5 0

5 0 1 4 2

4 2

3 3

3

0 3 1 5

0 2 5 1 3

3 1 0 5 2

5 3 2 0 1

2

4

4

4

4

4 4 4 4 4 4

5 1 3 0

1 0 3 2 4 5

0 4 1 2 3

4 1 3 0 2

3 0 2 4 1

1 2 4 3 0

2 3 0

5

5

5

5

51 4

5 5 5 5 5 523:57:23 60

Large sets of idempotent quasigroup 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 n≥3 , n≠6 n≥3 , n≠6 幂等对称拉丁方大集 幂等对称拉丁方超大集 二阶子群 n ≡1 mod 2, n≠5 n ≡1 mod 2,

n≥3 LMTS OLMTS 三阶子群 n≥3 , n≠6 n≥3 , n≠6 n≡1,3 mod 6 n≡1,3 mod

6 LSTS OLSTS 对称群 S3 n≥3, n≠7 n≥3

n≡1,3 mod 6 n≡1,3 mod 6 23:57:25 61

C B 0 1 4 A 5 2 3

A 3 1 0 B C 2 5 4

4 5 2 A C B 3 0 1

5 4 B C A 1 0 3 2

0 A C B 2 3 1 4 5

B C A 5 3 2 4 1 0

1 0 3 2 5 4

3 2 5 4 1 0

2 1 4 3 0 5

4 5 2 A C B 3 0 1

5 4 B C A 1 0 3 2

0 A C B 2 3 1 4 5

B C A 5 3 2 4 1 0

C B 0 1 4 A 5 2 3

A 3 1 0 B C 2 5 4

2 1 4 3 0 5

1 0 3 2 5 4

3 2 5 4 1 0

0 A C B 2 3 1 4 5

B C A 5 3 2 4 1 0

C B 0 1 4 A 5 2 3

A 3 1 0 B C 2 5 4

4 5 2 A C B 3 0 1

5 4 B C A 1 0 3 2

3 2 5 4 1 0

2 1 4 3 0 5

1 0 3 2 5 4

3 2 A C B 5 0 1 4

2 B C A 5 4 1 0 3

A C B 3 1 0 4 5 2

C A 3 2 0 B 5 4 1

B 1 5 4 A C 2 3 0

1 0 4 B C A 3 2 5

4 5 0 1 2 3

0 3 2 5 4 1

5 4 1 0 3 2

A C B 3 1 0 4 5 2

C A 3 2 0 B 5 4 1

B 1 5 4 A C 2 3 0

1 0 4 B C A 3 2 5

3 2 A C B 5 0 1 4

2 B C A 5 4 1 0 3

5 4 1 0 3 2

4 5 0 1 2 3

0 3 2 5 4 1

B 1 5 4 A C 2 3 0

1 0 4 B C A 3 2 5

3 2 A C B 5 0 1 4

2 B C A 5 4 1 0 3

A C B 3 1 0 4 5 2

C A 3 2 0 B 5 4 1

0 3 2 5 4 1

5 4 1 0 3 2

4 5 0 1 2 3

For 0 , ( , ) ( , ) (2,1), (6,5).a n LDILS n a a n a

(6 3, 3)LDILS ：

Large set of disjoint incomplete Latin squares

J. Lei, Q. Kang, Y. Chang 2001

0 : 018 246 348 6 3 8

8 1 4 4 0 2

12 57 04 23 07 15

56 02 37 45 68 27

78 35 16 67 13 05

ab ad af

cb cd cf eb ed ef

a a a b b b

c c c d d d

e e e f f f

9 Points:

Groups: {{ , , },{ , , },{0,3,6},{1,4,7}

{ , , , , ,

,{2,5,8

}

}}

a

a c e b

b d f Z

d

e

f

c

53- (3 )LGDD

2 3- ( ) 6 | ( 1) , 2 | ( 1) , ( , ) (1,7)uLGDD t u u t u t t u

0 (mod 9)i i

3- ( )uThe existence of LGDD t

23:57:25 63

J. Lei, 1997

(t,t+1,v)-decomposition( , , ) ( , , )- ( , , )t v k design S t k v LS t k v

* gcd( , {1,2, , 1})v t lcm t

*( , 1, , 1,)- ( )decomposition LS t vt v tt

1 2(1,2,2 ) & - : (1, 2,2 1(1,2, ) )LS t LS tv

1

2 2

3 6

: (2,3, ) ( )

(2,3,6 ) & (2,3,6 4)

(2,

(2

3,

,3,6 5) & (2,3,6

2)

-) LS v LSTS v

LS t LS t

LS t S t

v

L

3 6 12(3,4,6 ), (3,4,12 9- ) & (3,4,1( : 3, 4,3 ) 2 3)t LS t LS t LS t

1 4 4(2,3,7), (3,4,8), (3,4,10)LS LS LS23:57:25 64

4 4(3,4,11), (3, 4, 2 3)LS LS

(t,t+1,v)- decomposition

2LS (1,2,7)

(0123456) {01 12 23 34 45 56 60}

(0246135) {02 24 46 61 13 35 50}

(0362514) {03 36 62 25 51 14 40}

2 LS (2,3,6)

{012 013 024 035 045 125 134 145 234 235}

{014 015 023 0

25 034 123 124 1

35 245

345}

1LS (1,2,6)

{12 34 56}

{13 25 46}

{14 26 35}

{15 24 36}

{16 23

45}

Examples

23:57:25 65

Thanks !

23:57:25 66