部分ラテン方格の拡張可能性について

Abstract

A latin square is an n x n square matrix each of which cells contains a symbol chosen from the set [1, 2, . . . , n] ; each symbol occurs exactly once 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 conditions 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.