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Kirkman Cubes
699
Annals of Discrete Mathematics 18 (1983) 699-712 North-Holland Publishing Company
KIRKMAN CUBES
A. Rosa* and S.A.
Vanstone
1. INTRODUCTION P r e s e n t l y t h e r e appears t o be q u i t e a l o t o f i n t e r e s t i n v a r i o u s g e n e r a l i z a t i o n s o f Room squares, such as Howell designs [ lI, g e n e r a l i z e d Room squares 131, Kirkman squares [ 6, 111 , g e n e r a l i z e d Howell designs [ 91, Room r e c t a n g l e s [ 41- f o r a survey on these and v a r i o u s f u r t h e r g e n e r a l i z a t i o n s o f Room squares i n c l u d i n g those t o h i g h e r dimensions see [ l o ] .
The concept o f o r t h o g o n a l i t y o f r e s o l u t i o n s
o f t h e associated u n d e r l y i n g design i s common t o a l l these g e n e r a l i z a t i o n s .
By
r e l a x i n g t h e usual concept o f o r t h o g o n a l i t y , several a d d i t i o n a l i n t e r e s t i n g obj e c t s can be d e f i n e d t h a t s t i l l r e t a i n t h e e s t h e t i c a l l y and o t h e r w i s e d e s i r a b l e p r o p e r t i e s o f u n i f o r m i t y , r e g u l a r i t y and balance.
I n t h i s paper we study t h e
e x i s t e n c e o f one k i n d o f these o b j e c t s i n t h r e e dimensions, which we c a l l Kirkman cubes [ 101 (as t h e i r u n d e r l y i n g design i s a Kirkman t r i p l e system).
2. BASIC DEFINITIONS AND PROPERTIES A uniform multidimensional generalized Room design o f degree k, dimension d, m u l t i p l i c i t y A and o r d e r v ( b r i e f l y UMGRD (k,d,A,v),
o f [ l o ] ) i s a d-dimensional
a r r a y such t h a t ( i ) every c e l l o f t h e a r r a y i s e i t h e r empty o r c o n t a i n s a k-subset o f a v-set
V,
( i i ) every element o f V i s contained i n e x a c t l y one c e l l of any (d-1)-dimens i o n a l subarray ( i i i ) every 2-subset o f V i s contained i n e x a c t l y h c e l l s o f t h e a r r a y . Thus t h e s e t o f k-subsets i n t h e nonempty c e l l s o f a UMGRD i s t h e s e t o f b l o c k s o f a (v,k,h)-design.
A UMGRD (k,d,h,v)
i s regular of index t ( 2 G t G d )
i f i t s p r o j e c t i o n on any t dimensions i s a UMGRD (k,t,A,v)
but p r o j e c t i o n on any
( t - 1 ) dimensions i s never a UMGRD. Examples of UMGRDs i n c l u d e multidimensional
700
A . Rosa and S.A. Vanstone
[lo]),
Room designs (cf.
o r , f o r instance, a r e g u l a r UMGRD (8,12,1,24)
obtained from t h e unique S(5,8,24)
o f index 2
(see [ 4 1 ) .
I t i s sometimes t o our advantage t o view UMGRDs as block designs possessing
m u l t i p l e resolutions.
If (V,B)
To e s t a b l i s h t h i s equivalence, we need a few d e f i n i t i o n s .
i s a (v,k,h)-BIBD
i s c a l l e d a puraZZeS class.
then any s e t C o f blocks i n B t h a t p a r t i t i o n s V
Any s e t o f d i s j o i n t p a r a l l e l classes R = {C1,.,.,Cr}
t h a t p a r t i t i o n s B i s c a l l e d a r e s o l u t i o n ; a B I B D a d m i t t i n g a t l e a s t one r e s o l u t i o n
i s c a l l e d rcsolvablc.
...,
Suppose now t h a t a (v,k,h)-BIBD admits a s e t R = IR1,R2, R S ] o f s resolui i i t i o n s where Ri = { C l,...,C 1 , C.’s a r e p a r a l l e l classes. We F a l l !he s e t R a dn~ 1 m$hogmzal szt; of resolutions i f f o r any d p a r a l l e l classes C . l , C’ ‘,...,Cid 1 J1 j2 Jd E Rk), (where C Jk
i
jcjl
1
i
n c.2 n
J2
... n c i d l Jd
< 1.
A d-orthogonal s e t R of r e s o l u t i o n s i s s a i d t o be regular o f index t ( 2 i f R i s a t-orthogonal s e t b u t c o n t a i n s no (t-1)-orthogonal
<
t d d)
subset.
The aforementioned equivalence i s then almost immediate (cf. [ 101 ) :
A r e g u l a r UMGRD (k,d,x,v)
o f index t e x i s t s i f and o n l y if t h e r e e x i s t s a d-
orthogonal s e t o f d r e s o l u t i o n s o f a (v,k,x)-BIBD
t h a t i s o f index t.
I n what f o l l o w s we w i l l be concerned w i t h a s p e c i a l case o f UMGRDs when k = 3, d = 3,
= 1 and index t = 3; these w i l l be henceforth c a l l e d Kirkman cubes
and denoted by KC3(v).
[ I n c o n t r a s t , a UMGRD w i t h k = 3, d = 3, h = 1 and index
t = 2 w i l l be c a l l e d a strong Kirkman cube SKC ( v ) ;
u n t i l very r e c e n t l y , no exam2 p l e s o f s t r o n g Kirkman cubes were known, however, j u s t p r i o r t o t h i s meeting, t h e second author succeeded i n c o n s t r u c t i n g an SKC (255).].
3
Thus, a Kirkman cube KC ( v ) i s a 3-dimensional a r r a y on v elements such t h a t 3 the s e t o f t r i p l e s i n t h e nonempty c e l l s o f each o f i t s planes i s a p a r a l l e l class, and the s e t o f t r i p l e s i n t h e nonempty c e l l s o f t h e whole a r r a y i s t h e s e t o f t r i p l e s o f an STS(v) w h i l e t h e p r o j e c t i o n on any two dimensions never y i e l d s a Kirkman square ( i . e .
a UMGRD w i t h k = 3, d = 2, h = 1, t = 2, c f . [ 2 ] ) .
70 1
Kirkman cubes 3. KIRKMAN CUBES OF SMALL ORDERS
Obviously the t r i v i a l necessary c o n d i t i o n f o r the existence o f a KC3(v) i s that v
3 (mod 6), v
> 3.
Since the a f f i n e plane o f order 3 has a unique resolu-
t i o n there e x i s t s no KC3(9).
I t i s well-known t h a t there are f o u r resolvable
Steiner t r i p l e systems o f order 15. and Cumnings [12] (= PG(3,2)),
O f these,
No. 1.1 on the l i s t o f White, Cole
as w e l l as No. 1.7 on the same l i s t , admit a s e t
( a c t u a l l y , several sets) o f three 3-orthogonal resolutions, while the remaining two resolvable STS(15) do n o t admit such a set. example o f such i s i n Table 1.
Thus a KC3(15) e x i s t s , and an
There a l s o e x i s t several KC3(21)'s (see [ 5 ] ) ,
an example o f a KC3(27) i s easy t o o b t a i n from AG(3,3)
(see Table 2.)
and
The e x i s -
tence o f a KC3(33) i s undecided a t t h i s p o i n t .
A general method o f constructing a KC3(v) d i r e c t l y i s the well-known starter-adder method (see, e.g.,
[3]). The smallest order f o r which we succeeded
i n constructing a Kirkman cube by t h i s method i s v = 39 (see Table 3). P o t e n t i a l l y important f o r b u i l d i n g up small order Kirkman cubes i s a " t r i p l i c a t i o n method" producing a KC3(v) from a KC3(v).
U n l i k e i n the case o f
Kirkman squares, t h i s method i s a t l e a s t f e a s i b l e and has indeed been used t o obt a i n a KC3(45) and a KC3(63). "twisted" d i r e c t product.
The construction involves, as i s t o be expected, a
Unfortunately, t h e general features o f one o f the
components o f t h i s construction remain e l u s i v e so t h a t a t present a t l e a s t the t r i p l i c a t i o n i s o n l y a hypothetical construction i n general (although i t has been successful on every order on which i t has been tested).
I n view o f the r e s u l t s o f
the next section, t h i s may be a l l t h a t i s needed t o completely s e t t l e the e x i s tence question f o r Kirkman cubes.
