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A uniqueness result for -homogeneous latin trades
summary:A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A -homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either or times. In this paper, we show that a construction given by Cavenagh, Donovan and Drápal for -homogeneous latin trades in fact classifies every minimal -homogeneous latin trade. We in turn classify all -homogeneous latin trades. A corollary is that any -homogeneous latin trade may be partitioned into three, disjoint, partial transversals
Latin bitrades derived from groups
A latin bitrade is a pair of partial latin squares which are disjoint, occupy
the same set of non-empty cells, and whose corresponding rows and columns
contain the same set of entries. Dr\'apal (\cite{Dr9}) showed that a latin
bitrade is equivalent to three derangements whose product is the identity and
whose cycles pairwise have at most one point in common. By letting a group act
on itself by right translation, we show how some latin bitrades may be derived
from groups without specifying an independent group action. Properties of latin
trades such as homogeneousness, minimality (via thinness) and orthogonality may
also be encoded succinctly within the group structure. We apply the
construction to some well-known groups, constructing previously unknown latin
bitrades. In particular, we show the existence of minimal, -homogeneous
latin trades for each odd . In some cases these are the smallest known
such examples.Comment: 23 page
Partitioning 3-homogeneous latin bitrades
A latin bitrade is a pair of partial latin
squares which defines the difference between two arbitrary latin squares
and
of the same order. A 3-homogeneous bitrade has
three entries in each row, three entries in each column, and each symbol
appears three times in . 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.Comment: 13 pages, 11 figures, fixed the figures. Geometriae Dedicata,
Accepted: 13 February 2008, Published online: 5 March 200
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