$(2,10)GWhD(10n+1)$-Existence Results

Abstract

GWhD($v$)s, or Generalized Whist Tournament Designs on $v$ players, are a relatively new type of design. GWhD($v$)s are (near) resolvable ($v,k,k-1$) BIBDs. For $k=et$, each block of the design is considered to be a game involving $e$ teams of $t$ players each. The design is subject to the requirements that every pair of players appears together in the same game exactly $t-1$ times as teammates and exactly $k-t$ times as opponents. These conditions are referred to as the Generalized Whist Conditions, and when met, we refer to the (N)RBIBD as a ($t,k$) GWhD($v$). When $k=10$, necessary conditions on $v$ are that $v\equiv 0,1(\mathrm{mod}10)$. In this study, we focus on the existence of ($2,10$) GWhD($v$), $v\equiv 1(\mathrm{mod}10)$. It is known that a ($2,10,9$)-NRBIBD does not exist. Therefore, it is impossible to have a ($2,10$) GWhD($21$). It is established here that ($2,10$) GWhD($10n+1$) exist for all other $v$ with at most 42 additional possible exceptions.