4. M A I N RECURSIVE CONSTRUCTIONS I n t h i s section, we describe various recursive constructions f o r KC3(v)s. The constructions are, o f course, more general and w i l l apply t o any UMGRD (k,t,X,v).
The most powerful o f these constructions i s PBD-closure
I131 ). L e t B be a s e t o f k elements. Define VG = B from the elements i n B the symbol s e t VB. KB(x,y,z)
x
{l,ZI.
x
I1,2} u { = I where
Suppose t h e r e e x i s t s a KC3(21Bl
+
Coordinatize t h i s cube KB by the elements o f B.
be the block o f K contained i n the c e l l (x,y,z);of
(see Wilson
-
i s distinct
1) defined on Let
course, KB(x,y.z)
=
0
702
A , Rosa and S.A. Vanstone
i f the c e l l (x,y,z)
Without l o s s o f g e n e r a l i t y , we assume t h a t
i s empty.
i s the b l o c k {m,x1,x2]
KB(x,x,x)
(NOTE: For convenience we w r i t e xi i n
o f K.
place o f ( x , i ) . ) .
THEOREM 4.1: ist D be a PBD(v;K) such t h a t for each k
f
;i:eri tizezv e x i s t s a KC ( Z V + 1 ) . 3
KC3(2k+l).
PROOF: L e t V be the p o i n t s e t o f D and 6 t h e block set. where
9 V
For each B
K there e d s t s a
x
(1,2}.
Coordinatize a v
x
v
x
DefineV" = V
x
{l,Z}u{m)
v cube K"' w i t h t h e elements of V. The
consider t h e subcube o f K* determined by t h e elements o f B.
E 6,
e n t r i e s ( c e l l s ) i n t h i s subcube a r e f i l l e d i n by t h e e n t r i e s i n KB.
We now show
t n a t K" i s a K C ~ ( E V t 1).
15 V".
Consider any p a i r {ai,b.
b . i s i n a unique c e l l o f K and, hence, a J B I f i # j and a # b, t h e same argument holds. I f i # j and
i n a unique block B o f 8 . unique c e l l o f Kn.
I f i = j , then a # b and I a , b } i s contained
J Then ai,
a = b then {a.,b.} i s contained i n c e l l (a,a,a) o f K". 7 .'. J !-,ai ), ai f V" -{-I , i s contained i n a unique c e l l .
I t i s e a s i l y seen t h a t
Therefore, t h e nonempty
c e l l s o f K" are t h e blocks o f a (Ev+1,3,1)-BIBD. Consider any plane o f K". form a plane o f K" f o r f i x e d a. a.
For instance, t h e c e l l s (a,x,y), L e t B1,B z,...,Bt
f o r a l l x,y E V,
be t h e blocks o f D which c o n t a i n
Then, iBi-{a 1: 1
i G tl
i s a p a r t i t i o n o f V-{al. (x,y,z)
9f
of K P(Bi)
.
be t h e block of D contained i n c e l l
This block i s t h e empty s e t i f t h e c e l l i s empty.
= IK"(a,x,y):
1
Then, iP(Bi):
L e t KR(x,y,z)
Q
x,y
f
Bill
Let
1 G i d t.
i Q t } induces a p a r t i t i o n o f the plane through a i n K
into
, 1 < i d t. Since each plane through a i n KB . Bi contains each element o f V p r e c i s e l y once and the o n l y common c e l l t o a l l o f Bi these planes i s (a,a,a) which contains Im,a a 1 then the plane through a i n K 1' 2 ;5 c o n t a i n s each element o f V p r e c i s e l y once. F i n a l l y , consider t h e p r o j e c t i o n o f planes through a i n each K
any KB, B E 6, o n t o the corresponding face. i s a KC3(2v
t
1) and t h e p r o o f i s complete.
Since KB i s a KC3(21Bl + 1 ) then K"
Kirkman cubes
703
I n s e c t i o n 5, we d i s p l a y t h e power o f t h i s c o n s t r u c t i o n . The n e x t r e c u r s i v e c o n s t r u c t i o n i s one which i s t h e analogue o f t h e s i n g u l a r d i r e c t p r o d u c t c o n s t r u c t i o n f o r Room squares ( [ 71).
Before g i v i n g t h e construc-
t i o n , we r e q u i r e s e v e r a l d e f i n i t i o n s .
A KC ( v ) i s s a i d t o be n o r m a l i z e d i f a l l c e l l s o f t h e form (x,x,x) (the 3 main d i a g o n a l ) c o n t a i n a common element. C l e a r l y , any KC ( v ) can be n o r m a l i z e d by 3 s u i t a b l e p e r m u t a t i o n s o f c o o r d i n a t e s . L e t K be a KC3(v) whose c o o r d i n a t e s a r e
A KC3(u) K ’ i s s a i d t o be a subcube o f K i f t h e r e e x i s t s subsets
indexed by V.
o f V such t h a t t h e s u b a r r a y
S1,S2,S3 y E
S2, z
E
x E Sly
determined by t h e c e l l s {(x,y,z):
S 3 } i s K’.
D e f i n e an orthogonu2 cube o f o r d e r n and b l o c k s i z e 3 (OC,(n))
as a 3-dimen-
s i o n a l a r r a y d e f i n e d on a s e t o f 3n elements V p a r t i t i o n e d i n t o n - s e t s G1,G2
and
G3 such t h a t
( 1 ) each c e l l i s empty o r c o n t a i n s a 3-subset o f V ( 2 ) t h e u n d e r l y i n g s e t o f b l o c k s forms a t r a n s v e r s a l d e s i g n TD(3,n). ( 3 ) each element o f V i s c o n t a i n e d i n e x a c t l y one c e l l o f any 2-dimensiona subarray.
LEMMA 4.1:
n # 6,lO or 14.
There e x i s t s an OC ( n ) f o r a22 n 2 4, 3
PROOF: F o r each v a l u e o f n 2 4 , n # 6,lO o r 14, t h e r e e x i s t t h r e e p a i r w i s e o r thogonal l a t i n squares o f o r d e r n.
Let L
9.
R
= (a..),
squares d e f i n e d on d i s j o i n t symbol s e t s V1,V2
x,y
E
I}.
and V3.
o r 3 be t h e l a t i n
We f o r m an n
R
symbol s e t V s e t {s,t,u)
x
ed i n c e l l (x,y,z)
o f 0.8.
n cube
and l e t
D e f i n e OoB t o be t h e OC3(n) d e f i n e d on t h e
B and h a v i n g p a r t i t i o n G1
i s c o n t a i n e d i n c e l l (x,y,z)
x
I t i s r e a d i l y checked t h a t 0 i s an
L e t 0 be an OC3(n) d e f i n e d on s e t V w i t h p a r t i t i o n G1,G2,G3 be any 3 - s e t o f elements.
n
o f Pi p l a c e t h e empty s e t i f x f i;
I n c e l l (x,y,i)
i f x = i, p l a c e t h e t r i p l e { a . * 1 G R G 31. 1Y. 0C3(”. B = {a,b,c}
x
L e t Pi be t h e p l a n e c o n s i s t i n g o f t h e
0 which i s indexed by I = {l,Z,...,n}. c e l l s {(x,y,i):
R = 1,2,
1J
x
{a),G2
x
{ b ) , G3
x
{ c } and such t h a t i f
o f 0 t h e n {(s,a),(t,b),(u,c)
(We a r e assuming h e r e t h a t S
€
G1,
t
€
I i s containG2, u
€
G3.)
We a r e now i n a p o s i t i o n t o s t a t e t h e s i n g u l a r d i r e c t p r o d u c t c o n s t r u c t i o n f o r Kirkman cubes.
704
A . Raso and S.A. Vanstone
THEOREM 4.2: If there exists a KC ( V ) and if there e x i s t s a KC ( v ) which 3 1 3 2 '2-'3 c0ntair.s a subcube KC3(v3) ad if there e x i s t s an OC (-) then there e x i s t s a 3 2 (v2-~3) KC ( v ) where v = (vl-l)-+ v 3 2 3' v.-1
Let r. = 1
1 ,1
In = {1,2y...yn}
G i
< 3,
Ki be a KC3(vi),
f o r each p o s i t i v e i n t e g e r n.
on symbol s e t Vi,
For an n
x
n
x
1
< i & 3 and
n array coordinat-
d e f i n e the i t h plane t o be t h e s e t o f a l l c e l l s o f the i z e d by the elements o f In a r r a y o f the form ( x y y y i ) , x,y E In. L e t K1 normalized w i t h r e s p e c t t o an element y. then c e l l ( i , i , i )
I.
c o n t a i n s t h e t r i p l e Iyyaiybi
I f K1
i s indexed by Ir
L e t 0 be t h e g i v e n OC3(r2
-
r3)
defined on a s e t U and l e t K i be t h e KC3(v2) obtained from K2 by w r i t i n g K2 on t h e symbol s e t U
x
{ajybj } u V3.
