WurnaI of Stat&W PIarming md Inference 3 (197%1 367-379, @ Wrth~Hoiland PubIishing Company

S.S. sHItIIamwE* kfichigan $trrOe ~nitn%&y, &st Lanshg, MI, U.S~A.

N.M. SINGHI TM h&We of Fun&zmental Research, Bombcty, India

Received 2 October 1978 Recommended by Esther Seiden

Abstmct: Let x z CI and n 2 2 be integers. Suppose there exists an orthogonal array A (n, q, g *) of strength 2 in n symbolswith q rows 2nd n$* columns where q = q*- Cs, q* = n2x + n + 1, p*=(n-1)x+1 and d is a positive integer. Then d is called the deficiency of the orthogonal array. The question of embedding such an iarray into a complete array A(n, q’, p*) is considered for the case d L 3. It is shown that iFar d = 3 such an embedding is always possible if n ~2(d - 1)2(2d2- 2d + 1). Pa&al results are indicated if d ~4 for the embedding of a related design, in a corresponding balanced incomplete block design.

Key words: Orthogonal array, balanced incomplete block design, partial geometric design, edge regular multigraph.

An orthogonal array of strength 2 is a matrix A = A@, q, p) of’ q rows and n2p columns in n 2 2 symbols such that in any two rows of A each of the rt* ordered pairs of these symbols occurs exactly cd time:s where p 2 1. We will use the term array for an orthogonal array. From Bose and Bush (1952) we have

The right-hand side of the inequality i:; an inlfeger if and only if p = p*, where p * is defined by

y*=(n-1)x+1 (1.1)

and x ~0 is an integer. In this ease

Gn array A(n, q*, F*) is called a complete array and the integer

may be called the deficiency of the array. In this paper we consider the problem of er&edding 80 array J~(R, 9% v*),into a

complete array A (S 4*, $@). If x = 0, Le. p * = 1, the problem is quivaient ta the embedding of a set of n - 1 - d mutually orthogonal latin squares of order n into a complete set of 1;1- 1 such squares af order it. The general saluticrn to the problem was given by Bruck (1963) and tthe particular case for d = 2 was given by Sh~khande (l%l).

The above result has been generalized to the case x z 1, i.e. I.C’ :> I, for d = I,2 by Shrikhande and Bhagwandas (4969) and Shrikhande and Singhi (1978). It has been shown there that

(i) for any vaiue of n an array A(n,q*- 1, p”), and (ii) for any n # 4, a~ array A (n., 4* - 2, g*) can each be embedded in the

correspondin complete array A(n, 4*, or; *). WC, therefore, consider here the question of embedding A(n, 4* - d F*) into the complete array A(n, 4’. #) where .K 2: 1 and d 23.

rl, design D is a pair (X, t) where X is a firrite set of v symbols (called treatments) and L pi (1lI i tz- I) is a finite family of not necessarily distinct subsets & (caHied blocks) of X with each Ii of size k, where 2 1~ k < v. If A and I3 are any two treatments of D, then m(A, B) will denote the number of blocks containing both A and f3. By definition then rn(A, A) is the number of bfocks in D con- taining A.

A balanced incomplete block design (BIBD) is a design D = (X, L) such that m (A, B) = A for any two distinct treartments. It then follows that. m(A, A,) = r for any A and we have

VI = bk, A(v - 1) t’ r(& - 1).

it is we!1 known that a necessary condition for the existence of a BIBD with parameters i v, 6, r, k, A) is that

62 v.

A resolvable RIED (RBIBD) is a BIBD in which the set of blocks can be artitianed into r sets of TV blocks each forming a complete replication of the v

treittments. Then c = nk and b = nr. Bose (1942) strengthened the above inequal- ity to

369

showed tha,t the parameters of an ARBIBD can be expressed in terms of two integers ~2 2 2, x 2 0 as

v = nk = n*p?, b=nr=nq*,

Jh.=~*+.l* #Pp” (14) *

where b* and q* are given by (1.1) and (1.2). A design (YC, L) is etiiled WI afine resolvable design (ARD) if the blocks can be

partitioned into’ r replication classes such that any two blocks of different replications intersect irr the same number p of treatments, The parameters of an ARD can then be written as

