Journal of Combinatorial Theory, Series A 88, 162�166 (1999)

NOTE

New Product Theorems for Z-Cyclic Whist Tournaments

Ian Anderson

University of Glasgow, Glasgow, Scotland G12 8QW

Norman J. Finizio

University of Rhode Island, Kingston, Rhode Island 02881

and

Philip A. Leonard

Arizona State University, Tempe, Arizona 85287

Communicated by the Managing Editors

Received December 23, 1997

The aim of this note is to show how existing product constructions for cyclic and1-rotational block designs can be adapted to provide a highly effective method ofobtaining product theorems for whist tournaments. � 1999 Academic Press

1. INTRODUCTION

A whist tournament Wh(v) on v players is a (v, 4, 3)-BIBD meeting addi-tional conditions. An ordered block (a, b, c, d ) denotes a game (or table) inwhich the partner pair [a, c] opposes the partner pair [b, d]. The whistconditions include that every player partners each other player exactlyonce, and opposes every other player exactly twice. Moreover, the gamesare divided into rounds. If v#0 (mod 4) each player plays in one gamein each round (so that the Wh(v) is a resolvable design), while ifv#1 (mod 4) each player plays in one game in all but one of the rounds(so that the Wh(v) is a near resolvable design). It is known [4] that aWh(v) exists for all v#0 or 1 (mod 4).

Recently interest has been centered on Z-cyclic whist tournaments. AWh(4m+1) is Z-cyclic if the players are the elements of Z4m+1 and round

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j+1 is obtained from round j by adding 1 (mod 4m+1) to each element.A Wh(4m) is Z-cyclic if the players are the elements of Z4m&1 _ [�] andthe rounds are similarly cyclically generated (�+1=�). Thus a Z-cyclicWh(4m) is a 1-rotational design. By convention, in a Z-cyclic Wh(4m+1),0 is missing from the initial round.

Special types of whist tournaments have been studied extensively. In agame (a, b, c, d ), b is called a's left-hand opponent and c's right-hand oppo-nent. Considering a, b, c, d to be seated round a table, similar definitionsapply to each of a, c, and d. A directed whist tournament is a whist tourna-ment in which every player plays each other player as a left- (and as aright-) hand opponent exactly once. Opponent pairs [a, b], [c, d] arecalled pairs of opponents of the first kind, while [a, d], [b, c] are calledpairs of opponents of the second kind. A triple whist TWh(v) tournamentis a whist tournament in which every player opposes every other playeronce as an opponent of the first kind and once as an opponent of thesecond kind. Finally, a Z-cyclic Wh(4m+1) is called a Z-cyclic patternedstarter whist tournament, denoted ZCPSWh(4m+1), if its initial roundgames are all of the form (a, b, &a, &b). These concepts (with examples)are included in [3], a convenient reference for material on whist tour-naments.

2. DIFFERENCE MATRICES AND FAMILIES

We introduce the special cases of (cyclic) difference matrices and dif-ference families needed for our results. See [8] and [1], respectively, forfuller treatments of these concepts.

A (cyclic) difference matrix, (v, k; 1)-DM, is a k_v matrix A=[aij], aij

in Zv , such that for r{s the differences arj&asj , 1� j�v, comprise all theelements of Zv . If, in addition, the elements of each row comprise all theelements of Zv , we speak of a homogeneous (v; k; 1)-DM. When (v, 6)=1a homogeneous (v, 4; 1)-DM exists; its four rows can be given asRv=(0, 1, ..., v&1), &Rv , 2Rv , and &2Rv . For certain odd multiples of 3examples also exist; see [2, Section 2.4], where cyclic examples providewhat is needed for v=15, 21, 33, and 51.

Other cases such as v=105 follow from [12] and the following lemma.

Lemma 2.1. If v#1 (mod 4) and a Z-cyclic triple whist tournamentTWh(v) exists, then a (v, 4; 1)-DM exists.

Proof. Take aiv=0 for each i, a1j= j, and, for each j, 1� j�v&1, takea2j= j 's initial round partner, a3j= j 's initial round opponent of the firstkind, and a4j= j 's initial round opponent of the second kind. K

163NOTE

A family D=[Di | i # I] of subsets of Zv is called a difference family ifthe multiset of differences d&d $, d, d $ in Di , i in I, gives each element ofZv the same number, say *, of times. If the Di are disjoint in pairs, thefamily is called disjoint. Let |Di |=ki , 1�i�t; we refer to disjoint dif-ference families using the notation (v, [k1 , k2 , ..., kt], *)-DDF (or (v, k, *)-DDF when ki=k, 1�i�t).