Without loss o f g e n e r a l i t y , we assume t h a t
t h e subarray K3 a r e normalized w i t h r e s p e c t t o an element
K g has t h e form:
where R!
= Ri
I
and
and t h e i t h plane of
I
i s the i t h plane of K3 f o r 1 G i G r3.
We now c o n s t r u c t a new a r r a y K" o f s i d e l e n g t h r 1 ( r 2 w r i t t e n on t h e symbol s e t V * = U x ( v l - { y l )
1 G i 6; r3 has t h e form:
my
Kj
U
V3.
-
r )
3
+ r3 which
The i t h plane o f Ka f o r
is
705
Kirkman cubes where
0
r e f e r s t o a 2-dimensional a r r a y o f empty c e l l s .
*
The j t h plane of K f o r r t (i-1)u t 1 3
I; j
G r3 t i u where u = r2
-
r3 and i
E
I
r
1 i t has the f o l l o w i n g form: L e t t = j r3 ( i - l ) u 1 and l e t OCt(k,k) = Pi(.t,k)oOC 3 t where Pi(!L,k) i s the block i n c e l l (a,k,i) o f K1, and OC3 i s the t - t h plane o f 0.
-
-
I
I
...
I
IN;
OCt(rl,i)
I
I
I
I
I
li
I
-
1
...
I ... I
1
-
r ) t r planes form a KC3(v). I t i s a s t r a i g h t f o r w a r d b u t tedious 3 3 task t o v e r i f y t h i s statement and so we omit the proof.
These rl(r2
I n order t o s t a t e a g e n e r a l i z a t i o n o f t h i s r e s u l t , we r e q u i r e the f o l l o w i n g definition. n
x
n
G1,G2,G3
x
An incomplete orthogonal cube o f order n and d e f i c i e n c y s i s an
n a r r a y 0 defined on a s e t o f elements V which i s p a r t i t i o n e d i n t o groups each o f c a r d i n a l i t y n along w i t h s-sets Hi
5 Gi,
1 G i d 3, such t h a t
(1) every c e l l o f 0 i s e i t h e r empty o r contains an unordered t r i p l e from V
( 2 ) every p a i r o f d i s t i n c t elements, one from Gi-Hi from Hi,one
,one from G.-H. o r one J J ( i # j ) i s contained i n a unique c e l l o f the array.
from G.-H. J J ( 3 ) there e x i s t s an s
x
s
x
s empty subarray S of 0 such t h a t the nonempty
c e l l s o f each plane o f 0 which contains no c e l l of S gives a p a r t i t i o n o f V and
706
A . Rosa and S.A. Vanstone
t h e nonempty c e l l s o f each plane o f O which c o n t a i n a c e l l o f S g i v e a p a r t i t i o n 3 Denote such an a r r a y by IOC(n,s). Examples o f these arrays a r e o f V-(iYIHi). e a s i l y constructed.
THEOREM 4.3: subcube
3
1
3
and i f f o r some non-negative integer u there e x i s t s an
v2-1
IOC(---
If t h e w e x i s t s a KC ( V ), a KC3(v2) which contains a KC ( v ) as a
2
v -1
-'' 2
-
a d i f there e x i s t s a KC ( v -1 3
v,-1 L *;:ere e x i s t s a K C 3 ( V 1 - 1 ) ( F
-
U)
1
v -1 3 2
-
+
u ) t 2u
1) then
+ 2~ + 1). M u l l i n [7,8]
This r e s u l t g e n e r a l i z e s a c o n s t r u c t i o n o f R.C. product. f o r skew Room squares.
I(-
3
f o r an i n d i r e c t
Again, t h e p r o o f o f t h e r e s u l t i s tedious and
s i n c e i t f o l l o w s the s p i r i t o f M u l l i n ' s p r o o f we omit i t . The hypotheses o f Theorem 4.2 and 4.3 can be weakened s l i g h t l y .
The
KC3(v1) or the KC3(v2) ( b u t n o t b o t h ) can be replaced by UMGRD(3,2,1,v1)
2 o r an UMGRD(3,2,1,v2)
o f index 2, r e s p e c t i v e l y ,
An UMGRD(3,2,1,v)
o f index
o f index 2 i s
a l s o denoted KS3(v) f o r a Kirkman square. We conclude t h i s s e c t i o n w i t h a r e s u l t on t r i p l i c a t i o n .
If there e x i s t s a KS ( v ) then tkere e x i s t s a KC (3v).
THEOREM 4.4:
3
3
PROOF: I f t h e r e e x i s t s a KS3(v), then v
=
3(mod 6 ) and v i s a t l e a s t 21.
t h e r e e x i s t 3 p a i r w i s e orthogonal l a t i n squares o f order v.
Hence,
Therefore, t h e r e
u G u G 1 2 3 Also, the blocks o f
e x i s t s a t r a n s v e r s a l design TD(3,v) d e f i n e d on a symbol s e t V = G where Giy
1 G i G 3 i s a v - s e t and i s a group of the design.
T admit a t l e a s t t h r e e r e s o l u t i o n s R1,R2
and R3 and t h i s s e t o f r e s o l u t i o n s i s 2-
orthogonal. L e t D. be a KS3(v) d e f i n e d on t h e symbol s e t Giy
1
1
S(i,j)
= {Sh(i,j):
1
Q
v-1 h
be a p a i r o f orthogonal r e s o l u t i o n s o f
1S j
S
t h e union o f t h e blocks o f T and t h e b l o c k s of D1,DE STS(3v) D.
i
Q
3, and l e t
2
D. f o r each i, 1 1
<
G i Q 3.
I f we consider
and D3 then we o b t a i n a
We now show t h a t t h i s i s t h e u n d e r l y i n g design f o r a KC3(3v). v-1 Let Let r =
-.
Kirkman cubes 3
Q,(j)
=
:$
Sh(i,j),
1
Q
r, 1
Q j Q
2.
and Q(j) = {Qh(j) I t i s n o t d i f f i c u l t t o check t h a t
M. = R. J J
U
Q(j , 1 Q j Q 2
a r e 2 r e s o l u t i o n s 0 , 0. Q h ( 3 ) = sh(lY1)
Let sht1(2Y1)
sht2(3Y1)
where s u b s c r i p t s a r e reduced modulo r and 1
Q(3) = { Q h ( 3 ) : 1 Q h
Q
Q
Let
h Q r.
rl
and M3 = R
3
u 4(3). We now show t h a t Mi,
M3 i s a r e s o l u t i o n o f D. orthogonal resolutions.
R1,R2.
Q
i Q 3, f o r m a s e t o f 3-
To p r o v e t h i s we need o n l y c o n s i d e r t h e i n t e r s e c t i o n o f
r e s o l u t i o n classes from Q ( j ) , i Q j S h ( i , l ) and S c ( i , 2 )
1
Q
3.
NOW,
Consider Q h ( 1 ) , Q c ( 2 ) , and Q f ( 3 ) .
have a t most one b l o c k i n comnon by t h e 2 - o r t h o g o n a l i t y o f
I f t h e i n t e r s e c t i o n i s empty t h e n
Qh(1) n Qc(2) n Qf(3) =
0.
I f t h e i n t e r s e c t i o n c o n t a i n s a b l o c k B ( i ) E Di,
t h e n I Q h ( l ) n Q,(2)I
= 3.
NOW,
t h e o n l y p o s s i b i l i t y f o r a nonempty i n t e r s e c t i o n o f Q n ( l ) n Q R ( 2 ) n Q f ( 3 ) r e q u i r e s f = h.
9<
There i s p r e c i s e l y one v a l u e i E {1,2,31
9,
such t h a t S h ( i ,1) E Q n ( 3 ) .
Hence
IQh(1)
Qc(2)
Qh(3)1 = 1 -
T h i s completes t h e p r o o f t h a t t h e r e s o l u t i o n s a r e 3-orthogonal and f o r m a KC3(3v).
708
A. Rosa and S.A. Vanstone
5 . EXISTENCE The main r e s u l t s on existence a r e d e r i v e d from Theorem 4.1 and
R.M.
L e t K be a s e t o f p o s i t i v e i n t e g e r s .
