Content uploaded by Malcolm Greig

Author content

All content in this area was uploaded by Malcolm Greig on Mar 07, 2022

Content may be subject to copyright.

Discrete Mathematics 268 (2003) 1 – 19

www.elsevier.com/locate/disc

Generalized whist tournament designs

R. Julian R. Abela, Norman J. Finiziob;∗, Malcolm Greigc,

Scott J. Lewisd

aSchool of Mathematics, University of New South Wales, Sydney, 2052, Australia

bDepartment of Mathematics, University of Rhode Island, 9 Greenhouse Road,

Suite 3, Kingston, RI 02881-0806, USA

cGreig Consulting, 317-130 East 11th Street, North Vancouver, BC, Canada V7L 4R3

dDepartment of Mathematics and Statistics, Murray State University, Murray, KY 42071, USA

Received 8 February 2002; received in revised form 29 July 2002; accepted 12 August 2002

Abstract

In this study a new class of tournament designs is introduced. In particular, each game of the

tournament involves several (two or more) teams competing against one another. The tournament

is also required to satisfy certain balance conditions that are imposed on each pair of players.

These balance conditions are related to both the total number of players on each team and the

number of teams in each game. In one sense, these balance conditions represent a generalization

of the balance requirements for whist tournaments although the games in a whist tournament

involve, exclusively, two two-player teams. Several techniques for constructing these new tour-

nament designs are developed and theorems guaranteeing innite classes of such designs are

proven.

c

2002 Elsevier Science B.V. All rights reserved.

Keywords: Generalized whist tournaments; Pitch tournaments; Resolvable BIBDs; Near resolvable BIBDs;

Nested designs

1. Introduction

Ageneralized whist tournament design is a schedule of games for a tournament

involving vplayers to be played in v−1 (or v) rounds. A game involves kplayers

in a multi-team game with teams of tplayers competing; a round consists of v=k (or

(v−1)=k) simultaneous games, with a player playing in at most one of these. One

∗Corresponding author.

E-mail address: nizio@uriacc.uri.edu (N.J. Finizio).

0012-365X/03/$ - see front matter c

2002 Elsevier Science B.V. All rights reserved.

PII: S0012-365X(02)00743-4

2R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19

also requires that the schedule be balanced in the sense that each pair of players play

together as teammates in a(t−1) games, and as opponents in a(k−t) games; we

will normally consider a= 1 unless otherwise specied. Such a schedule of games

will be denoted by (t; k) GWhD(v). Note that a (1;2) GWhD(v) is a resolvable (or

near resolvable) round robin tournament [11, Remark V.7.1]. A (2;4) GWhD(v)isthe

standard whist tournament (see e.g., [6]), and a (4;8) GWhD(v) is a pitch tournament

[2,4,12]. Some of the team tournaments investigated recently by Berman et al. [8] are

(t; 2t) GWhD(v)s; however, their generalization diers from ours in that they allow up

to 2t−1 players to “sit-out” a round (in this paper a maximum of 1 is allowed), and

all of their games involve exactly two teams.

As a matter of notation, each game of a (t; k) GWhD(v) is written in the form

(a11;:::;a

1t;a21;:::;a

2t;:::;ae1;:::;a

et ) wherein the semicolons separate the teams and

k=et.

Ina(t; k) GWhD(v), if one ignores the composition of teams within a game, and

treats the game as a block, the schedule is a (v; k; k −1) BIBD, and the arrangements

of the games into rounds means that this schedule is either a resolvable BIBD, i.e., a

(v=kn; k; k −1) RBIBD, or a near-resolvable BIBD, i.e., a (v=kn+1;k;k−1) NRBIBD.

Treating the team as the block, again we have a (v; t; t −1) RBIBD or NRBIBD.

Since the designs we are discussing are the standard RBIBD or NRBIBD designs with

additional properties, we are able to make use of many known constructions provided

that they preserve the additional balance properties. For the constructions employed in

this study we will need several standard designs which, for completeness, are described

below.

The notation (K; ) GDD with group type (s1;s

2;:::;s

n) denotes a group divisible

design with ngroups, with the ith group of size si, and block sizes taken from K, with

being the pairwise occurrence parameter, or index, for points from dierent groups.

We will sometimes use exponential notation for the groups, so that, if the group sizes

were constant, say si=s, then we might give the group type as sn. A pairwise balanced

block design is a GDD with a group type of 1v. It is denoted as a (v; K; ) PBD. If the

block size is uniform, with K={k}, then this block design is referred to as a balanced

incomplete block design, and denoted as a (v; k; ) BIBD.

A transversal design, TD(k; s), is a (k; ) GDD with group type sk; when = 1, the

subscript will be dropped.

A resolvable design is a design that admits a partition of the block set into subsets

(of blocks) that contains every point exactly once. A resolvable design will be denoted

by the prex R. A TD(k+1;m) can be converted into an RTD(k; m), and vice versa.

A(k; ) Frame with group type (s1;s

2;:::;s

n)isa(k; ) GDD with group type

(s1;s

2;:::;s

n) with the additional property that the block set admits a partition into

holey resolution sets, where a holey resolution set contains every point outside its

hole exactly once, and the holes coincide with the groups. Note that the block size is

constant.

A near-resolvable block design, (v; k; k −1) NRBIBD is a (k; k −1) Frame with

group type 1v. Since it is also a BIBD, we must have v≡1 (mod k).

The (t; k)ageneralized whist property (GWhP

a)ofa(k; a(k−1)) design is the

property that we can subdivide each of its blocks (games) into sub-blocks (teams)of

R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19 3

size t, so that any pair of points which occurs together a(k−1) times does so a(t−1)

times as members of the same team and a(k−t) times as members of opposing

teams. We will drop the subscript if a= 1. It should be noted that there is no general

restriction that abe integral in the v≡0 (mod k) case, although this is implied by the

near-resolvability in the v≡1 (mod k) case. Also, there is no restriction that a¿1inthe

basic concept of our generalization, which is that each player-pair should play equally

often as teammates, and also equally often as opponents, with frequencies proportional

to the number of places of that role available, and that the games be scheduled in

rounds.Ift−1 and k−tare not co-prime, then non-integral values of asatisfy the

general integrality conditions for some values of v, specically when t−1, k−1 and

v−1 have a common factor, but co-primality forces integrality of a; however, we will

later be considering designs other than GWhDs with the GWhP. Example 15 below

is an illustration of a (3;9) GWhD(27) wherein we could have taken a=1

2, since

each round (down to the table and team arrangements within the round) is duplicated

but, as stated above, we will usually only consider a= 1. (In this article, as in most

combinatorial articles, there is an implicit convention that all parameters are integral;

we will only contravene this convention for the parameter a.)

