Griggs, Terry and Drapal, Ales
(2009).
Homogeneous toroidal Latin bitrades.
In: *22nd British Combinatorial Conference*, 05-10 Jul 2009, St Andrews, Scotland.

## Abstract

Let T be a partial Latin square. Then T is a Latin trade
if there exists a partial Latin square T^{1}, called a trade mate of T, with the properties that
(i) a cell is filled in T^{1} if and only if it is filled in T,
(ii) no entry occurs in the same cell in T and T^{1},
(iii) in any given row or column, T and T^{1} contain the same elements.
The pair {T, T^{1}} is called a Latin bitrade.
A Latin trade T (and T^{1}) is said to be (r, c, e)-
homogeneous if each row contains precisely r entries,
each column contains precisely c entries, and each entry occurs precisely e times. An (r, c, e)-homogeneous Latin bitrade can be embedded on the torus only for three parameter sets, namely (r, c, e) = (3, 3, 3), (4, 4, 2) or (6, 3, 2). In this talk I will present classifications for all three cases.

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