Wilson.
a(K) = gcd{k-1: k
E
6(K) = gcd { k ( k - l ) :
Define
K}
k
E
K).
THEOREM 5.1: There e x i s t s a positive integer c such that for a l l v v
l(mod a(K)),
v(v-1)
E
and KC3(21), take K = t7,lO).
Applying Theorem 4.1,
THEOREM 5.2: There e z i s t s a constant c
.
Since t h e r e e x i s t s a KC3(15)
From Theorem 5.1, we
Then a(K) = 3 and B(K) = 6.
h a v e t h a t a c o n s t a n t c e x i s t s such t h a t f o r a l l v
t k r e e x i s t s a KC ( v ) 3
> c and
0 (mod B(K)) there e x i s t s a PBD(v;K).
A p r o o f o f t h i s r e s u l t can be found i n [ 1 3 ] .
e x i s t s a PBD(v;K).
a result of
0
>c
and v
5
1 (mod 3), t h e r e
we have
such that f o r a l l v
>
c
0
and v
5
3 (mod62
The co i n t h e above theorem i s n o t s p e c i f i e d o t h e r than t h e f a c t t h a t i t i s a p o s i t i v e integer.
The computation o f an e x p l i c i t value f o r co w i l l we s u b j e c t
o f a subsequent a r t i c l e . We conclude w i t h a l i s t o f small values o f v f o r which a KC3(v) e x i s t s and how they are obtained. V
METHOD
15
DIRECT (SEE TABLE 1 )
21
DIRECT (SEE [ 5 1 )
27
DIRECT (SEE TABLE 2)
33
UNKNOWN
39
DIRECT (STARTER-ADDER, SEE TABLE 3 )
45
TRIPLICATION
51
UNKNOWN
57
UNKNOWN
63
TRIPLICATION
69
UNKNOWN
75
UNKNOWN
81
TRIPLICATION (THEOREM 4.4)
Kirkman cubes 87
UNKNOWN
93
UNKNOWN
99
THEOREM 4.1 OR THEOREM 4.2
ACKNOWLEDGEMENT: Research supported by NSERC G r a n t No. A7268 and by NSERC G r a n t
No. A9258.
709
A. Rosa and S.A. Vanstone
710 1 2 3 4 5 6 7
1 2 3 1 4 5 1 6 7 1 8 9 11011 11213 11415
41015 21214 21315 2 4 6 2 5 7 2 9 1 1 2 8 1 0
5 9 1 3 3 8 1 1 3 910 31317 31215 3 4 7 3 5 6
61112 6 9 1 5 4 812 51012 4 9 1 4 5 8 1 5 41113
7 8 1 4 71013 51114 71115 6 8 1 3 61014 7 9 1 2
1 2 3 4 5 6 7
1 2 3 1 4 5 1 6 7 1 8 9 11011 11213 11415
41113 21214 21315 2 4 6 2 5 7 2 9 1 1 2 8 1 0
5 8 1 5 3 910 3 8 1 1 31215 31314 3 5 6 3 4 7
61014 6 813 4 9 1 4 51114 4 8 1 2 41015 5 9 1 3
7 9 1 2 71115 51012 71013 6 9 1 5 7 8 1 4 61112
1 2 3 4 5 6 7
1 2 3 1 4 5 1 6 7 1 8 9 11011 11213 11415
4 8 1 2 21315 21214 2 5 7 2 4 6 2 8 1 0 2 9 1 1
51114 3 8 1 1 3 9 1 0 31314 31215 3 5 6 3 4 7
6 9 1 5 61014 41113 41015 5 9 1 3 4 9 1 4 51012
71013 7 9 1 2 5 8 1 5 61112 7 8 1 4 71115 6 8 1 3
t h r e e 3-orthogonal r e s o l u t i o n s of an STS(15). i n the l i s t of 1121.)
A 8et of
(1.7
TABLE 1.
Adderl:
0
4
2
M d e r 2:
0
2
16
8
8 11
1 8 15
4
0
8 17 18
5
6
A S t a r t e r and two a d d e r s f o r a
Table 3.
7
18 11 1 2
KC3(39).
10
2
8 18
.
1
2 5 8 11 14 17 20 23 26 13 14 4 18 9 11
4 7 10 13 16 19 22 25 2 7
3 6 9 12 15 18 21 24 27 27 21 20 22
5 1 1 1 7 8 3 8 6 3 11 12 5
16 21 15 26 17 24 5 9 12 23 10 24 14 25 15 19 17 19 16 23 15 16 13 17 12 25
9 20 21 3 6 4 5 6 8 10 4 7 2
13 24 22 18 10 11 11 18
14 14 10 10 17
20 25 26 24 26 27 26 21 20 18 25 22 23
19 7 6 2 3 2 9 2 3 9 20 21 3
23 13 12 14 4 6 12 4 5 15 22 23 15
27 19 27 26 8 7 24 9 7 21 27 25 27
10 15 17
8 4 8 1 1 1 1 1 5 5 3 7
11 16 16 16 13 6 12 11 17 13 11 18
23 19 27 22 25 8 20 21 20 24 19 26
6 12 11 6 9 5 4 11 12 19 2 9 4
14 14 13 13 17 15 15 14 15 24 12 16 14
22 16 18 23 25 22 23 17 18 26 19 26 24
8 2 3 19 21 20 7 5 2 1 3 2 8
18 10 10 22 24 23 17 10 16
4
6 5 13
25 21 20 25 27 26 27 27 24 7 9 8 21
4 3 2 7 5 9 10 3 9 6 1 1 6
12 17 15 15 18 14 13 13 10 11 18 14 16
26 22 25 20 19 19 16 26 23 25 23 27 20
1 1 24 18 27 14 23 17 21 1 1 20 12 21 18 22 16 25 13 22 12 22 17 26 15 24 1 10 19
7 9 4 4 2 3 2 7 4 8 8 6
9 f 3
I?
0-
2
Resolution 2: Apply permutation Resolution 3: Apply permutation TABLE 2
A. Rosa and S.A. Vanstone
712
BIBLIOGRAPHY
1. 2. 3.
4. S.
6. 7. 8.
9. 10. 11. 12. 13.
Anderson, S t a r t e r s , digraphs and Howell designs, U t i l i t a s Math., 14 (1978), 219-248. R. Fuji-Hara and S.A. Vanstone, On the spectrum of doubly resolvable Kirkman sys terns. Congressus Nmerantiwn, 28 (1 980), 399-407. F. Hoffman, P.J. Schellenberg and S.A. Vanstone, A s t a r t e d - a d d e r approach t o e q u i d i s t a n t permutation a r r a y s and generalized Room squares, Ars Cornhinutoria, 1 (1976), 307-319. E.S. Kramer and S.S. Magliveras, A 57 x 57 x 57 Room-type design, ilrs Com3inatoria, 9 ( 1980) , 163-1 66. R . Mathon, K. Phelps and A. Rosa, A c l a s s of Steiner t r i p l e systems of order 21 and a s s o c i a t e d Kirkman systems, Math. Cornput. (1981) ( t o appear). R. Mathon and S.A. Vanstone, On t h e e x i s t e n c e o f doubly r e s o l v a b l e Kirkman systems and e q u i d i s t a n t permutation a r r a y s , Discrete Math., 30 (1980), 157-1 72, R.C. Mullin, A s i n g u l a r i n d i r e c t product f o r Room squares, J.C.T. A ( t o appear). R.C. Mullin, O.R. Stinson and S.A. Vanstone, Kirkman t r i p l e systems cont a i n i n g maximum subdesigns, U t i l i t a s Math. ( t o appear). A. Rosa, Generalized Howell designs, Annals iVew York Acad. Sci., 319 ( 1 979) , 484-489. A. Rosa, Room squares generalized, Annals Discrete Math., 8 (1980), 45-57. S.A. Vanstone, Doubly resolvable designs, Discrete Math., 29 (1980), 77-86. H.S. White, F.N. Cole and L.D. Cumnings, Complete c l a s s i f i c a t i o n of t h e t r i a d systems on f i f t e e n elements, Memoirs Nut. Acad. Sci. U.S.A. 14, 2nd memoir (1919), 1-89. R.M. Wilson, An e x i s t e n c e theory f o r pairwise balanced designs 111: Proof o f the e x i s t e n c e conjecture, J.C.T. A 18 (1975), 71-79. B.A.