1)=?Zk=F12p, b=nr; _ p (W

where n 2 2 is an integer and is the number of blocks in each replication class. A group divisible design (GDD) is a design (X, L) with u = mn treatments

(partitioned into m sets of n treatments each) such that m(A, B) = AI (or A2 # .&) according as A f: B belong to the same set (or ditierent sets) of II treatments, It then follows that m(A, A) = r for every A. A GDD is called a semi-regular GDD (SRGDD) if rk - h,tr = 0. Bose and Connor (1952) have shown that in a SRGDD each block contains the same number of treatments f;am each of the m sets. The parameters of a GDD may be denoted by (u, b, r, k; Al, A,; m, n).

The following lemmas are contained in Shrikhande and Bhagwandas (1969).

Lemma 1.1. The existence of any one of the following conjigurations implies the existence of the other two

(i) ARE?: u = nk = n”p, b = nr; p, (ii) an away A(n, r, p), (iii) SRGD with parameters v1 = nk, = FV, b, = nrl = n’p; Al = 0, A2 = p*’ m = r, n = 12.

It is to be noted that configurations (i) and (iii) are duals of each other.

Lemma 1.2* ,4n array A(n, q, CA. *) with q = q * - d, d > 0 c~zrz be embedded in the compktti array A(n, q*, p*) if and only if tzn ARD I) with parameters

v = rrk = n2p*, b==nq; /A.*

can be embedded in a RBIBD with parGmt*ters

Vu= nk = n*p*, b = nq*, A=nx+1.

We note that the above is aetu~ally an

lemma in is obvious that in the

ARD D if A # B arc any two treatments, then

nx+ i-drm(A, B)=ax+ I

must necessarily be satisfied. It is easy to verify from I~mma 13 in Shrikhande and Bhagwandas (1969) and

Theor~lem 2 in Shrikhande and Sing& (1978) that the ‘Following result is true,

1.3. bt A P B be any hrs treatments of an ARD with parameters

v = nk = n*p', b = n(ip-4); p*

where d 2 3 and x 2 1, then

The results in the remaining portion of this section are contained in Bose, Shrikhande and Sislghi (11976).

pi finite muitigraph G is a triple (V, E, m) where V is a finite set of elements called vertices of G; E is a suhse! of unordered pairs of distinct elements of V ca!‘,ed the edges of G; and m is the multiplicity function which maps the edges into the set of positive integers. If AJB belongs to E, then m(A, B)= m is said to be the multiplicity of the edge AB. 23~ definition m(A, B) ;4 I if 1QB is an edge. We can extend the multiplicity function to all unordered pairs by setting m(A. B) = 0 if AB is not an edge. We say that ,A and B# A are adjacent if m (A, I?) 2 I and lion-adjacent otherwise. tit v be the number of vertices in V.

The degree q(A) of thr: vertex A is defined by

s(A) = C ni(Aq Bi)

where Bi’s are all the vmtic‘es of G other than A. If nt (A, B) = m we define

ItA, B) = fm(m - 1)

its the loop multipheity liof thle edge AB. The loop degree S(A) of the vestcx A is defined by

where as before Bi ‘s are all the vertices other than A. C&en Tao distinct vertices A and I3 we define

p(A, B)= 1 m&S, G)m(B, G)

Embedding of orthogonal arrays 371

A multigraph G is said to be regular of degree q and loop degree 6 if q(A) = q and 6(A) = S for each vertex A,

A regular muldgraph is said to be edge regular if for any set of adjacent vertices A and B, p(A, B) depends only on the value of an(A, B). Note that no assumption is made regarding the value: p(A, B), if A and I3 are nonadjacent,

In what follows we explicitly assume that iS f 0, i.e. G is not a ‘graph,. An edge regular multigraph is said to be of type

Gkk 6; ao, a1 l . . y 4

if it satisfies the following properties:

( ) a1 m(A, B) s r for any edge A.8 (a ) 2 q(A)= r(k - 1) for all vertices A.