Remark 1. As we have noted, the GWhP is essentially a nesting property, with the

teams forming the sub-blocks, and the games forming the containing blocks. Thus a

GWhD can be viewed as a nested BIBD with the additional property of resolvability

or near-resolvability.

Remark 2. Note that a (t; k) GWhD(v)isa(v; k; k −1) RBIBD or NRBIBD with

the GWhP; (the underlying design is an RBIBD if v≡0 (mod k), and an NRBIBD if

v≡1 (mod k)).

For convenience, we take the view that we are dealing with some pre-specied pair

(t; k), and omit reference to these parameters when referring to the (t; k ) GWhP.

A design is a (t; k)aGWhGDD (GWhRGDD, GWhFrame), if the underlying design

isa(k; a(k−1)) GDD (RGDD, Frame). The GWhP does not aect the group structure,

as it applies to pairs of points that occur together a(k−1) times, and pairs that do not

occur together are unaected.

Incomplete designs are denoted by the prex I, with the size of the missing subde-

sign appearing as a subscript. For incomplete resolvable designs on Vpoints missing

a subdesign on Wpoints, we also require that the parallel classes of the missing sub-

design lie within the parallel classes of the whole design, so, for example, an IRGDD

has parallel classes that either span Vor span V\W. Note we do not require that

the missing subdesign actually exists. Also, trivially, any complete design can be said

to be an incomplete design missing a subdesign on one of its points; since a subde-

sign on one point needs no blocks as there are no pairs needed, one can always say

one has already removed the subdesign’s blocks, and one really has the incomplete

design.

We will sometimes say that a design contains disjoint subdesigns. For clarity, the

disjointedness refers to the point sets of the subdesigns, and so also to the block sets,

4R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19

but not to the containing parallel classes, which will often coincide. In practice, one is

often concerned with a construction that uses the containing design as a component,

and which destroys one of its subdesigns in the construction process. Furthermore, it

is of importance whether one can conclude that the constructed design has the other

subdesign of that containing component as its own subdesign. In general, this will not

happen unless the subdesigns are disjoint in the component. However, a construction

does not usually destroy a subdesign in every component, so if one is looking for a

particular subdesign in an intact component, disjointedness is not a concern.

We conclude this section by stating the most noted existence result for any sub-class

of GWhDs, namely Wilson and Baker’s now classic result for whist designs. For a

more accessible proof than Wilson and Baker’s, see [6, Chapter 13].

Theorem 3. If v≡0or 1 (mod 4), then there exists a (2;4) GWhD(v), (i.e.,there

exists the classical whist design on v points).

2. Some classic designs

As we noted in Remark 2, a GWhD is an NRBIBD or RBIBD with the GWhP.

There is an obvious consequence.

Theorem 4. A(tn; t; t −1) RBIBD exists if and only if a (t; tn) GWhD(tn)exists,and

a(tn +1;t;t−1) NRBIBD exists if and only if a (t; tn) GWhD(tn +1) exists.

Proof. The blocks of the BIBD form the teams, and the (partial) parallel classes of

the (near) resolvable design form the games of the GWhD. There is just one game per

round.

Corollary 5. A(k; ks) GWhD(ks)and a (k; ks) GWhD(ks +1) exist for all integral

s¿1for k=2;3and 4, except for the non-existent (3;6) GWhD(6).

Proof. This follows from Theorem 4. The required (ks; k; k −1) RBIBDs and NRBIBDs

are well known for k= 2; for k= 3 and 4, see [14, 4.1.7–8].

As we have just seen, existence results for (N)RBIBDs are of interest here. We

next note that a Hadamard matrix is an n×nmatrix, H(n), with all its entries being

±1, that satises H(n)HT(n)=nI. The following result is known, see Kocay and van

Rees [16].

Lemma 6. The Hadamard matrix H(n)exists if and only if an (n; n=2;n=2−1) RBIBD

exists.

A simple counting argument yields a well-known result on the order of a Hadamard

matrix, namely that if the order exceeds 2, then it must be a multiple of 4. This result

has a consequence that is of interest here.

R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19 5

Theorem 7. Ifkisodd,then no (t; k ) GWhD(2k)exists for any t and no (k; 2k)

GWhD(2k)exists.

On the positive side, a fair amount is known about the existence of Hadamard

matrices, and we select a couple of facts for the following theorem.

Theorem 8. A(2s; 4s) GWhD(4s)exists for all s¡107, and also whenever 4s−1is

a prime power.

Proof. The existence result for small sis from [10]. If 4s−1 is a prime power, then

the quadratic residues in GF(4s−1) form a dierence set for the (4s−1;2s−1;s−1)

BIBD. Augmenting these blocks with an innite element forms one team, and the other

is given by their complement.

Theorem 7can be considered as a special case of Bose’s criterion [9].

Theorem 9. Ifa(v; k; ) RBIBD with b blocks and a replication count of r satises

b=v+r−1(or equivalently r=k+or k(k−1) = (v−k)), then =k2=v is integral.

The only other general non-existence result we know of concerns the non-existence

of certain ane planes which are perforce embeddable in projective planes that are

excluded by the Bruck–Ryser–Chowla criterion, but these are not relevant for this

article. However, there is one result of an exhaustive search worth noting [15].

Theorem 10. No (15;5;4) RBIBD exists.

3. Direct constructions

In this section we will give several constructions of GWhDs, both general and spe-

cic. We also tabulate what we know of the existence for small v. Our rst theorem

is an adaptation of a well-known construction for NRBIBDs.

Theorem 11. If v=kn+1 is a prime power,and k=et,then a (t; k) GWhD(v)exists.

