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.