“School of Mathematics”
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| Paper IPM / M / 7876 |
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| Abstract: | |||||||||
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Let T be a partial latin square and L a latin square such that
T ⊂ L. Then T is called a latin trade, if there
exists a partial latin square T* such that T* ∩T = ϕ
and (L \T)∪T* is a latin square. We call T* a
disjoint mate of T. A latin trade is called
k-homogeneous if each row and each column contains exactly k
elements, and each element appears exactly k times. The number
of elements in a latin trade is referred to as its volume.
It is shown by Cavenagh, Donovan, and Drapal (2003 and 2004) that
3-homogeneus and 4-homogeneous latin trades of volume 3m and
4m, respectively, exist for all m ≥ 3 and m ≥ 4,
respectively. We show that k-homogeneous latin trades of volume
km exist for all 3 ≤ k ≤ 8 and m ≥ k. Also we show
that for each given k ≥ 3 and m ≥ k, all k-homogeneous
latin trades of volume km exist except possibly for finitely
many m, i.e. k < m < 2k+20.
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