( ) a3 S(A) = 6 for all vertices A. ( ) a4 If m(A, B) = m ZE 1, then ,p(A, B) = an(k - 2) + U,

( 1 a5 If m(A, B) = 0, then p(A, B) s q,. (a6) a0229 cxo~cu,~q, ii=2,3.,4...,r.

Here r, k, 8, tie, . . . , cu,,, are fixed non-negative integer-s, r la 2, k 2: 2. We note that if for any m, there exists no pair A, B of vertices wi:h m(A, B) = FTI, then (a,) is assumed to be vacuosly satisfied.

We shall abbreviate C&{r, 6; ao% . . . , G} to Gk where r, 6, ayg, . . . , a, remain fixed.

A subgraph of C& any two vertices ca,f which are adjacent is called a clique. A clique K is said to be complete, if we c mnot find a vertex of Gk not in K which is adjacent to each vertex of K. The clique K is said to be a major clique if

IK(rk-(r-1)q.

A grand clique is a complete major clique. An edge regular multigraph C&{r, S; QI~, . . .4, cu,) is said to be strongly regular

if m(A, B) = 0 implies p(A, B) = QC~. Let D = (X, L) be a design. If A is a treatment ami I a block of D, then by

n(A, I) we will deslste the sum Cs m(& B) =xP 12 TsI pi where B runs over all the treatment of 2 anf3 p runs over all the h!ocks containing the treatment A.

Given a dzign D = (X, L) we can define the multigraph G(D) of the design 6) as follows. The vertex set of G(D) is tht! set X of treatments of . 1[f A and R are

distinct treatments of D occurring in %(A, 8) bloc&s of C, then in G(D) tht=

unordered pair {A, S] has the multiplicity rrt(A. $3). ‘Thus in G(D), AB is not an edge if and only if A ..-.nd B do not occw together in a block of D.

If D is a design then one can define in, a natural way its dual design D!. The spt of treatments and the set of blocks of D are respectively thz set of blocks .and the set of treatments of ‘. A treatment d of

block 1 of 63 cant A design D(X,

sat~sfi~ the follo~~ing axioms: (A,) E&t blc~k has exactly k treatments.

atrnent c~wrs in exactly r blo&s. and kL, then n(A,I) is t or r+k-l+(c awarding as A&i or

1 s (Thds inplies that r ZE 2, k ;sr 2 mci c &O), that if 0 is a partial geametric desi~gn D(r, k, 6 c), then its dual

Pa’ is Al garti;al gwmetlric design D’(k, r, #, c). The following Lemma is obvious.

* lz.kh,+(k m \?

c =: (-&- l)(Al- 1)-u- (k--t)(A2- 1).

A mu%igraph G will be calted a geometric farultigraph (r, k, t, c) if there exists a partial gGametric design D(r, k, I?, c’) such that G = G(D). We now have:

lb0tmt 1.1. Zf G is a gfomettic multigraph (r, k, t, c) with c < t + r - 1, then (i) G ktus exactly (k/~)i$ - l)(bk - I)+ t - c] vertices and vr=O (mod k).

(ii) 6; rk a stro~tgly regular multigr*aph of typ Gk [T, 8; aO, . a . , cu,] where 6 = $rc artd lx, =: rit-m(t=W- l-c).

If P, k, 1, c are nonnegative integen r - -2, kr2, tzl and c<t+r-I, then a muMgraph G wit! be called a p3eudo-geometric multigraph if it satisfies the conditions (i) and (ii) of the above theorem.

1.2. Zf G is a pseudo -geomearic muCtigraph (I; k, t, c) with r 2 3, c c t+~- 1 and kBQ(r, t,c) where

~(3,Z,c)=max(3+10t~~c,-10+13t+Sc),

00. t,c)=;:&(r-- l)+t(r+ l)(+-2r+2)+c(r2-2)] if rZ4.

always denote an AND with parameters

B= flF= n(q*-d); p* t

In this section & will1

D = nk = n2b*,

where

(2.U

&&*=+I- X)x+1, q*=n2x “-n+a, Xkl, d&3.