Department o f Mathematics McMaster University Hami 1ton, Ontario Canada L8S 4K1
Department of Combinatorics & Optimization University of Waterloo Waterloo, Ontario Canada N2L 361
Annals of Discrete Mathematics 18 (1983) 699-712 North-Holland Publishing Company
KIRKMAN CUBES
A. Rosa* and S.A.
Vanstone
1. INTRODUCTION P r e s e n t l y t h e r e appears t o be q u i t e a l o t o f i n t e r e s t i n v a r i o u s g e n e r a l i z a t i o n s o f Room squares, such as Howell designs [ lI, g e n e r a l i z e d Room squares 131, Kirkman squares [ 6, 111 , g e n e r a l i z e d Howell designs [ 91, Room r e c t a n g l e s [ 41- f o r a survey on these and v a r i o u s f u r t h e r g e n e r a l i z a t i o n s o f Room squares i n c l u d i n g those t o h i g h e r dimensions see [ l o ] .
The concept o f o r t h o g o n a l i t y o f r e s o l u t i o n s
o f t h e associated u n d e r l y i n g design i s common t o a l l these g e n e r a l i z a t i o n s .
By
r e l a x i n g t h e usual concept o f o r t h o g o n a l i t y , several a d d i t i o n a l i n t e r e s t i n g obj e c t s can be d e f i n e d t h a t s t i l l r e t a i n t h e e s t h e t i c a l l y and o t h e r w i s e d e s i r a b l e p r o p e r t i e s o f u n i f o r m i t y , r e g u l a r i t y and balance.
I n t h i s paper we study t h e
e x i s t e n c e o f one k i n d o f these o b j e c t s i n t h r e e dimensions, which we c a l l Kirkman cubes [ 101 (as t h e i r u n d e r l y i n g design i s a Kirkman t r i p l e system).
2. BASIC DEFINITIONS AND PROPERTIES A uniform multidimensional generalized Room design o f degree k, dimension d, m u l t i p l i c i t y A and o r d e r v ( b r i e f l y UMGRD (k,d,A,v),
o f [ l o ] ) i s a d-dimensional
a r r a y such t h a t ( i ) every c e l l o f t h e a r r a y i s e i t h e r empty o r c o n t a i n s a k-subset o f a v-set
V,
( i i ) every element o f V i s contained i n e x a c t l y one c e l l of any (d-1)-dimens i o n a l subarray ( i i i ) every 2-subset o f V i s contained i n e x a c t l y h c e l l s o f t h e a r r a y . Thus t h e s e t o f k-subsets i n t h e nonempty c e l l s o f a UMGRD i s t h e s e t o f b l o c k s o f a (v,k,h)-design.
A UMGRD (k,d,h,v)
i s regular of index t ( 2 G t G d )
i f i t s p r o j e c t i o n on any t dimensions i s a UMGRD (k,t,A,v)
but p r o j e c t i o n on any
( t - 1 ) dimensions i s never a UMGRD. Examples of UMGRDs i n c l u d e multidimensional
700
A . Rosa and S.A. Vanstone
[lo]),
Room designs (cf.
o r , f o r instance, a r e g u l a r UMGRD (8,12,1,24)
obtained from t h e unique S(5,8,24)
o f index 2
(see [ 4 1 ) .
I t i s sometimes t o our advantage t o view UMGRDs as block designs possessing
m u l t i p l e resolutions.
If (V,B)
To e s t a b l i s h t h i s equivalence, we need a few d e f i n i t i o n s .
i s a (v,k,h)-BIBD
i s c a l l e d a puraZZeS class.
then any s e t C o f blocks i n B t h a t p a r t i t i o n s V
Any s e t o f d i s j o i n t p a r a l l e l classes R = {C1,.,.,Cr}
t h a t p a r t i t i o n s B i s c a l l e d a r e s o l u t i o n ; a B I B D a d m i t t i n g a t l e a s t one r e s o l u t i o n
i s c a l l e d rcsolvablc.
...,
Suppose now t h a t a (v,k,h)-BIBD admits a s e t R = IR1,R2, R S ] o f s resolui i i t i o n s where Ri = { C l,...,C 1 , C.’s a r e p a r a l l e l classes. We F a l l !he s e t R a dn~ 1 m$hogmzal szt; of resolutions i f f o r any d p a r a l l e l classes C . l , C’ ‘,...,Cid 1 J1 j2 Jd E Rk), (where C Jk
i
jcjl
1
i
n c.2 n
J2
... n c i d l Jd
< 1.
A d-orthogonal s e t R of r e s o l u t i o n s i s s a i d t o be regular o f index t ( 2 i f R i s a t-orthogonal s e t b u t c o n t a i n s no (t-1)-orthogonal
<
t d d)
subset.
The aforementioned equivalence i s then almost immediate (cf. [ 101 ) :
A r e g u l a r UMGRD (k,d,x,v)
o f index t e x i s t s i f and o n l y if t h e r e e x i s t s a d-
orthogonal s e t o f d r e s o l u t i o n s o f a (v,k,x)-BIBD
t h a t i s o f index t.
I n what f o l l o w s we w i l l be concerned w i t h a s p e c i a l case o f UMGRDs when k = 3, d = 3,
= 1 and index t = 3; these w i l l be henceforth c a l l e d Kirkman cubes
and denoted by KC3(v).
[ I n c o n t r a s t , a UMGRD w i t h k = 3, d = 3, h = 1 and index
t = 2 w i l l be c a l l e d a strong Kirkman cube SKC ( v ) ;
u n t i l very r e c e n t l y , no exam2 p l e s o f s t r o n g Kirkman cubes were known, however, j u s t p r i o r t o t h i s meeting, t h e second author succeeded i n c o n s t r u c t i n g an SKC (255).].
3
Thus, a Kirkman cube KC ( v ) i s a 3-dimensional a r r a y on v elements such t h a t 3 the s e t o f t r i p l e s i n t h e nonempty c e l l s o f each o f i t s planes i s a p a r a l l e l class, and the s e t o f t r i p l e s i n t h e nonempty c e l l s o f t h e whole a r r a y i s t h e s e t o f t r i p l e s o f an STS(v) w h i l e t h e p r o j e c t i o n on any two dimensions never y i e l d s a Kirkman square ( i . e .
a UMGRD w i t h k = 3, d = 2, h = 1, t = 2, c f . [ 2 ] ) .
70 1
Kirkman cubes 3. KIRKMAN CUBES OF SMALL ORDERS
Obviously the t r i v i a l necessary c o n d i t i o n f o r the existence o f a KC3(v) i s that v
3 (mod 6), v
> 3.
Since the a f f i n e plane o f order 3 has a unique resolu-
t i o n there e x i s t s no KC3(9).
I t i s well-known t h a t there are f o u r resolvable
Steiner t r i p l e systems o f order 15. and Cumnings [12] (= PG(3,2)),
O f these,
No. 1.1 on the l i s t o f White, Cole
as w e l l as No. 1.7 on the same l i s t , admit a s e t
( a c t u a l l y , several sets) o f three 3-orthogonal resolutions, while the remaining two resolvable STS(15) do n o t admit such a set. example o f such i s i n Table 1.
Thus a KC3(15) e x i s t s , and an
There a l s o e x i s t several KC3(21)'s (see [ 5 ] ) ,
an example o f a KC3(27) i s easy t o o b t a i n from AG(3,3)
(see Table 2.)
and
The e x i s -
tence o f a KC3(33) i s undecided a t t h i s p o i n t .
A general method o f constructing a KC3(v) d i r e c t l y i s the well-known starter-adder method (see, e.g.,
[3]). The smallest order f o r which we succeeded
i n constructing a Kirkman cube by t h i s method i s v = 39 (see Table 3). P o t e n t i a l l y important f o r b u i l d i n g up small order Kirkman cubes i s a " t r i p l i c a t i o n method" producing a KC3(v) from a KC3(v).
U n l i k e i n the case o f
Kirkman squares, t h i s method i s a t l e a s t f e a s i b l e and has indeed been used t o obt a i n a KC3(45) and a KC3(63). "twisted" d i r e c t product.
The construction involves, as i s t o be expected, a
Unfortunately, t h e general features o f one o f the
components o f t h i s construction remain e l u s i v e so t h a t a t present a t l e a s t the t r i p l i c a t i o n i s o n l y a hypothetical construction i n general (although i t has been successful on every order on which i t has been tested).
I n view o f the r e s u l t s o f
the next section, t h i s may be a l l t h a t i s needed t o completely s e t t l e the e x i s tence question f o r Kirkman cubes.
