\(\mathcal{Z}\)-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 \(p \in \{7, 13, 19, 37, 61, 67\}\).