Proof. Let xbe a primitive element of GF(v). A dierence family for the tournament

is

B={(i(mod e);x

in+a): 06i6k−1}for a=0;1;:::;n −1:

Each block of Brepresents a game, with the players represented by pairs; the rst

element of the pair, i(mod e), is the team identier within each game and serves a

labelling purpose only. All arithmetic is performed (in GF(v)) on the second element

of the pair. The second elements of the base blocks of the dierence family span

GF(v)\{0}, and adding yto all the second elements, for y∈GF(v), generates the yth

round.

6R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19

Table 1

Examples of Theorem 11 with v633

v(t; k)

5(2;4).

7(2;6), (3;6).

9(2;4), (2;8), (4;8).

11 (2;10), (5;10).

13 (2;4), (2;6), (3;6), (2;12), (3;12), (4;12), (6;12).

16 (3;15), (5;15).

17 (2;4), (2;8), (4;8), (2;16), (4;16), (8;16).

19 (2;6), (3;6), (3;9), (2;18), (3;18), (6;18), (9;18).

23 (2;22), (11;22).

25 (2;4), (2;6), (3;6), (2;8), (4;8), (2;12), (3;12), (4;12),

(6;12), (2;24), (3;24), (4;24), (6;24), (8;24), (12;24).

27 (2;26), (13;26).

29 (2;4), (2;14), (7;14), (2;28), (4;28), (7;28), (14;28).

31 (2;6), (3;6), (2;10), (5;10), (3;15), (5;15),

(2;30), (3;30), (5;30), (6;30), (10;30), (15;30).

The construction in Theorem 11 is both powerful and exible. We illustrate with a

couple of examples.

Example 12. (1) For v=17;x=3;k=8;t= 2 the initial round of a (2;8) GWhD(17)

is given by the following two blocks:

{(0;1);(0;16); (1;9);(1;8); (2;13);(2;4); (3;15);(3;2)};

{(0;3);(0;14); (1;10);(1;7); (2;5);(2;12); (3;11);(3;6)}:

(2) For v=17;x=3;k=8;t= 4 the initial round of a (4;8) GWhD(17) (i.e., a

Pitch(17)) is given by the following two blocks:

{(0;1);(0;13);(0;16);(0;4); (1;9);(1;15);(1;8);(1;2)};

{(0;3);(0;5);(0;14);(0;12); (1;10);(1;11);(1;7);(1;6)}:

In Table 1we tabulate all examples of Theorem 11 where v633.

Lemma 13. Let q=pnbe a prime power,with n¿1, let k=pufor 0¡u¡n.Then

a(q; k; k −1) RBIBD exists.

Proof. Let xbe a primitive generator for GF(q), and represent the elements of GF(q)

as Rixu+Cjfor Cj∈GF(pu), the subeld of GF(q). Now lay out the discrete logarithm

table for GF(q)asap(n−u)by putable, with log(0) = ∞. We claim the rows of this

table form the base blocks of a 1-rotational dierence family over Zq−1∪ {∞}. Since

the entire table covers every element once, the resolvability is obvious.

Consider the dierence d= log(Rixu+Cj)−log(Rixu+(Cj+c)) for some xed

c= 0. Now d=−log(1 + c=(Rixu+Cj)) so as Rixu+Cjranges over the eld GF(q),

R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19 7

every dierence (including ∞when Rixu+Cj= 0) occurs exactly once, and our result

follows as we let crange over the k−1 non-zero values of GF(pu).

Theorem 14. If p is a prime,then a (pn;p

n+m) GWhD(pn+m+s)exists,where n and

m are positive integers,and s is a non-negative integer.

Proof. Let q=pm+n+s. Now lay out the discrete logarithm table of GF(q)asaps

by pn+mtable. By Lemma 13 the rows give a 1-rotational dierence family for the

tables of our GWhD. Next, lay out the discrete logarithm table of GF(q)asapm+sby

pntable. By Lemma 13 the rows give a 1-rotational dierence family for the teams

of our GWhD. Finally, note that the rows of this log table are subdivisions of the

rows in the earlier log table, so we can make the team assignments based on the

column of the psby pn+m, and retain these assignments throughout the development

over GF(q).

Theorem 14 is illustrated in the following example wherein we construct a (3;9)

GWhD(27), noting that x3+2x2+ 1 is a primitive polynomial in GF(27). This example

also serves to illustrate the comments made in Section 1relative to the possible choice

a=1

2.

Example 15. The following three blocks serve as the initial round of a 1-rotational

(3;9) GWhD(27):

{∞;0;13; 1;18;11; 14;24;5};

{2;7;3; 19;22;8; 12;10;4};

{15;16;20; 25;17;23; 6;21;9}:

Note that the full (3;9) GWhD(27) is obtained by developing the base blocks modulo

26. Observe however that the set of base blocks is invariant when 13 is added to each

element. Thus, if one stopped the development exactly halfway through, one would

obtain a design wherein each pair of players are partners once and opponents three

times, i.e., exactly half the total frequencies of the (3;9) GWhD(27).

In Table 2we tabulate all examples of Theorem 14 where v633.

In [18], a triplewhist tournament was introduced, and in [12], this concept was

extended to a multipitch tournament. For (2n;2n+m) GWhDs we extend this as follows.

What is required is that one be able to assign a tag to each factor of a complete

1-factorization of K2n+m, assigning the tag “opponent of the ith kind” to the ith factor

(for i¿2n), and “teammate of the ith kind” to the ith factor (for i¡2n), and that each

possible pairing of players has their pairings span the tags. It seems likely that one

is quite restricted to which 1-factorizations can be used, and furthermore within that

1-factorization, which factors can receive teammate=opponent tags. However, the most

natural one does work. For p= 2, 1-factorizations consist of spanning partitions into

K2s. For p¿2 a more natural analogue might be spanning partitions into Kps; but we

will not pursue this further here.

8R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19

Table 2

Examples of Theorem 14 with v633

v(t; k)

4(2;4).

8(2;4), (2;8), (4;8).

9(3;9).

16 (2;4), (2;8), (4;8), (2;16), (4;16), (8;16).

25 (5;25).

27 (3;9), (3;27), (9;27).

