Necessary Conditions for the Completion of Partial Latin Squares·
نویسنده
چکیده
A latin square is an n x n square matrix each of which cells contains a symbol chosen from the set 11,2, ... ,n]; each symbol occurs exactly oncl~ in each row or column of the matrix. A partial latin square is a latin square in which some cells are unoccupied. We consider the problem of obtaining necessary and sufficient conditions for a partial latin square to be completed to a latin square. For this problem A. B. Cruse has recently given a necessary condition associated with triply stochastic matrices. In this paper two sets of necessary condtions are given, one developed from network flow theory and another obtained from matroid theory. It is shown that the network condition is equivalent to Cruse's condition and that the matroid condition is strictly stronger than either of the former.
منابع مشابه
A note on the completion of partial latin squares
The problem of completing partial latin squares to latin squares of the same order has been studied for many years. For instance, in 1960 Evans [9] conjectured that every partial latin square of order n containing at most n− 1 filled cells is completable to a latin square of order n. This conjecture was shown to be true by Lindner [12] and Smetaniuk [13]. Recently, Bryant and Rodger [6] establi...
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تاریخ انتشار 2009