New Z-cyclic triplewhist frames and triplewhist tournament designs

AbstractTriplewhist tournaments are a specialization of whist tournament designs. The spectrum for triplewhist tournaments on v players is nearly complete. It is now known that triplewhist designs do not exist for v=5,9,12,13 and do exist for all other v≡0,1(mod4) except, possibly, v=17. Much less is known concerning the existence of Z-cyclic triplewhist tournaments. Indeed, there are many open questions related to the existence of Z-cyclic whist designs. A (triple)whist design is said to be Z-cyclic if the players are elements in Zm∪A where m=v, A=∅ when v≡1(mod4) and m=v-1, A={∞} when v≡0(mod4) and it is further required that the rounds also be cyclic in the sense that the rounds can be labelled, say, R1,R2,… in such a way that Rj+1 is obtained by adding +1(modm) to every element in Rj. The production of Z-cyclic triplewhist designs is particularly challenging when m is divisible by any of 5,9,11,13,17. Here we introduce several new triplewhist frames and use them to construct new infinite families of triplewhist designs, many for the case of m being divisible by at least one of 5,9,11,13,17

Some difference matrix constructions and an almost completion for the existence of triplewhist tournaments TWh(v)

AbstractA necessary condition for the existence of a triplewhist tournament TWh(v) is v≡0 or 1(mod4); this condition is known to be sufficient except for v=5,9,12,13 and possibly v=17,57,65,69,77,85,93,117,129,153. In this paper, we remove all the possible exceptions except v=17. This provides an almost complete solution for the more than 100 year old problem on the existence of triplewhist tournaments TWh(v). By applying frame constructions and product constructions, several new infinite classes of Z-cyclic triplewhist tournaments are also obtained. A couple of new cyclic difference matrices are also obtained

One frame and several new infinite families of Z-cyclic whist designs

AbstractIn 2001, Ge and Zhu published a frame construction which they utilized to construct a large class of Z-cyclic triplewhist designs. In this study the power and elegance of their methodology is illustrated in a rather dramatic fashion. Primarily due to the discovery of a single new frame it is possible to combine their techniques with the product theorems of Anderson, Finizio and Leonard along with a few new specific designs to obtain several new infinite classes of Z-cyclic whist designs. A sampling of the new results contained herein is as follows: (1) Z-cyclic Wh(33p+1), p a prime of the form 4t+1; (2) Z-cyclic Wh(32n+1s+1), for all n⩾1, s=5,13,17; (3) Z-cyclic Wh(32ns+1), for all n⩾1, s=35,55,91; (4) Z-cyclic Wh(32n+1s), for all n⩾1, and for all s for which there exist a Z-cyclic Wh(3s) and a homogeneous (s,4,1)-DM; and (5) Z-cyclic Wh(32ns) for all n⩾1, s=5,13. Many other results are also obtained. In particular, there exist Z-cyclic Wh(33v+1) where v is any number for which Ge and Zhu obtained Z-cyclic TWh(3v+1)

Some constructions of combinatorial designs

The objects of study of this thesis are combinatorial designs. Chapters 2 and 3 deal with various refinements of whist tournament, while Chapters 3 and 4 focus on terraces. Chapter 2 is devoted to the investigation of Z-cyclic ordered triplewhist tournaments on p elements, where p = 5 (mod 8); Z-cyclic ordered triplewhist and directed triplewhist tournaments on p elements, where p = 9 (mod 16); and Z-cyclic directed moore (2,6) generalised whist tournament designs on p elements, where p = 7 (mod 12). In each of these cases, p is prime. In an effort to prove the existence of an infinite family of each of these tournaments, constructions are introduced and the conditions under which they give the initial round of a tournament of the kind we desire are found. A bound above which these conditions are always satisfied is then obtained, and we try to fill in the appropriate gaps below that bound. In Chapter 3 we investigate the existence of tournaments of the type seen in Chapter 2 which involve four players per game, with an additional property. This is known as the three person property and is defined in Chapter 1. Here, we focus on one of the constructions introduced in Chapter 2 for each type of tournament. Then we find a new bound using only that construction with the additional conditions introduced by the three person property, and again try to fill in the appropriate gaps below the bound. Chapter 4 is an investigation of logarithmic terraces and their properties. Very little work has been done on them previously, so this was really an opportunity to look at them more closely in an effort to find as many interesting properties as possible. Some general results and examples are given, with the focal point of the chapter being the study of terraces which are simultaneously logarithmic for two different primitive roots. In Chapter 5, a more specific problem is addressed which involves training schedules for athletes. Here we want n(n - 1) athletes to carry out n tasks in some order, then keep repeating them in different orders in blocks of n as many times as possible so that certain conditions are satisfied. These conditions are listed in Chapter 5. We make use of the Williams terrace and the Owens terrace in our attempt to find a general method which allows the given conditions to be satisfied and gets as close as possible to the theoretical limit where each athlete carries out the n tasks n - 1 times

Existence of Z-cyclic triplewhist tournaments for a prime number of players

A $Z$-cyclic triplewhist tournament for $4n+1$ players is equivalent to a set of $n$ quadruples $(a_i,b_i,c_i,d_i)$ such that $\bigcup_i\{a_i,b_i,c_i,d_i\}=\bigcup_i\{\pm(a_i-b_i),\pm(c_i-d_i)\}=\bigcup_i\{\pm(a_i-c_i),\pm(b_i-d_i)\}=\bigcup_i\{\pm(a_i-d_i),\pm(b_i-c_i)\}=Z_{4n+1}-\{0\}$. The case where $4n+1$ is a prime $p$ is considered. The existence of such tournaments for all $p\equiv 5\pmod 8,\ p\geq 29$, was established by I. Anderson, S. D. Cohen and N. J. Finizio [Discrete Math. 138 (1995), no. 1-3, 31--41], and the case $p\equiv 9\pmod{16}$ was fully dealt with by Y. S. Liaw [J. Combin. Des. 4 (1996), no. 4, 219--233] and G. McNay [Utilitas Math. 49 (1996), 191--201]. In this paper, besides giving easier proofs of these authors' results, we solve the problem also for primes $p \equiv 1$ (mod 16). The final result is the existence of a Z-cyclic TWh(v) for any v whose prime factors are all $\equiv 1$ (mod 4) and distinct from 5, 13, and 17