4. M A I N RECURSIVE CONSTRUCTIONS I n t h i s section, we describe various recursive constructions f o r KC3(v)s. The constructions are, o f course, more general and w i l l apply t o any UMGRD (k,t,X,v).
The most powerful o f these constructions i s PBD-closure
I131 ). L e t B be a s e t o f k elements. Define VG = B from the elements i n B the symbol s e t VB. KB(x,y,z)
x
{l,ZI.
x
I1,2} u { = I where
Suppose t h e r e e x i s t s a KC3(21Bl
+
Coordinatize t h i s cube KB by the elements o f B.
be the block o f K contained i n the c e l l (x,y,z);of
(see Wilson
-
i s distinct
1) defined on Let
course, KB(x,y.z)
=
0
702
A , Rosa and S.A. Vanstone
i f the c e l l (x,y,z)
Without l o s s o f g e n e r a l i t y , we assume t h a t
i s empty.
i s the b l o c k {m,x1,x2]
KB(x,x,x)
(NOTE: For convenience we w r i t e xi i n
o f K.
place o f ( x , i ) . ) .
THEOREM 4.1: ist D be a PBD(v;K) such t h a t for each k
f
;i:eri tizezv e x i s t s a KC ( Z V + 1 ) . 3
KC3(2k+l).
PROOF: L e t V be the p o i n t s e t o f D and 6 t h e block set. where
9 V
For each B
K there e d s t s a
x
(1,2}.
Coordinatize a v
x
v
x
DefineV" = V
x
{l,Z}u{m)
v cube K"' w i t h t h e elements of V. The
consider t h e subcube o f K* determined by t h e elements o f B.
E 6,
e n t r i e s ( c e l l s ) i n t h i s subcube a r e f i l l e d i n by t h e e n t r i e s i n KB.
We now show
t n a t K" i s a K C ~ ( E V t 1).
15 V".
Consider any p a i r {ai,b.
b . i s i n a unique c e l l o f K and, hence, a J B I f i # j and a # b, t h e same argument holds. I f i # j and
i n a unique block B o f 8 . unique c e l l o f Kn.
I f i = j , then a # b and I a , b } i s contained
J Then ai,
a = b then {a.,b.} i s contained i n c e l l (a,a,a) o f K". 7 .'. J !-,ai ), ai f V" -{-I , i s contained i n a unique c e l l .
I t i s e a s i l y seen t h a t
Therefore, t h e nonempty
c e l l s o f K" are t h e blocks o f a (Ev+1,3,1)-BIBD. Consider any plane o f K". form a plane o f K" f o r f i x e d a. a.
For instance, t h e c e l l s (a,x,y), L e t B1,B z,...,Bt
f o r a l l x,y E V,
be t h e blocks o f D which c o n t a i n
Then, iBi-{a 1: 1
i G tl
i s a p a r t i t i o n o f V-{al. (x,y,z)
9f
of K P(Bi)
.
be t h e block of D contained i n c e l l
This block i s t h e empty s e t i f t h e c e l l i s empty.
= IK"(a,x,y):
1
Then, iP(Bi):
L e t KR(x,y,z)
Q
x,y
f
Bill
Let
1 G i d t.
i Q t } induces a p a r t i t i o n o f the plane through a i n K
into
, 1 < i d t. Since each plane through a i n KB . Bi contains each element o f V p r e c i s e l y once and the o n l y common c e l l t o a l l o f Bi these planes i s (a,a,a) which contains Im,a a 1 then the plane through a i n K 1' 2 ;5 c o n t a i n s each element o f V p r e c i s e l y once. F i n a l l y , consider t h e p r o j e c t i o n o f planes through a i n each K
any KB, B E 6, o n t o the corresponding face. i s a KC3(2v
t
1) and t h e p r o o f i s complete.
Since KB i s a KC3(21Bl + 1 ) then K"
Kirkman cubes
703
I n s e c t i o n 5, we d i s p l a y t h e power o f t h i s c o n s t r u c t i o n . The n e x t r e c u r s i v e c o n s t r u c t i o n i s one which i s t h e analogue o f t h e s i n g u l a r d i r e c t p r o d u c t c o n s t r u c t i o n f o r Room squares ( [ 71).
Before g i v i n g t h e construc-
t i o n , we r e q u i r e s e v e r a l d e f i n i t i o n s .
A KC ( v ) i s s a i d t o be n o r m a l i z e d i f a l l c e l l s o f t h e form (x,x,x) (the 3 main d i a g o n a l ) c o n t a i n a common element. C l e a r l y , any KC ( v ) can be n o r m a l i z e d by 3 s u i t a b l e p e r m u t a t i o n s o f c o o r d i n a t e s . L e t K be a KC3(v) whose c o o r d i n a t e s a r e
A KC3(u) K ’ i s s a i d t o be a subcube o f K i f t h e r e e x i s t s subsets
indexed by V.
o f V such t h a t t h e s u b a r r a y
S1,S2,S3 y E
S2, z
E
x E Sly
determined by t h e c e l l s {(x,y,z):
S 3 } i s K’.
D e f i n e an orthogonu2 cube o f o r d e r n and b l o c k s i z e 3 (OC,(n))
as a 3-dimen-
s i o n a l a r r a y d e f i n e d on a s e t o f 3n elements V p a r t i t i o n e d i n t o n - s e t s G1,G2
and
G3 such t h a t
( 1 ) each c e l l i s empty o r c o n t a i n s a 3-subset o f V ( 2 ) t h e u n d e r l y i n g s e t o f b l o c k s forms a t r a n s v e r s a l d e s i g n TD(3,n). ( 3 ) each element o f V i s c o n t a i n e d i n e x a c t l y one c e l l o f any 2-dimensiona subarray.
LEMMA 4.1:
n # 6,lO or 14.
There e x i s t s an OC ( n ) f o r a22 n 2 4, 3
PROOF: F o r each v a l u e o f n 2 4 , n # 6,lO o r 14, t h e r e e x i s t t h r e e p a i r w i s e o r thogonal l a t i n squares o f o r d e r n.
Let L
9.
R
= (a..),
squares d e f i n e d on d i s j o i n t symbol s e t s V1,V2
x,y
E
I}.
and V3.
o r 3 be t h e l a t i n
We f o r m an n
R
symbol s e t V s e t {s,t,u)
x
ed i n c e l l (x,y,z)
o f 0.8.
n cube
and l e t
D e f i n e OoB t o be t h e OC3(n) d e f i n e d on t h e
B and h a v i n g p a r t i t i o n G1
i s c o n t a i n e d i n c e l l (x,y,z)
x
I t i s r e a d i l y checked t h a t 0 i s an
L e t 0 be an OC3(n) d e f i n e d on s e t V w i t h p a r t i t i o n G1,G2,G3 be any 3 - s e t o f elements.
n
o f Pi p l a c e t h e empty s e t i f x f i;
I n c e l l (x,y,i)
i f x = i, p l a c e t h e t r i p l e { a . * 1 G R G 31. 1Y. 0C3(”. B = {a,b,c}
x
L e t Pi be t h e p l a n e c o n s i s t i n g o f t h e
0 which i s indexed by I = {l,Z,...,n}. c e l l s {(x,y,i):
R = 1,2,
1J
x
{a),G2
x
{ b ) , G3
x
{ c } and such t h a t i f
o f 0 t h e n {(s,a),(t,b),(u,c)
(We a r e assuming h e r e t h a t S
€
G1,
t
€
I i s containG2, u
€
G3.)
We a r e now i n a p o s i t i o n t o s t a t e t h e s i n g u l a r d i r e c t p r o d u c t c o n s t r u c t i o n f o r Kirkman cubes.
704
A . Raso and S.A. Vanstone
THEOREM 4.2: If there exists a KC ( V ) and if there e x i s t s a KC ( v ) which 3 1 3 2 '2-'3 c0ntair.s a subcube KC3(v3) ad if there e x i s t s an OC (-) then there e x i s t s a 3 2 (v2-~3) KC ( v ) where v = (vl-l)-+ v 3 2 3' v.-1
Let r. = 1
1 ,1
In = {1,2y...yn}
G i
< 3,
Ki be a KC3(vi),
f o r each p o s i t i v e i n t e g e r n.
on symbol s e t Vi,
For an n
x
n
x
1
< i & 3 and
n array coordinat-
d e f i n e the i t h plane t o be t h e s e t o f a l l c e l l s o f the i z e d by the elements o f In a r r a y o f the form ( x y y y i ) , x,y E In. L e t K1 normalized w i t h r e s p e c t t o an element y. then c e l l ( i , i , i )
I.
c o n t a i n s t h e t r i p l e Iyyaiybi
I f K1
i s indexed by Ir
L e t 0 be t h e g i v e n OC3(r2
-
r3)
defined on a s e t U and l e t K i be t h e KC3(v2) obtained from K2 by w r i t i n g K2 on t h e symbol s e t U
x
{ajybj } u V3.