We consider the problem of embed&g D in an ARI3IBB with parameters;

(l4). As indicated in Section 1, the prdtblem has been solved for x = 0 as also for x2:1 and d=l or 2,

By virtue of Lemma 1.1 and 1.4 it LS easy to see that D is a partial geometric design D(r, k, t, c) with t = (r- lj@, c = (r- l)(~*- 1). Eet G = C(D) be the corresponding geometric multigraph. In this section G will always denote G(D). Then from Theorem 1.1 we have:

r=q*-d, k = n@,

t=(r-l)@*, c = (P l)(p*- l),

6 = 4X, cu, = r+m(t+r--1-c).

We now assume that n~2d(d - 1). Then from Lemma 1.3 we have

nx+l-dsm(A,B)%nx+l

where A and B are any two distinct treatments of LX In this section a will denote a multigraph on the vertex set of G and is defined as follows. If A and 13 are any two distinct vertices of G, then the multilplicity function ti (A, B) in e is defined

by

?%(A, B) = nx + 1- m(A, B). (2.2)

Then

Qsrii(A, B)rd.

It is obvious that in e the degree @(A\) of any ~crtex, the loop multiplicity i(A, B) etc. are all uniquely determined because of (2.2). To calculate the par;mneters of G we can therefore, assume that there exists an AlRID F given by

2, :z nk =2 $$$ bz +-+zz nd; P*

at a se &al Qe.$t wit

374

parameters (1.4). The parameters of d will the3 pm~kd~ be the parameter% of G(F). Hence using Lemmas 1.1 and 1.4 we get:

Grti; 8, - G”, * ..,&]

i; = d, ff:z k z% np”,

i’= (d - l)y*, z = (d-a l)(p* - 1),

s’=&?=~dE, Grn =di^-fi(hd-1-c’); rii=l,2,...,d.

r9 grand clique K in e is a complete clique with IlKI 2 np” - (d ‘- V(&* - 2). This fo!!ows the definition of a grand clique in the previous section.

We now appiy Theorem 2.2 to the pseudo-geometric graph 0. For d = 3, the value of U(39 c E) is 36~“. 20. The condition k > Q(3, c c’) is

easily seen to imply that either n > “tl or n < n, where n, is in the open interval (35,36) and n2 is in the open interval (0,l). Thus if PY 2 36, k > Q(3, E E).

Similarly f0r d 2 4, the condition k > Q(d, i, c’) is easily seen to imply that n 4:~($-. I)$

We note that the lower bounds for n in both the cases are independent of x. We then have the following temm;is.

I.,emnaa 2.3. In the pseudo-geometric multigraph a(3, k, c E) if n 2 36, then: (i) Each vertex of G is contained in exactly 3 grand cliques.

(ii) If A and B are two distinct vertices of d with fi(A, B) = vii, then A and B occur together in exactly fi grand cliques.

(iii) lf kl, k2k3 ure the 3 grand c&pes containing a given vertex A of G, then C_: ‘I&! = 3k.

a 2.44. In the pseudo-geometric multigraph e( d, k, c c’), if d 2 4 and n 2 $(d-- l)d”, then

(i) Em/t vertex of (? is contained in exactly d grand cliques. (iii) If A and B are wo distinct vertices of c with *(A, B) = r?‘i, then A and B

occur together in exacr ly 61 grand cliques. (iii) If Ka, &, . . . , PCd are the d grmd cY@es containing a given vertex A of G,

then Ct IK,i = dk.

Now consider G where n 2 36 or n $(d - l)d3 according to d = 3 or d 24. Let

Em6edding of orthogonuf umys 375

Then from El-e, Shrik;hande and Singhi (1976) we have

OsPlJT~(dfiS+k-;-l)ek.

Now since IRI k k + 23, we can find 25 vertices Q in R which are not in 7’. The contribution to NT from each of these vertices contribution from each of the remaining vertices is

2c’(k - i)ls(d + c’+ k - i- 1)Ek.

is at least (k - i)” and the nonnegative. Hence

k2-d(d+E+3i- 1)+2i%O.

Substituting the values of k, 5, i we get

n s(d - 1)(2+&).