32 (2;4), (2;8), (4;8), (2;16), (4;16), (8;16),

(2;32), (4;32), (8;32), (16;32).

Corollary 16. If p=2, then the (pn;p

n+m) GWhD(pn+m+s)of Theorem 14 has a

generalized triplewhist,or multipitch property.

Proof. Now lay out the discrete logarithm table of GF(q)asapsby pn+mtable. For

any pair of columns, Cand C, we may dene their relationship based on the value

of c=C−C. Since we are dealing with a GF of characteristic 2 here, c=C−Cand

this relationship is reexive. For c∈GF(2n)\{0}, we say the pair of columns denes

“teammates of the cth kind”, and for c∈GF(2n+m)\GF(2n), the pair of columns denes

“opponents of the cth kind”. The proof of Lemma 13 shows that every dierence

occurs once for any particular value of c, and so every pair of points occurs once in

this particular c-relationship.

In the nal general construction of this section we construct a GWhFrame.

Theorem 17. If k¿2is a power of 2, then a (k; k −1) GWhFrame of type k2k−1

exists. This design has the (t; k) GWhP for any t that divides k.

Proof. The standard construction of a (k; 1) RGDD of type k2k−1is that of Seiden

[19], based on a {0;k}arc in PG(2;2k), with the resolvability dened by taking an

external 0-line, and using its points to indicate the resolution classes, one class of

which we delete to form groups (this arc is formed as the dual of the external lines of

a hyperoval). We shall take k−1 copies of this design. By Theorem 14, we can build

a(t; k) GWhD(k) on each block using these k−1 copies, so we have the required

GWhP. It now remains to exhibit the Frame resolvability. This is done by using all the

points o the arc, except those on the pair of external lines through the point dening

the groups, with each point dening a holey parallel class. It is easy to verify that each

of the group lines meets kdening points, and that each of the lines of our blocks

meets k−1 dening points, and that each dening point lies on one group line, so

the dening points do dene partial parallel classes, and these classes do partition the

block set.

R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19 9

In the next two examples, we give several instances of (t; k) GWhD(v)s for small

values of v.

Example 18. The initial round tables are given below for several specic 1-rotational

(t; k) GWhD(v); all are developed over Zv−1.

(1) A (2;6) GWhD(12):

{∞;4; 1;3; 5;9};{0;6; 7;8; 2;10}:

(2) A (3;6) GWhD(12):

{∞;0;1; 2;4;7};{3;8;10; 5;6;9}:

(3) A (2;10) GWhD(20):

{∞;6; 1;4; 16;7; 9;17; 11;5};{0;2; 3;10; 8;12; 14;15; 13;18}:

(4) A (2;6) GWhD(24):

{∞;0; 1;20; 15;7};{5;14; 4;10; 2;3};

{16;21; 9;12; 8;18};{17;19; 6;13; 11;22}:

(5) A (3;6) GWhD(24):

{∞;7;20; 0;1;15};{2;4;5; 3;10;14};

{6;11;22; 13;17;19};{8;16;21; 9;12;18}:

(6) A (2;12) GWhD(24):

{∞;9; 1;16; 2;3; 4;18; 8;12; 6;13};

{0;5; 22;10; 7;20; 17;19; 11;14; 15;21}:

(7) A (3;12) GWhD(24):

{∞;3;13; 1;6;12; 2;9;18; 4;8;16};

{0;21;22; 5;7;20; 10;15;19; 11;14;17}:

(8) A (4;12) GWhD(24):

{∞;3;9;18; 1;6;13;16; 2;4;8;12};

{0;7;10;11; 15;20;21;22; 5;14;17;19}:

(9) A (3;9) GWhD(28):

{∞;16;23; 1;7;14; 2;5;24};{9;19;20; 12;15;17; 13;25;26};

{3;11;21; 4;8;10; 6;18;22};{0;9;18; 3;12;21; 6;15;24}:

The rst three base blocks cover all points except {0}. The last block has a short

orbit (of length 3); its development covers all points except {∞}.

10 R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19

(10) A (2;6) GWhD(30):

{∞;6; 4;22; 12;15};{0;19; 7;11; 16;25};

{8;14; 9;21; 23;24};{5;18; 10;17; 13;27};{1;3; 2;26; 20;28}:

(11) A (3;6) GWhD(30):

{∞;15;22; 4;6;12};{0;16;19; 7;11;25};

{8;9;21; 14;23;24};{5;10;17; 13;18;27};{1;3;26; 2;20;28}:

(12) A (2;8) GWhD(33):

{∞;26; 2;21; 4;22; 17;28};{5;14; 7;19; 8;25; 9;10};

{1;11; 12;15; 18;24; 23;31};{3;30; 6;13; 16;20; 27;29};

{0;16; 4;20; 8;24; 12;28}:

The rst four base blocks cover all points except {0}. The last block has a short

orbit (of length 4); its development covers all points except {∞}.

Example 19. A(2;14) GWhD(28): here we give the initial round tables for a

1-rotational design developed over Z3×Z3×Z3.

{∞;001; 100;122; 220;112; 121;120; 020;201; 011;202; 111;021};

{000;010; 102;022; 101;222; 012;211; 002;200; 110;210; 221;212}:

We conclude this section with Tables 3–6in which the status of possible (t; k)aGWhD

(v) for v633 are cataloged. In the table we omit those designs for which vis a prime

power, since such designs were discussed earlier, either in Table 1using Theorem 11,

or in Table 2using Theorem 14. We have also excluded all the classical (2;4) GWhDs

which all exist as noted in Theorem 3, and the one game per round cases with teams

of size 2, 3 or 4, which, with the exception of (3;6) GWhD(6), all exist as noted

in Corollary 5. Unless otherwise mentioned, we only consider a= 1. In forming this

table, we used [17, Table I.1.3] and [14, Chapter 4] to determine the known status of

some RBIBDs and NRBIBDs. As a matter of notation, “?” indicates an open case.

