Latin squares with one subsquare
We look at two classes of constructions for Latin squares
which have exactly one proper subsquare. The first class includes
known squares due to McLeish and to Kotzig and Turgeon, which had not
previously been shown to possess unique subsquares. The second class
is a new construction called the corrupted product. It uses
subsquare-free squares of orders m and n to build a Latin square
of order mn whose only subsquare is one of the two initial squares.
We also provide tight bounds on the size of a unique
subsquare and a survey of small order examples. Finally, we foreshadow
how our squares might be used to create new Latin squares devoid of
proper subsquares -- so called N∞ Latin squares.
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