Thus if n > (d - 1)(2+fi), we cannot have IR[ or k + 2c’ and hence IRI I k + 2c’- 1. We now assume that n or 36 when d = 3 and n zi(d - 1)d3 when d ~4. Then

Lemmas 2.2, 2.3 and 2.4 continue to hold and the size of each grand clique R satisfies the inequality

lR(sk+2c’-1.

Let RI, R2,. . . , & be the grand cliques of e containing a vertex A. Then

IR&k+2c’-1 for i=l,2...,d

and

Hence l.RJ ;;3r k -- (d - l)(2e - 1). Suppose &+ 1, . . . , Rb the set of ail grand cliques of 6 not containing A where

b is the total number of grand cliques in 6. Let (R,( = ki and e, = k - k,. Note that ei may be positive, negative or zero. Then fram Lemmas 2.3 and 2.4

b

T. ki = ud, 1

i k,(k,-l)=ud(k-1). 1

(2.3)

(2s4)

Since eger M be defined

k). Let the

b

c ei = Mk. 1

(2 5) l

From (2.3) and (2.4),

Vdk = f (k- i?i)“=C e:-2k2M+bk2, 1

which implies

k(bk-Mk)=ze’-2k2M+bk2.

Hence

and hence M is nonnegative. Then using Lemma 3.5 in Bose, Shrikhande and Singhi (1976) we get

J ‘(A)++-.

l + es 5 d(d - 1)(2? - 1j2

since k, 5 k + 22 . 1. Let S be the set of all vertices in G, then

c f(A) 2~ u&d - 1)(2E - 1)2. A+S

l(2.6)

In (2~53 each ei occur k, times, i = 1,2 *. . , b. Since,

it follows that in the sum on the left-hand side of (2.6) each ei occurs at least k - (d - 1)(2? - 1) times. Hence

[k --W- 1)(2c’- l)] f e:Cf&(d-- 1)(2C- 1)2 1

or

Substituting the value

k U=;[(d-1)(k-lj+i+]

WC get MCI 1 and hence M = 0 as .M is a nonnegzitive integer if

d(d - 1)2(2E - 3)2 (d-1)(2+1)+- t‘ -

I rg

I (2 1 .7

Then each ei = 0 and hems l&I = k for i = 1,2 . . . , b and b f= u&k. Now take the vertices of 0 and the grand cliques Ri’s as treatme ts and blocks respectively of a design 0. Then D has n2p* treatments and nd blocks of size k = np* such that each treatment occurs in d blocks and any two treatments which occurred together in m(A, 8) blocks of D occur in fi(A, B) = nx + I. - m(A, B) blocks of .6. Then the blocks of D together with those of D give rise to a BIBD E with parameters

u = nk = n2@*, b = PV = r&4*, h = nx -e 1.

Sub5tuting the values of k, E c’ in (2.7) we get

cw

The left-hand side of the above inequality is a cubic in n when the value of p* is substituted. It is cumbersome to find a lower bound for re for which the inequality holds. We however, note that if n z 2(d - 1)“(2d2 -- 2d + 1) the inequality is satisf&l for all x. With this lower bound for n, Lemmas 2.2, 2.3, 2.4 are valid.

We now show that 15 is a partial geometric design. (d, k, 5; 5) with i= (d - I)@* and e =:{d - l)(fi*- I). The values of d and k are obvious. Now let A be a treatment not in a block of fi. Then the value of n(A, I) for the BIBD E is np*(4* - 1)/n = (4” - 1)~” from Lemma 1.4. Now from Bose (1942) it follows that I intersects each block of B in p* treatments: Hence the contribution from D to the value of n(A, E) is (4*- d)p*. Thus the value of n(A, 1) for fi is (d-- 1)~” which is i.

Now let A and iF be treatments and blocks of D where A is in I. Then the: value of n(A, I) for the BIBD E is 4*-t k - 1 + (k - l)(h - 1). Again the conttitsution from D to the value of n(A, 1) is (4*- d)p* simce each of the 4” - d blocks of D through A intersect I in M* treatments. Thus the value of n(A, 1) for D when A lies on the block 1 of D is the difference of these two values. On the other hand this value for D is d + k - ]I + c’(A, 1). IIence we have

q”+(k-l)+(k-l)(A-l)-(q*-d)fi*=d+k--1 tZ(A, I).

