Partitioning 3-homogeneous latin bitrades

Abstract

A latin bitrade \({(T^{\diamond},\, T^{\otimes})}\) is a pair of partial latin squares that define the difference between two arbitrary latin squares \({L^{\diamond} \supseteq T^{\diamond}}\) and \({L^{\otimes} \supseteq T^{\otimes}}\) of the same order. A 3-homogeneous bitrade \({(T^{\diamond},\, T^{\otimes})}\) has three entries in each row, three entries in each column, and each symbol appears three times in \({T^{\diamond}}\). Cavenagh [2] showed that any 3-homogeneous bitrade may be partitioned into three transversals. In this paper we provide an independent proof of Cavenagh’s result using geometric methods. In doing so we provide a framework for studying bitrades as tessellations in spherical, euclidean or hyperbolic space. Additionally, we show how latin bitrades are related to finite representations of certain triangle groups.