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Jointly extendable Latin rectangles. (English) Zbl 0687.05009

A well-known theorem of M. Hall says that an arbitrary (k\(\times n)\)- latin rectangle can be extended to a \(((k+1)\times n)\)-Latin rectangle by adjoining a row. In this paper the authors ask when can a set \(\{L_ 1,...,L_ t\}\) of Latin rectangles be jointly extended, that is, when can a single row be found which extends each of the individual \(L_ i?\) They prove a simple sufficient condition for a set of rectangles to be jointly extendable, namely that \(k_ 1+...+k_ t\leq n/2\), where \(k_ i\) is the number of rows in the ith rectangle and n is the number of columns. Moreover, they show by construction that the result is best possible when only the orders of the given Latin rectangles are taken into account.
Reviewer: D.S.Archdeacon

MSC:

05B15 Orthogonal arrays, Latin squares, Room squares
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