This gives

C(A,O)=(d-l)(p*--1)-E.

Mence fi is a partial geometry (d, k, c i!) and G another partial geometry & such that if = B(d,b phic. This follows from the fact that the b size k in G which are also the blocks of fi.

uppose there exists and .D are isomor-

378

geometric D giw a

2A If n 2: 2(6 - 1)2(262 - 2d + l), &nr there exists a unkp4e p~rtia5 clefsign D such that G = G(n) and the bbcks of D together with those of EWBDE with parameters (2.8).

in genera! we cannot prove that the design fi is reqolvable* Howeve? fgr the ~a d = 3 we show that this is actually the case, Suppose ~WCI .blWks of fi intersect in a treatment, say, A. Let II, la, Zs be the 3 blo&s or" B ttiugh A. L)efi.ne 14 n \I= mij, i, j = 1,2,3. From ithe value ii =2(@*- 2) we Arave from the pair A, I;,

nIll i rtflaz+ Ml3 - -k+2+c’=kt2# .

Since ml, = k we get

ml*+ rn~, = 2cc”.

Similarly,

3 * .%I2 -?- Mz, = L&& ,

Hence

ml3 +m23= 2,1;*.

w2 *

=vn~3=m23=p.

Thus acy two blocks of fi intersect in either 0 or p* treatments. Let y. and yW+ be the number of blocks in 0 which intersect a given block 2 in 0 and EL* treatments respectively. Then

p*y,. = 2np* or

y12* =2n, yc= n-l.

Mow let B be a treatment which lies on one of tlze n - 1 blocks disjtoint from 1. Then IB does not lie on 1. If B lies in 2 ur 3 blocks of the set of II! - 1 block:lr disjoint from 1 and hence in 1 or 0 blocks respectively of the set of 2n blocks intersecting I in F* treatment, then from the value of F= 2~* we get either 2y”z p* or 0 which is a contradiction. Hence any treatment not in 2 occurs

precisely once in the n - 1 blocks disjoint from !. Then I ana the n - ! blocks disjoint from 2 from a complete replication of the treatments of fi. It is now easy to see that 6 is resolvable i.e. the set of 3n blocks of fi can be partitioned into 3 se% of n blocks each set forming a complete replication.

Since for td = 3, 13 is resolvable, the BIUD IZ is resolvable and hence from Base 0942) E is an ARBIBD. We thus have the following.

Embedding of’ orthogonal arrays 379

C~aja Far d 24, there exists a plynomiaZ

f(d) 2 2(d - 1)2(2d” - 2d -!- I),

hr witiw of Lemma 1.1, the above conjecture and Theorem 2.2 can be stated in terms of embedding an array of deficiency d 2 3 into a complete array.

Bcxe, R.C. (1942). A note on the resolvability of incompleee block d<:signs. Sankhya 6, 105-110. Bose, R.C. and K.A. Bush (1952). Orthogonal arrays of strength two and three. Ann. Math. Statist.

23, 508-524. Bose, R.C. and W.S. Connor (1952). Combinatorial properties of group divisible incomplete block

designs. Ann. Math. Statist. 23, 367-383.. Bose, R.C, S.S. Shrikhande and N.M. Singhii (1976). E!dge regular multigraphs and part ial geometric

designs with an application to tIie embedding of quasi-residual designs. Academia Fqationale Dei Lincei, Atti dei come@ Lincei. 17, 49-8 1.

Bruck, R.H. (1963). Pinite nets 11, uniqueness and embedding. Pacific J. Math. 13, 421-457. Shrikhande, S.S. (1961). A note on mutually orthogonal latin squares, Sankhvli ser. A23, 115-l 16. Shrikhande, S.S. and Bhagwandas (1969). On embedding of orthogonal ‘arrays of strength two.

Combinatorial Mathematics and its Application. Univ. of N. Carolina Press, 256-273. Shtikhande, S.S. and N-M. Singhi (1979). A note on embedding of orthogonal arrays of strength two.

J. Statist. Planning Inf. 3 267-271.