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Cropper's question and Cruse's theorem about partial Latin squares by J. Goldwasser; A. J. W. Hilton; D. Patterson is a Engineering article available to read on EtoBox.

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In 1974 Cruse gave necessary and suf cient conditions for an r×s partial latin square P on symbols r 1 , r 2 ,...,r t , which may have some unfilled cells, to be completable to an n×n latin square on symbols r 1 , r 2 ,...,r n , subject to the condition that the unfilled cells of P must be lled with symbols chosen from {r t+1 , r t+2 ,...,r n }. These conditions consisted of r+s+t+1 inequalities. Hall's condition applied to partial latin squares is a necessary condition for their completion, and is a generalization of, and in the spirit of Hall's Condition for a system of distinct representatives. Cropper asked whether Hall's Condition might also be suf cient for the completion of partial latin squares, but we give here a counterexample to Cropper's speculation. We also show that the r+s+t+1 inequalities of Cruse's Theorem may be replaced by just four inequalities, two of which are Hall inequalities for P (i.e. two of the inequalities which constitute Hall's Condition for P), and the other two are Hall inequalities for the conjugates of P.

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Author

J. Goldwasser; A. J. W. Hilton; D. Patterson

Publisher

John Wiley and Sons; Wiley (John Wiley & Sons); John Wiley & Sons Inc.; Wiley (ISSN 1063-8539)

Published

2011

Language

EN

Field

Engineering (Physical Sciences)