The Existence of Balanced Tournament Designs and Partitioned Balanced Tournament Designs

Abstract

A balanced tournament design of order <I>n</I>, BTD(<I>n</I>), defined on a 2<I>n</I>-set<I> V</i>, is an arrangement of the all of the (2<I>n</i>2) distinct unordered pairs of elements of <I>V</I> into an <I>n</I> X (2<I>n</i> - 1) array such that (1) every element of <I>V</i> occurs exactly once in each column and (2) every element of <I>V</I> occurs at most twice in each row. We will show that there exists a BTD(<i>n</i>) for <i>n</i> a positive integer, <i>n</i> not equal to 2. For <I>n</i> = 2, a BTD (<i>n</i>) does not exist. If the BTD(<i>n</i>) has the additional property that it is possible to permute the columns of the array such that for every row, all the elements of<I> V</I> appear exactly once in the first <i>n</i> pairs of that row and exactly once in the last <i>n</i> pairs of that row then we call the design a partitioned balanced tournament design, PBTD(<I>n</I>). We will show that there exists a PBTD (<I>n</I>) for <I>n</I> a positive integer, <I>n</I> is greater than and equal to 5, except possibly for <I>n</I> an element of the set {9,11,15}. For <I>n</I> less than and equal to 4 a PBTD(<I>n</I>) does not exist.