Homogeneous Toroidal Latin Bitrades

Abstract

Let $\{T,T^{\prime }\}$ be a Latin bitrade. Then $T$ (and $T^{\prime }$) is said to be $(r,c,e)$-homogeneous if each row contains precisely $r$ entries, each column contains precisely $c$ entries, and each entry occurs precisely $e$ times. An $(r,c,e)$-homogeneous Latin bitrade can be embedded on the torus only for three parameter sets, namely $(r,c,e)=(3,3,3),(4,4,2)$, or $(6,3,2)$. The first case has been completely classified by a number of authors. We present classifications for the other two cases.