4. General constructions

In this section we look at adapting standard recursive constructions to our generalized

whist tournament designs, which can be treated as BIBDs with their blocks collectively

having the GWhP, and also having near=full parallel classes. Our basic strategy will

be to take a standard construction, restrict our components by imposing some sort of

GWhP on them, and ensure that the construction produces a design that inherits that

GWhP. If the original construction produced near=full parallel classes with unrestricted

components, then the construction will do so with the restricted components, and this

part of our construction will follow the standard construction without change.

There are several key steps in developing this approach. Firstly, we have separated

the concepts of the GWhP and resolvability. Secondly, in Theorem 20 below we give a

R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19 11

Table 3

Status of possible (t; k)GWhD(v) for v633: the k=vcase

tk v Exists Comments

5 10 10 No Theorem 7

6 12 12 Yes Theorem 8

7 14 14 No Theorem 7

5 15 15 No Theorem 8

(a=1=2 does not exist either)

6 18 18 Yes Theorem 4with [14, Lemma 4.4.2.1]

9 18 18 No Theorem 7

5 20 20 Yes Theorem 4with [14, Theorem 4.3.3.11]

10 20 20 Yes Theorem 8

7 21 21 Yes Theorem 4with [14, Lemma 4.5.2.1]

(a=1=2 does not exist)

11 22 22 No Theorem 7

6 24 24 Yes Theorem 4with [14, Lemma 4.4.2.1]

8 24 24 ? No (24;8;7) RBIBD known

12 24 24 Yes Theorem 8

13 26 26 No Theorem 7

7 28 28 Yes Theorem 4with [14, Lemma 4.5.2.1]

(a=1=3 does not exist)

14 28 28 Yes Theorem 8

5 30 30 Yes Theorem 4with [14, Lemma 4.3.3.11]

6 30 30 Yes Theorem 4with [14, Lemma 4.4.2.1]

10 30 30 ? No (30;10;9) RBIBD known

15 30 30 No Theorem 7

11 33 33 ? No (33;11;10) RBIBD known

(a=1=2 does not exist)

Table 4

Status of possible (t; k)GWhD(v) for v633: the k=v−1 case

tk v Exists Comments

7 14 15 Yes Theorem 4with [14, Example 2.6.10]

5 20 21 Yes Theorem 4with [14, Theorem 4.3.1.7]

10 20 21 ? No (21;10;9) NRBIBD known

7 21 22 Yes Theorem 4with [14, Lemma 4.5.1.1]

5 25 26 Yes Theorem 4with [14, Lemma 4.3.3.6]

9 27 28 Yes Theorem 4with [5, Example I.6.30]

8 32 33 Yes Theorem 4with [14, Lemma 4.6.1.1]

16 32 33 ? No (33;16;15) NRBIBD known

version of Wilson’s Fundamental Construction that allows the inheritance of the GWhP,

although it says nothing about resolvability. So now we must examine the standard

constructions, often reinterpreting their proofs, to see that they can be viewed as an

application of WFC (and allow our variant of WFC), followed by a demonstration of

12 R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19

Table 5

Status of possible (t; k)GWhD(v) for v633: the v≡1 (mod k) Case

tk v Exists Comments

2 10 21 ? No (21;10;9) NRBIBD known

5 10 21 ? No (21;10;9) NRBIBD known

3 9 28 Yes Example 18.9

2 8 33 Yes Example 18.12

4 8 33 ?

2 16 33 ? No (33;16;15) NRBIBD known

4 16 33 ? No (33;16;15) NRBIBD known

8 16 33 ? No (33;16;15) NRBIBD known

Table 6

Status of possible (t; k)GWhD(v) for v633: the v≡0 (mod k) case

tk v Exists Comments

2 6 12 Yes Example 18.1

3 6 12 Yes Example 18.2

2 6 18 ?

3 6 18 ?

3 9 18 No Theorem 7

2 10 20 Yes Example 18.3

51020?

2 6 24 Yes Example 18.4

3 6 24 Yes Example 18.5

2 8 24 ? No (24;8;7) RBIBD known

4 8 24 ? No (24;8;7) RBIBD known

2 12 24 Yes Example 18.6

3 12 24 Yes Example 18.7

4 12 24 Yes Example 18.8

61224?

2 14 28 Yes Example 19

71428?

2 6 30 Yes Example 18.10

3 6 30 Yes Example 18.11

2 10 30 ? No (30;10;9) RBIBD known

5 10 30 ? No (30;10;9) RBIBD known

3 15 30 No Theorem 7

5 15 30 No Theorem 7

resolvability. We were not able to follow this through with all standard constructions,

the most notable exception being Baker’s Uniform Base Factorization approach [7].

Theorem 20 (Wilson’s Fundamental Construction (WFC)). Suppose that we have a

“master”(K; ) GDD with g groups and a group type vector of (|Gj|:j=1;:::;g),

and a weighting that assigns a positive weight of w(x)to each point x. Let W(Bi)be

R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19 13

the weight vector of the ith block. If,for every block Bi,we have a (K;

) GDD with

a group size vector of W(Bi), an ingredient design,then there exists a (K;

) GDD

with a group size vector of (x∈G

jw(x): j=1;:::;g). Furthermore,if either the

master GDD has the GWhP

a,or all the ingredient GDDs have the GWhP

a,then the

resultant GDD has the GWhP

a,where a=a=(k−1) and k is the block size as-

sociated with the GWhP

a.

Proof. See [21]. Wilson’s proof for the case == 1 clearly carries over to our

theorem. If we denote the points of the resulting design by (x; yx) with 16yx6w(x),

then for the construction, for each block of the master design, Bi, we use one of the

ingredient GDDs to construct a design on all the points of form (x; yx) with x∈Bi.

Sometimes this theorem is stated with a non-negative weighting. Points receiving a zero

weight could eectively be deleted from the master GDD, and the theorem applied to

this modied master GDD, with all weights being positive. If the master has the GWhP,

then when a pair x1;x

2appears in a block (with its teammate=opponent designation),

then that designation will be propagated to all pairs with rst components as x1and x2,

and the GWhP of the master design is seen to carry over to the resultant. Alternatively,

every time an x1;x

2pair appears in the master, the GWhP of the ingredient used for

that block will generate the GWhP in the resultant for that appearance of the pair in

the master.

