Partitioning 3-homogeneous latin bitrades

TL;DR

This paper offers a geometric proof that 3-homogeneous latin bitrades can be partitioned into three transversals, expanding the understanding of their structure through tessellations in various geometric spaces.

Contribution

It provides an independent geometric proof of Cavenagh's partitioning result and introduces a framework for analyzing bitrades as tessellations in different geometries.

Findings

3-homogeneous bitrades can be partitioned into three transversals

A geometric approach to studying bitrades as tessellations

Framework applicable to spherical, euclidean, and hyperbolic spaces

Abstract

A latin bitrade $(T^{\diamond}, T^{\otimes})$ is a pair of partial latin squares which defines the difference between two arbitrary latin squares $L^{\diamond} \supseteq T^{\diamond}$ and $L^{\diamond} \supseteq T^{\otimes}$ 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 (2006) 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 of spherical, euclidean or hyperbolic space.