Some new Z-cyclic whist tournament designs Original Research Article

Whist tournaments on image players are known to exist for all image. A whist design is said to be Z-cyclic if the players are elements in image where image, image when image and image, image when image and the rounds of the tournament are arranged so that each round is obtained from the previous round by adding image. 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 image players for every prime p of the form image. In 1992, nearly a century after the appearance of Mooreʹs 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 Mooreʹs construction to the case of image 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 image for a class of values of image and Finizio (Discrete Math. 279 (2004) 203–213) obtained Z-cyclic solutions for image 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 image for all image and a class of image values that includes the class of s values of Ge and Zhu. It is also established that there exists a Z-cyclic solution when image for all image and for a class of image 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 image for all image, and for image for all image. Furthermore, in the former case the designs are triplewhist tournaments. Our results, as are those of the above-mentioned studies, are constructive in nature.