Without loss o f g e n e r a l i t y , we assume t h a t
t h e subarray K3 a r e normalized w i t h r e s p e c t t o an element
K g has t h e form:
where R!
= Ri
I
and
and t h e i t h plane of
I
i s the i t h plane of K3 f o r 1 G i G r3.
We now c o n s t r u c t a new a r r a y K" o f s i d e l e n g t h r 1 ( r 2 w r i t t e n on t h e symbol s e t V * = U x ( v l - { y l )
1 G i 6; r3 has t h e form:
my
Kj
U
V3.
-
r )
3
+ r3 which
The i t h plane o f Ka f o r
is
705
Kirkman cubes where
0
r e f e r s t o a 2-dimensional a r r a y o f empty c e l l s .
*
The j t h plane of K f o r r t (i-1)u t 1 3
I; j
G r3 t i u where u = r2
-
r3 and i
E
I
r
1 i t has the f o l l o w i n g form: L e t t = j r3 ( i - l ) u 1 and l e t OCt(k,k) = Pi(.t,k)oOC 3 t where Pi(!L,k) i s the block i n c e l l (a,k,i) o f K1, and OC3 i s the t - t h plane o f 0.
-
-
I
I
...
I
IN;
OCt(rl,i)
I
I
I
I
I
li
I
-
1
...
I ... I
1
-
r ) t r planes form a KC3(v). I t i s a s t r a i g h t f o r w a r d b u t tedious 3 3 task t o v e r i f y t h i s statement and so we omit the proof.
These rl(r2
I n order t o s t a t e a g e n e r a l i z a t i o n o f t h i s r e s u l t , we r e q u i r e the f o l l o w i n g definition. n
x
n
G1,G2,G3
x
An incomplete orthogonal cube o f order n and d e f i c i e n c y s i s an
n a r r a y 0 defined on a s e t o f elements V which i s p a r t i t i o n e d i n t o groups each o f c a r d i n a l i t y n along w i t h s-sets Hi
5 Gi,
1 G i d 3, such t h a t
(1) every c e l l o f 0 i s e i t h e r empty o r contains an unordered t r i p l e from V
( 2 ) every p a i r o f d i s t i n c t elements, one from Gi-Hi from Hi,one
,one from G.-H. o r one J J ( i # j ) i s contained i n a unique c e l l o f the array.
from G.-H. J J ( 3 ) there e x i s t s an s
x
s
x
s empty subarray S of 0 such t h a t the nonempty
c e l l s o f each plane o f 0 which contains no c e l l of S gives a p a r t i t i o n o f V and
706
A . Rosa and S.A. Vanstone
t h e nonempty c e l l s o f each plane o f O which c o n t a i n a c e l l o f S g i v e a p a r t i t i o n 3 Denote such an a r r a y by IOC(n,s). Examples o f these arrays a r e o f V-(iYIHi). e a s i l y constructed.
THEOREM 4.3: subcube
3
1
3
and i f f o r some non-negative integer u there e x i s t s an
v2-1
IOC(---
If t h e w e x i s t s a KC ( V ), a KC3(v2) which contains a KC ( v ) as a
2
v -1
-'' 2
-
a d i f there e x i s t s a KC ( v -1 3
v,-1 L *;:ere e x i s t s a K C 3 ( V 1 - 1 ) ( F
-
U)
1
v -1 3 2
-
+
u ) t 2u
1) then
+ 2~ + 1). M u l l i n [7,8]
This r e s u l t g e n e r a l i z e s a c o n s t r u c t i o n o f R.C. product. f o r skew Room squares.
I(-
3
f o r an i n d i r e c t
Again, t h e p r o o f o f t h e r e s u l t i s tedious and
s i n c e i t f o l l o w s the s p i r i t o f M u l l i n ' s p r o o f we omit i t . The hypotheses o f Theorem 4.2 and 4.3 can be weakened s l i g h t l y .
The
KC3(v1) or the KC3(v2) ( b u t n o t b o t h ) can be replaced by UMGRD(3,2,1,v1)
2 o r an UMGRD(3,2,1,v2)
o f index 2, r e s p e c t i v e l y ,
An UMGRD(3,2,1,v)
o f index
o f index 2 i s
a l s o denoted KS3(v) f o r a Kirkman square. We conclude t h i s s e c t i o n w i t h a r e s u l t on t r i p l i c a t i o n .
If there e x i s t s a KS ( v ) then tkere e x i s t s a KC (3v).
THEOREM 4.4:
3
3
PROOF: I f t h e r e e x i s t s a KS3(v), then v
=
3(mod 6 ) and v i s a t l e a s t 21.
t h e r e e x i s t 3 p a i r w i s e orthogonal l a t i n squares o f order v.
Hence,
Therefore, t h e r e
u G u G 1 2 3 Also, the blocks o f
e x i s t s a t r a n s v e r s a l design TD(3,v) d e f i n e d on a symbol s e t V = G where Giy
1 G i G 3 i s a v - s e t and i s a group of the design.
T admit a t l e a s t t h r e e r e s o l u t i o n s R1,R2
and R3 and t h i s s e t o f r e s o l u t i o n s i s 2-
orthogonal. L e t D. be a KS3(v) d e f i n e d on t h e symbol s e t Giy
1
1
S(i,j)
= {Sh(i,j):
1
Q
v-1 h
be a p a i r o f orthogonal r e s o l u t i o n s o f
1S j
S
t h e union o f t h e blocks o f T and t h e b l o c k s of D1,DE STS(3v) D.
i
Q
3, and l e t
2
D. f o r each i, 1 1
<
G i Q 3.
I f we consider
and D3 then we o b t a i n a
We now show t h a t t h i s i s t h e u n d e r l y i n g design f o r a KC3(3v). v-1 Let Let r =
-.
Kirkman cubes 3
Q,(j)
=
:$
Sh(i,j),
1
Q
r, 1
Q j Q
2.
and Q(j) = {Qh(j) I t i s n o t d i f f i c u l t t o check t h a t
M. = R. J J
U
Q(j , 1 Q j Q 2
a r e 2 r e s o l u t i o n s 0 , 0. Q h ( 3 ) = sh(lY1)
Let sht1(2Y1)
sht2(3Y1)
where s u b s c r i p t s a r e reduced modulo r and 1
Q(3) = { Q h ( 3 ) : 1 Q h
Q
Q
Let
h Q r.
rl
and M3 = R
3
u 4(3). We now show t h a t Mi,
M3 i s a r e s o l u t i o n o f D. orthogonal resolutions.
R1,R2.
Q
i Q 3, f o r m a s e t o f 3-
To p r o v e t h i s we need o n l y c o n s i d e r t h e i n t e r s e c t i o n o f
r e s o l u t i o n classes from Q ( j ) , i Q j S h ( i , l ) and S c ( i , 2 )
1
Q
3.
NOW,
Consider Q h ( 1 ) , Q c ( 2 ) , and Q f ( 3 ) .
have a t most one b l o c k i n comnon by t h e 2 - o r t h o g o n a l i t y o f
I f t h e i n t e r s e c t i o n i s empty t h e n
Qh(1) n Qc(2) n Qf(3) =
0.
I f t h e i n t e r s e c t i o n c o n t a i n s a b l o c k B ( i ) E Di,
t h e n I Q h ( l ) n Q,(2)I
= 3.
NOW,
t h e o n l y p o s s i b i l i t y f o r a nonempty i n t e r s e c t i o n o f Q n ( l ) n Q R ( 2 ) n Q f ( 3 ) r e q u i r e s f = h.
9<
There i s p r e c i s e l y one v a l u e i E {1,2,31
9,
such t h a t S h ( i ,1) E Q n ( 3 ) .
Hence
IQh(1)
Qc(2)
Qh(3)1 = 1 -
T h i s completes t h e p r o o f t h a t t h e r e s o l u t i o n s a r e 3-orthogonal and f o r m a KC3(3v).
708
A. Rosa and S.A. Vanstone
5 . EXISTENCE The main r e s u l t s on existence a r e d e r i v e d from Theorem 4.1 and
R.M.
L e t K be a s e t o f p o s i t i v e i n t e g e r s .
