$\mathcal{Z}$-Cyclic Generalized Ordered Whist Tournaments

Abstract

Generalized whist tournament designs and ordered whist tournament designs are relatively new specializations of whist tournament designs, having first appeared in $2003$ and $1996$, respectively. In this paper, we extend the concept of an ordered whist tournament to a generalized whist tournament and introduce an entirely new combinatorial design, which we call a generalized ordered whist tournament. We focus specifically on generalized whist tournaments for games of size $6$ and teams of size $3$, where the number of players is a prime of the form $6n+1$, and prove that these tournaments exist for all primes $p$ of the form $p=6n+1$, with the possible exception of $p\in \{7,13,19,37,61,67\}$.