Steiner Trades That Give Rise to Completely Decomposable Latin Interchanges

In this paper we focus on the representation of Steiner trades of volume less than or equal to nine and identify those for which the associated partial latin square can be decomposed into six disjoint latin interchanges. 1 Background information In any combinatorial configuration it is possible to identify a subset which uniquely determines the structure of the configuration and in some cases is minimal with respect to this property. For example such subsets can be found by studying the literature on critical sets in latin squares (see Donovan and Howse [2]) and defining sets in block designs (see Street [7]), as well as in the study of premature partial latin squares (see Brankovic, Horak, Miller and Rosa [1]). Research has shown that computer analysis of critical sets, defining sets and premature partial latin squares is computationally expensive. This fact has led to a study of the inherent nature of the configuration in order to obtain information for refining searches and associated algorithms. In the past, critical sets, defining sets and premature partial latin squares have been studied in isolation and, in many cases, using different techniques. However, there is much to be gained by studying these configurations in unison. A crucial element in the identification of defining sets or critical sets is the determination of interchangeable elements within the design or latin square. By representing the interchangeable sets in certain designs as associated partial latin squares, Donovan, Khodkar and Street [3], [4] have identified new families of defining sets. The work in the papers [3], [4] raises many new questions. For instance, can our knowledge of the interchangeable sets in latin squares (latin interchanges) be used to classify the interchangeable sets in the block designs (trades)? It is this interesting question that we focus on here. In Section 2 of this paper we give the appropriate definitions of (partial) latin squares and latin interchanges, (partial) Steiner triple systems and associated Steiner trades, and finally detail the connection between latin interchanges and Steiner trades. In Section 3 we take all Steiner trades of volume less than or equal to nine and classify them according to the structure of the associated latin interchanges. 2 Richard Bean et al.