Wilson.
a(K) = gcd{k-1: k
E
6(K) = gcd { k ( k - l ) :
Define
K}
k
E
K).
THEOREM 5.1: There e x i s t s a positive integer c such that for a l l v v
l(mod a(K)),
v(v-1)
E
and KC3(21), take K = t7,lO).
Applying Theorem 4.1,
THEOREM 5.2: There e z i s t s a constant c
.
Since t h e r e e x i s t s a KC3(15)
From Theorem 5.1, we
Then a(K) = 3 and B(K) = 6.
h a v e t h a t a c o n s t a n t c e x i s t s such t h a t f o r a l l v
t k r e e x i s t s a KC ( v ) 3
> c and
0 (mod B(K)) there e x i s t s a PBD(v;K).
A p r o o f o f t h i s r e s u l t can be found i n [ 1 3 ] .
e x i s t s a PBD(v;K).
a result of
0
>c
and v
5
1 (mod 3), t h e r e
we have
such that f o r a l l v
>
c
0
and v
5
3 (mod62
The co i n t h e above theorem i s n o t s p e c i f i e d o t h e r than t h e f a c t t h a t i t i s a p o s i t i v e integer.
The computation o f an e x p l i c i t value f o r co w i l l we s u b j e c t
o f a subsequent a r t i c l e . We conclude w i t h a l i s t o f small values o f v f o r which a KC3(v) e x i s t s and how they are obtained. V
METHOD
15
DIRECT (SEE TABLE 1 )
21
DIRECT (SEE [ 5 1 )
27
DIRECT (SEE TABLE 2)
33
UNKNOWN
39
DIRECT (STARTER-ADDER, SEE TABLE 3 )
45
TRIPLICATION
51
UNKNOWN
57
UNKNOWN
63
TRIPLICATION
69
UNKNOWN
75
UNKNOWN
81
TRIPLICATION (THEOREM 4.4)
Kirkman cubes 87
UNKNOWN
93
UNKNOWN
99
THEOREM 4.1 OR THEOREM 4.2
ACKNOWLEDGEMENT: Research supported by NSERC G r a n t No. A7268 and by NSERC G r a n t
No. A9258.
709
A. Rosa and S.A. Vanstone
710 1 2 3 4 5 6 7
1 2 3 1 4 5 1 6 7 1 8 9 11011 11213 11415
41015 21214 21315 2 4 6 2 5 7 2 9 1 1 2 8 1 0
5 9 1 3 3 8 1 1 3 910 31317 31215 3 4 7 3 5 6
61112 6 9 1 5 4 812 51012 4 9 1 4 5 8 1 5 41113
7 8 1 4 71013 51114 71115 6 8 1 3 61014 7 9 1 2
1 2 3 4 5 6 7
1 2 3 1 4 5 1 6 7 1 8 9 11011 11213 11415
41113 21214 21315 2 4 6 2 5 7 2 9 1 1 2 8 1 0
5 8 1 5 3 910 3 8 1 1 31215 31314 3 5 6 3 4 7
61014 6 813 4 9 1 4 51114 4 8 1 2 41015 5 9 1 3
7 9 1 2 71115 51012 71013 6 9 1 5 7 8 1 4 61112
1 2 3 4 5 6 7
1 2 3 1 4 5 1 6 7 1 8 9 11011 11213 11415
4 8 1 2 21315 21214 2 5 7 2 4 6 2 8 1 0 2 9 1 1
51114 3 8 1 1 3 9 1 0 31314 31215 3 5 6 3 4 7
6 9 1 5 61014 41113 41015 5 9 1 3 4 9 1 4 51012
71013 7 9 1 2 5 8 1 5 61112 7 8 1 4 71115 6 8 1 3
t h r e e 3-orthogonal r e s o l u t i o n s of an STS(15). i n the l i s t of 1121.)
A 8et of
(1.7
TABLE 1.
Adderl:
0
4
2
M d e r 2:
0
2
16
8
8 11
1 8 15
4
0
8 17 18
5
6
A S t a r t e r and two a d d e r s f o r a
Table 3.
7
18 11 1 2
KC3(39).
10
2
8 18
.
1
2 5 8 11 14 17 20 23 26 13 14 4 18 9 11
4 7 10 13 16 19 22 25 2 7
3 6 9 12 15 18 21 24 27 27 21 20 22
5 1 1 1 7 8 3 8 6 3 11 12 5
16 21 15 26 17 24 5 9 12 23 10 24 14 25 15 19 17 19 16 23 15 16 13 17 12 25
9 20 21 3 6 4 5 6 8 10 4 7 2
13 24 22 18 10 11 11 18
14 14 10 10 17
20 25 26 24 26 27 26 21 20 18 25 22 23
19 7 6 2 3 2 9 2 3 9 20 21 3
23 13 12 14 4 6 12 4 5 15 22 23 15
27 19 27 26 8 7 24 9 7 21 27 25 27
10 15 17
8 4 8 1 1 1 1 1 5 5 3 7
11 16 16 16 13 6 12 11 17 13 11 18
23 19 27 22 25 8 20 21 20 24 19 26
6 12 11 6 9 5 4 11 12 19 2 9 4
14 14 13 13 17 15 15 14 15 24 12 16 14
22 16 18 23 25 22 23 17 18 26 19 26 24
8 2 3 19 21 20 7 5 2 1 3 2 8
18 10 10 22 24 23 17 10 16
4
6 5 13
25 21 20 25 27 26 27 27 24 7 9 8 21
4 3 2 7 5 9 10 3 9 6 1 1 6
12 17 15 15 18 14 13 13 10 11 18 14 16
26 22 25 20 19 19 16 26 23 25 23 27 20
1 1 24 18 27 14 23 17 21 1 1 20 12 21 18 22 16 25 13 22 12 22 17 26 15 24 1 10 19
7 9 4 4 2 3 2 7 4 8 8 6
9 f 3
I?
0-
2
Resolution 2: Apply permutation Resolution 3: Apply permutation TABLE 2
A. Rosa and S.A. Vanstone
712
BIBLIOGRAPHY
1. 2. 3.
4. S.
6. 7. 8.
9. 10. 11. 12. 13.
Anderson, S t a r t e r s , digraphs and Howell designs, U t i l i t a s Math., 14 (1978), 219-248. R. Fuji-Hara and S.A. Vanstone, On the spectrum of doubly resolvable Kirkman sys terns. Congressus Nmerantiwn, 28 (1 980), 399-407. F. Hoffman, P.J. Schellenberg and S.A. Vanstone, A s t a r t e d - a d d e r approach t o e q u i d i s t a n t permutation a r r a y s and generalized Room squares, Ars Cornhinutoria, 1 (1976), 307-319. E.S. Kramer and S.S. Magliveras, A 57 x 57 x 57 Room-type design, ilrs Com3inatoria, 9 ( 1980) , 163-1 66. R . Mathon, K. Phelps and A. Rosa, A c l a s s of Steiner t r i p l e systems of order 21 and a s s o c i a t e d Kirkman systems, Math. Cornput. (1981) ( t o appear). R. Mathon and S.A. Vanstone, On t h e e x i s t e n c e o f doubly r e s o l v a b l e Kirkman systems and e q u i d i s t a n t permutation a r r a y s , Discrete Math., 30 (1980), 157-1 72, R.C. Mullin, A s i n g u l a r i n d i r e c t product f o r Room squares, J.C.T. A ( t o appear). R.C. Mullin, O.R. Stinson and S.A. Vanstone, Kirkman t r i p l e systems cont a i n i n g maximum subdesigns, U t i l i t a s Math. ( t o appear). A. Rosa, Generalized Howell designs, Annals iVew York Acad. Sci., 319 ( 1 979) , 484-489. A. Rosa, Room squares generalized, Annals Discrete Math., 8 (1980), 45-57. S.A. Vanstone, Doubly resolvable designs, Discrete Math., 29 (1980), 77-86. H.S. White, F.N. Cole and L.D. Cumnings, Complete c l a s s i f i c a t i o n of t h e t r i a d systems on f i f t e e n elements, Memoirs Nut. Acad. Sci. U.S.A. 14, 2nd memoir (1919), 1-89. R.M. Wilson, An e x i s t e n c e theory f o r pairwise balanced designs 111: Proof o f the e x i s t e n c e conjecture, J.C.T. A 18 (1975), 71-79. B.A.
Department o f Mathematics McMaster University Hami 1ton, Ontario Canada L8S 4K1
Department of Combinatorics & Optimization University of Waterloo Waterloo, Ontario Canada N2L 361