Homogeneous Toroidal Latin Bitrades

Abstract

Let \(\{T, T’\}\) be a Latin bitrade. Then \(T\) (and \(T’\)) 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.