Z-cyclic whist tournaments correspond directly to such differencefamilies. A Z-cyclic Wh(4m+1) is equivalent (via the blocks from its initialround) to a (4m+1, 4, 3)-DDF in which the whist conditions are satisfied.Similarly, a Z-cyclic Wh(4m) is equivalent to a (4m&1, [3, 4], 3)-DDFhaving a single block of size 3 and satisfying the whist conditions.

3. PRODUCT THEOREMS AND ILLUSTRATIONS

Several authors have given product constructions related to the struc-tures mentioned in Section 2. Examples include [13, 15] and the recentarticles [6, 7, 9, 14]. As a result of this work we have the following.

Theorem 3.1. Let v1 and v2 be integers for which (vi , k, k&1)-DF's,i=1, 2, and a (v1 ; k, 1)-DM exist. Then there exists a (v1v2 , k, k&1)-DF).

We note that the construction involved in obtaining this result impliesthat, if the constituent difference families are disjoint and the differencematrix is homogeneous, then the resulting difference family is disjoint.

When k=4, the conditions defining whist, directed whist, and triple whisttournaments also carry over, and if we start with ZCPSWh(vi), i=1, 2, and3 |% v1 , and we use the (v1 , 4; 1)-DM of Section 2, the resulting tournamentis a ZCPSWh(v1v2). Summarizing these observations we have

Theorem 3.2. If there exist Z-cyclic Wh(Pi), i=1, 2, where Pi #1 (mod 4), and if there exists a (P1 , 4; 1)-DM, then there exists a Z-cyclicWh(P1P2). This Wh(P1P2) is directed (triple whist, ZCPS) if both Wh(Pi)are, provided, in the ZCPS case, that 3 |% P1 .

By a similar argument we obtain

Theorem 3.3. Let Q>3, Q#3 (mod 4), P#1 (mod 4), where Z-cyclicWh(Q+1) and Wh(P) and a (Q, 4; 1)-DM exist. Then a Z-cyclicWh(PQ+1) exists. Further, if the Wh(Q+1) and Wh(P) are both triplewhist then so is Wh(PQ+1).

164 NOTE

The construction is as follows. For each initial round game (a, b, c, d) ofthe Wh(Q+1), take the game (aP, bP, cP, dP) (where �P=�). For eachinitial round game (a, b, c, d) of the Wh(P), construct games (a+Pa1j ,b+Pa2j , c+Pa3j , d+Pa4j), 1� j�Q, where A=(aij) is a (Q, 4; 1)-DM.This gives the initial round games of the Wh(PQ+1).

It is perhaps of interest to state a corresponding result for block designsthat does not appear to be explicitly stated in the literature. We state itsimply after noting two of its features. First, the required constructionprecisely parallels that in the preceding paragraph. Second, when the ``initialround games'' in that construction are replaced by the ``blocks forming aparallel class'' in the assumed designs, the construction produces the blocksin a parallel class of a resolvable design.

Theorem 3.4. Suppose that a 1-rotational (v+1, k, k&1)-BIBD, a cyclic(w, k, k&1)-BIBD, and a (v, k; 1)-DM exist. Then a 1-rotational (vw+1,k, k&1)-BIBD exists.

Theorems 3.2 and 3.3 generalize results already in the whist tournamentliterature (e.g., [3, 5, 11]) and give many new Z-cyclic Wh(v) as well assimpler constructions of many known families of tournaments. Forexample, in Theorem 3.3, P need not, as previously, be a product of primespi #1 (mod 4).

Example 3.1. Since Z-cyclic Wh(21) and Wh(57) exist [4], a Z-cyclicWh(32 } 7 } 19)=Wh(1197) exists.

Example 3.2. Since ZCPSWh(q2) exist for all primes q#3 (mod 4),3<q<500 [3, 16], a ZCPSWh(Q2) exists whenever Q is a product of suchprimes.

Example 3.3. Since Z-cyclic TWh(36) and TWh(105) exist [5, 12], aZ-cyclic TWh(3676)=TWh(3 } 52 } 72+1) exists.

Finally, we note a recent result of Lu [17], which asserts that if aZCPSWh(P) exists, P#1 (mod 4), 3 |% P, then a Z-cyclic TWh(3P+1)exists. We can now use this result to obtain many new triple whist tour-naments.

Example 3.4. Since a ZCPSWh(49 } 121) exists, a Z-cyclic TWh(17788)exists.

ACKNOWLEDGMENT

We are grateful to the referee for many helpful comments.

165NOTE

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