Constructions for regular-graph semi-Latin rectangles with block size two

Semi-Latin rectangles are generalizations of Latin squares and semi-Latin squares. Although they are called rectangles, the number of rows and the number of columns are not necessarily distinct. There are k treatments in each cell (row–column intersection): these constitute a block. Each treatment of the design appears a definite number of times in each row and also a definite number of times in each column (these parameters also being not necessarily distinct). When k=2, the design is said to have block size two. Regular-graph semi-Latin rectangles have the additional property that the treatment concurrences between any two pairs of distinct treatments differ by at most one. Constructions for semi-Latin rectangles of this class with k=2 which have v treatments, v/2 rows and v columns, where v is even, are given in Bailey and Monod (2001). These give the smallest designs when v is even. Here we give constructions for smallest designs with k=2 when v is odd. These are regular-graph semi-Latin rectangles where the numbers of rows, columns and treatments are identical. Then we extend the smallest designs in each case to obtain larger designs.