Remark 21. In applying the above theorem, we would usually assume either =1, or

= 1, but with fractional awe also have other options, e.g., a= 1. (As an instance,

Example 15 is essentially two copies of a (3;9)1=2GWhD(27); taking one of these

copies as the master design in Theorem 20 with each point having a weight of 8,

and using an RTD2(9;8) as the ingredient, produces a standard (i.e., a=1) (3;9)

GWhGDD of type 89.) Since many of the ensuing constructions employ WFC, they

can be adapted to deal with a= 1, but for simplicity we will henceforth only consider

a=1.

In the above proof, note that if we take a pair of disjoint blocks in the master design

then the set of blocks in the resulting design generated by the rst block will have no

point in common with any of the blocks generated by the second block. If blocks in

the ingredient designs are parallel, then the sets of blocks they generate will also be

parallel, consequently any parallelism in the constituent designs is carried through into

the resultant design. The real question is: can we put together enough of these parallel

blocks (on subsets of the points) into larger sets that span the whole point set, or all

but one group? We can do this in some circumstances.

Corollary 22 (Harison’s Theorem). Ifa(k;

1) RGDD of type umexists,and a (k; 2)

RGDD of type vkexists,then a (k; 12) RGDD of type (uv)mexists. If,in addition,

a(uv; k; 12) RBIBD containing a subdesign of order w exists (with w¿0), then a

(uvm; k; 12) RBIBD containing m disjoint subdesigns of order uv,each containing

a subdesign of order w exists.Furthermore,if either of the rst two RGDDs has

the GWhP, then so does the resultant RGDD; and if,additionally,the hypothesized

14 R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19

RBIBD with its subdesign have the GWhP, then so does the nal RBIBD and its

subdesigns.

Remark 23. Harison’s theorem (see e.g., [6, Theorem 7.1.6]), apart from a subsequent

lling for the groups’ operation, is actually a sub-case of this corollary, where the

input designs are an RGDD and an RTD, with 1=2= 1; note that in this case, the

existence of the RGDD follows from the existence of an RBIBD.

Stinson [20] gives a special case of the next corollary, using particular RGDDs,

namely RTDs.

Corollary 24. Ifa(k;

1)Frame of type umi

iexists,and a (k; 2) RGDD of type vk

exists,then a (k; 12)Frame of type (vui)miexists.Furthermore,if either the rst

Frame or the RGDD has the GWhP, then so does the resultant Frame.

Theorem 25 (The GDD construction for Frames). Suppose we have a “master”(K; 1)

GDD with g groups and a group type vector of (|Gj|:j=1;:::;g), and a weighting

that assigns a positive weight of w(x)to each point x. Let W(Bi)be the weight

vector of the ith block. If,for every block Bi,we have an ingredient (k; )Frame

with a group type vector of W(Bi), then there exists a (k; )Frame with a group size

vector of (x∈G

jw(x): j=1;:::;g). Furthermore,if all the ingredient Frames have

the GWhP, then so does the resultant Frame.

Proof. Considering the resulting Frame as just a GDD, the result follows from the WFC

of Theorem 20. The holey resolutions in the resulting GDD are exhibited by taking

the blocks through a point of the master GDD, and taking the holey resolution classes

in the resulting design that were generated from those blocks and had as their hole the

resulting points whose rst index was the common point. Letting the common point

range over a group of the master design generates the holey classes in the resulting

design whose rst index was a point of the chosen group.

Theorem 26 (Filling in Groups). Ifa(k; )Frame of type (S1;S

2;:::;S

n)exists,and

for each Si,a(k; )Frame of type (s0;s

i1;s

i2;:::;s

ini)exists,with Si=ni

j=1 sij and

s0¿0, then a (k; )Frame of type (s0∪ij sij)exists. Furthermore,if the rst Frame

and all of the second (lling)set of Frames have the GWhP, then so does the resultant

Frame.

Proof. Note that each of the lling Frames has a hole of size s0. In the target Frame

the holey resolutions associated with the group of size s0are obtained by combining,

from each of the lling Frames, the holey resolutions associated with the (common)

group of size s0.

In a Frame, note that a group of size sis the hole for s=(k−1) partial parallel

classes, but an (s; k; ) RBIBD has only (s−1)=(k−1) parallel classes, so we cannot

ll the holes of a Frame directly to get another RBIBD. However, a modication of the

R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19 15

last proof works, if we actually use some extra points, as we will see in Theorem 28

below, following the corollary.

Corollary 27. Ifa(k; k −1) Frame of type (kn)qexists,and a (kn+1;k;k−1) NRBIBD

exists,then a (knq +1;k;k −1) NRBIBD exists. Furthermore,if the Frame and the

lling NRBIBD have the GWhP, then so does the resultant NRBIBD.

Theorem 28. Ifa(k; )Frame of type (s1;s

2;:::;s

n)exists,and w¿1, and for each

i¿2an (si+w; k; ) IRBIBDwexists,then an (S+w; k; ) IRBIBDs1+wexists,where

S=j¿1sj.If,in addition,an (s1+w; k; ) RBIBD exists,then an (S+w; k; ) RBIBD

exists,which contains the design on s1+wpoints as a subdesign. Furthermore,if the

rst Frame and all of the second (lling)set of IRBIBDs and the lling design on

s1+wpoints have the GWhP, then so does the resultant design.

Proof. The lling IRBIBDs have (si+w−1)=(k−1) resolutions sets, with (w−1)=

(k−1) of them incomplete. Add winnite points, and ll in all but the rst of the

holes in the Frame with the si=(k−1) complete resolutions sets of the IRBIBDs,

aligning their missing subdesign with the winnite points, and then combine their

remaining (w−1)=(k−1) incomplete sets into parallel sets that span all but the rst

group and the innite points. Note this last step needs w−1¿0. This gives us an

(S+w; k; ) IRBIBDs1+w. For the additional part of the theorem, we complete this

design. The GWhP is fairly transparent.

Remark 29. In the previous theorem, if the design on s1+wpoints has a subdesign

of order msay, or if any of the IRBIBDs has a subdesign of order mthat is disjoint

from the subdesign of order wused in the theorem, then the design on S+wpoints

also has a subdesign of order m.

