The Existence of Partitioned Balanced Tournament Designs

Abstract A balanced tournament design, BTD(n) , defined on a 2n-set V is an arrangement of the 2n/2 distinct unordered pairs of the elements of V into an n × 2n– 1 array such that (1) every element of V is contained in precisely one cell of each column and (2) every element of V is contained in at most two cells of each row. If we can partition the columns of a BTD(n) into three sets C 1; C 2 ,C 3 of sizes 1 ,n– 1, n– 1 respectively so that the columns in C 1 U C 2 form an H(n,2n) and the columns in C 1 U C 3 form an H(n,2n) , then the BTD(n) is called partitionable. We denote a partitioned balanced tournament design of side n by PBTD(n). In this paper, we prove the existence of PBTD(n) for n ≥ 5 with 12 possible exceptions.