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Geometriae Dedicata

, Volume 133, Issue 1, pp 181–193 | Cite as

Partitioning 3-homogeneous latin bitrades

  • Carlo HämäläinenEmail author
Original Paper

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.

Keywords

Latin square Latin bitrade Triangle group Tessellation 

Mathematics Subject Classification (2000)

05B15 05B45 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsThe University of QueenslandBrisbaneAustralia

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