This remark is a vital observation for the construction of pitch designs, as it was for

the (v; 8;7) RBIBD constructions in [13]. In another article [4], it was demonstrated

that many pitch designs having disjoint subdesigns on 8 points can be constructed.

Since Theorem 28 is an important constructive tool, it is necessary to pay attention

to the existence of subdesigns in our constructions. We next exhibit several product

constructions.

Theorem 30. Ifa(k; k −1) Frame of type npexists,and an RTD(k; q)exists,and

a(k; k −1) Frame of type nqexists,then a (k; k −1) Frame of type npq exists.

Furthermore,if the both input Frames have the GWhP, then so does the resultant

Frame.

Proof. We may use the rst two designs, with Corollary 24, to produce a (k; k −1)

Frame of type (nq)p, and then ll the groups of this design using Theorem 26.

Lemma 31. Ifa(km; k; k −1) RBIBD exists,and an RTD(k; kn +1) exists,and a

(kn +1;k;k −1) NRBIBD exists,then there exists a (km(kn +1);k;k −1) RBIBD

containing kn +1 disjoint (km; k; k −1) RBIBDs as subdesigns. Furthermore,if the

input RBIBD and NRBIBD have the GWhP, then so does the resultant RBIBD.

16 R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19

Proof. For the BIBD part of the construction, treat the RBIBD as a GDD of type 1km,

and use this as the master design and give each point a weight of kn + 1, then use the

RTD as the ingredient design in WFC to get a GDD of type (kn +1)

km and use the

NRBIBD to ll the groups. This also shows the GWhP will carry through. Suppose

(X;A) is the RBIBD, and (Y;B) is the NRBIBD; for each block A∈A, the TD

will be dened on A×Y, with groups {a}×Y. If we assume the rst parallel class

of the RTD consists of the blocks {(1;y):::(k; y)}for y∈Y, this parallel class will

generate the disjoint subdesigns. For the needed resolvability of the resulting design,

see [14, Theorem 3.5.2], noting that a (kn +1;k;k −1) NRBIBD is (kn + 1)-block-

colorable.

Lemma 32. Ifa(km; k; k −1) RBIBD exists,and an RTD(k; kn)exists,and a (kn; k;

k−1) RBIBD exists,then there exists a (kmkn; k; k −1) RBIBD containing kn disjoint

(km; k; k −1) RBIBDs as subdesigns,and containing km disjoint (kn; k; k −1) RBIBDs

as subdesigns. Furthermore,if the input RBIBDs have the GWhP, then so does the

resultant RBIBD.

Proof. Similar to Lemma 31, we give the points of the 1km GDD weight kn in WFC

to get a GDD of type knkm which we can ll. In this guise, it is an application of

Corollary 22. Examining the actual construction carefully, as in Lemma 31,wesee

that we can alternatively consider that we have km subdesigns of size kn.

If the roles of the RBIBD and NRBIBD in Lemma 31 are interchanged, or equiv-

alently, if we have an RTD(k; km), then showing the resolvability is much harder.

However, if we assume some extra structure, we can exploit this extra structure to

obtain a powerful construction.

Theorem 33. Suppose that there exist the following:

(1) an RTD(km +1;kn−1) which is given by a km +1 by kn −1dierence matrix

over an additive Abelian group,G,of order kn −1,

(2) a(kn; k; k −1) RBIBD on G∪ {∞} which is generated by a dierence family

over G,

(3) a(km; k; k −1) RBIBD,

(4) a(km +1;k;k −1) NRBIBD,

(5) a(kw; k; k −1) RBIBD with 0¡w6n.

Then a (km(kn −1)+kw; k; k −1) RBIBD exists. This RBIBD contains a (kw; k;

k−1) RBIBD subdesign,and if w¡n,then this RBIBD contains a (km; k; k−1) RBIBD

subdesign. Furthermore,if the NRBIBD and all the input RBIBDs have the GWhP,

then so does the resultant RBIBD.

Proof. We can add kw −1 extra points to resolution sets of the RTD, and then delete

a complete group, to get a ({km; km +1};1) RGDD of type (kn −1)km (kw −1)1, which

we can ll with the aid of an innite point to get the resultant BIBD. From this the

GWhP is obvious; for the resolvability, see [14, Theorem 3.7.1].

R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19 17

Theorem 34. If we have an RTD(km +1;kn +1) which is given by a km +1 by

kn +1 dierence matrix over an additive Abelian group,G,of order kn +1, and a

(kn +1;k;k−1) NRBIBD on G which is generated by a dierence family over G,and

(km; k; k −1) RBIBD, (km +1;k;k−1) NRBIBD, and (kw +1;k;k −1) NRBIBD with

06w6nall exist,then a (km(kn +1)+kw +1;k;k−1) NRBIBD exists. Furthermore,

if the RBIBD and all the input NRBIBDs have the GWhP, then so does the resultant

NRBIBD.

Proof. We can add kw + 1 extra points to resolution sets of the RTD, and then delete

a complete group, to get a ({km; km +1};1) RGDD of type (kn +1)

km(kw +1)

1,

which we can ll to get the resultant BIBD. From this the GWhP is obvious, for the

resolvability, see [14, Theorem 2.5.5].

Under appropriate circumstances (v; k; 1) RBIBDs, when they exist, can be exploited

for our purposes as indicated in the next two lemmas.

Lemma 35. Ifa(ku; km; 1) RBIBD and a GWhD(km)exist,then there exists a

GWhD(ku)containing a spanning set of disjoint GWhD(km)subdesigns.

Proof. We use the GWhD to break the blocks of the RBIBD. A parallel class in the

RBIBD generates the subdesigns.

Lemma 36. If there exist a (kmu; km; 1) RBIBD and an RTD(km; n)and a (km; k; k −

1) RBIBD and a (kmn; k; k −1) RBIBD, then there exists a (kumn; k; k −1) RBIBD

containing u disjoint (kmn; k; k −1) RBIBDs, and nu disjoint (km; k; k −1) RBIBDs

as subdesigns. Furthermore,if the input RBIBDs with index k−1have the GWhP,

then so does the resultant RBIBD.

