Some new Z-cyclic whist tournament designs

Whist tournaments on v players are known to exist for all v≡0,1(mod4). A 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 the rounds of the tournament are arranged so that each round is obtained from the previous round by adding 1(modm). Despite the fact that the problem of constructing Z-cyclic whist designs has received considerable attention over the past 10-12 years there are many open questions concerning the existence of such designs. A particularly challenging situation is the case wherein 3 divides m. As far back as 1896, E.H. Moore, in his seminal work on whist tournaments, provided a construction that yields Z-cyclic whist designs on 3p+1 players for every prime p of the form p=4n+1. In 1992, nearly a century after the appearance of Moore's paper, the first new results in this challenging problem were obtained by the present authors. These new results were in the form of a generalization of Moore's construction to the case of 3pn+1 players. Since 1992 there have been a few additional advances. Two, in particular, are of considerable interest to the present study. Ge and Zhu (Bull. Inst. Combin. Appl. 32 (2001) 53-62) obtained Z-cyclic solutions for v=3s+1 for a class of values of s=4k+1 and Finizio (Discrete Math. 279 (2004) 203-213) obtained Z-cyclic solutions for v=33s+1 for the same class of s values. A complete generalization of these latter results is established here in that Z-cyclic designs are obtained for v=32n+1t+1 for all n⩾0 and a class of t=4k+1 values that includes the class of s values of Ge and Zhu. It is also established that there exists a Z-cyclic solution when v=32n+1w for all n⩾0 and for a class of w=4k+3 values. Several other new infinite classes of Z-cyclic whist tournaments are also obtained. Of these, two particular results are the existence of Z-cyclic whist designs for v=32n+1+1 for all n⩾0, and for v=32n for all n⩾2. Furthermore, in the former case the designs are triplewhist tournaments. Our results, as are those of the above-mentioned studies, are constructive in nature.