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Induced subarrays of Latin squares without repeated symbols
We show that for any Latin square L of order 2m, we can partition the rows and columns of L into pairs so that at most (m+3)/2 of the 2x2 subarrays induced contain a repeated symbol. We conjecture that any Latin square of order 2m (where m ≥ 2, with exactly five transposition class exceptions of order 6) has such a partition so that every 2x2 subarray induced contains no repeated symbol. We verify this conjecture by computer when m ≤ 4
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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