Proof. We can remove a parallel class of the (kmu; km; 1) RBIBD to get an RGDD of

type (km)u, and give its points a weight of nin WFC to get a (km; 1) RGDD of type

(kmn)u. We next break the blocks of this RGDD with the (km; k; k −1) RBIBD and

then we can ll the groups with the (kmn; k; k −1) RBIBD. The GWhP is obvious.

Theorem 37 (RB construction for Frames). Ifa(K∪H; 1) RPBD exists,and a point

p lies in blocks of size h1;h

2;:::;h

n,where H=i=n

i=1 hi,then a (K; 1) Frame of type

h1−1;h

2−1;:::;h

n−1exists.

Proof. Delete the point p. Dene the groups of the Frame by the resulting sets of

size h1−1;h

2−1;:::;h

n−1. Since each resolution class in the RPBD contained the

point p, its removal together with its line in that class automatically gives the holey

resolution class for the group dened by that line.

Lemma 38. Ifa(km +1;k;k −1) NRBIBD exists,an RTD(k; kn +1) exists,a

(kn +k; k; k −1) IRBIBDkexists and a (km +k; k; k −1) RBIBD exists,then there

exists a (k(kmn +m+n+1);k;k −1) RBIBD. Furthermore,if the NRBIBD, the set

18 R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19

of (lling) IRBIBDs, and the lling design on km +kpoints have the GWhP, then

so does the resultant design.

Proof. Treat the NRBIBD as a (k; k −1) Frame of type 1km+1, and inate using the

RTD with a block removed to produce a (k; k −1) IFrame of type (kn +1;1)km+1.

Add k−1 innite points, and ll using the IRBIBDs (aligning the subdesign with the

missing point of the group in the IFrame and the innite points), and then ll the

transverse hole of size km + 1 together with the innite points with the RBIBD to

produce the result.

Lemma 39. If there exist a (km +1;k;k −1) NRBIBD, an RTD(k; kn −1), and

a(kn; k; k −1) RBIBD, then a (k(m(kn −1) + n);k;k −1) RBIBD containing a

(kn; k; k −1) RBIBD subdesign exists. Furthermore,if the NRBIBD and the input

RBIBD have the GWhP, then so does the resultant design.

Proof. Treat the NRBIBD as a Frame of type 1km+1 and inate with the RTD, then

ll the groups with the RBIBD using one innite point.

The constructions embodied in the lemmas and theorems presented in this study can

be employed to produce a variety of generalized whist tournaments. In particular, appli-

cations to the existence of pitch tournaments, i.e., (4;8) GWhD(v)s, are investigated in

two separate papers [2,4], (2;6) GWhD(v)s are investigated in [3] and (3;6) GWhD(v)s

are investigated in [1]. These papers refer to lemmata in a preprint version of this paper

which had a dierent numbering.

Acknowledgements

This article beneted markedly from the comments of the referees, whom we thank.

References

[1] R.J.R. Abel, N.J. Finizio, M. Greig, (3;6) GWhD(v)—existence results, Discrete Math., to appear.

[2] R.J.R. Abel, N.J. Finizio, M. Greig, S.J. Lewis, Pitch tournament designs and other BIBDs—existence

results for the case v=8n+ 1, Congr. Numer. 138 (1999) 175–192.

[3] R.J.R. Abel, N.J. Finizio, M. Greig, S.J. Lewis, (2;6) GWhD(v)—existence results and some Z-cyclic

solutions, Congr. Numer. 144 (2000) 5–39.

[4] R.J.R. Abel, N.J. Finizio, M. Greig, S.J. Lewis, Pitch tournament designs and other BIBDs—existence

results for the Case v=8n, J. Combin. Des. 9 (2001) 334–356.

[5] R.J.R. Abel, S.C. Furino, Resolvable and near resolvable designs, in: C.J. Colbourn, J.H. Dinitz (Eds.),

Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 87–94.

[6] I. Anderson, Combinatorial Designs and Tournaments, Oxford University Press, Oxford, 1997.

[7] R.D. Baker, Resolvable BIBD and SOLS, Discrete Math. 44 (1983) 13–29.

[8] D.R. Berman, S. McLaurin, D.D. Smith, Team tournaments, Congr. Numer. 137 (1999) 33–46.

[9] R.C. Bose, A note on the resolvability of balanced incomplete block designs, Sankhy a 6 (1942)

105–110.

R.J.R. Abel et al. / Discrete Mathematics 268 (2003) 1 – 19 19

[10] R. Craigen, Hadamard matrices and designs, in: C.J. Colbourn, J.H. Dinitz (Eds.), Handbook of

Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 370–377.

[11] J. Dinitz, E. Lamken, W. Wallis, Scheduling a tournament, in: C.J. Colbourn, J.H. Dinitz (Eds.),

Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 565–578.

[12] N.J. Finizio, S.J. Lewis, Some specializations of pitch tournament designs, Utilitas Math. 56 (1999)

33–52.

[13] S. Furino, M. Greig, Y. Miao, J. Yin, Resolvable BIBDs with block size 8 and index 7, J. Combin.

Des. 2 (1994) 101–116.

[14] S. Furino, Y. Miao, J. Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence, CRC

Press, Boca Raton, FL, 1996.

[15] P. Kaski, P.R.J.

Osterg

ard, There exists no (15;5;4) RBIBD, J. Combin. Des. 9 (2001) 357–362.

[16] W. Kocay, G.H.J. van Rees, Some non-isomorphic (4t+4;8t+6;4t+3;2t+2;2t+1)-BIBDs, Discrete

Math. 92 (1991) 159–172.

[17] R. Mathon, A. Rosa, 2-(v; k; ) designs of small order, in: C.J. Colbourn, J.H. Dinitz (Eds.), Handbook

of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 3–41.

[18] E.H. Moore, Tactical memoranda I–III, Amer. J. Math. 18 (1896) 264–303.

[19] E. Seiden, A method of construction of resolvable BIBD, Sankhya Ser. A 25 (1963) 393–394.

[20] D.R. Stinson, Frames for Kirkman triple systems, Discrete Math. 65 (1987) 289–300.

[21] R.M. Wilson, An existence theory for pairwise balanced designs, Part II: the structure of PBD-closed

sets and the existence conjectures, J. Combin. Theory Ser. A 13 (1972) 246–273.