The Size of the Smallest Strong Critical Set in a Latin Square

A critical set in a latin square is a set of entries in a latin square which can be embedded in only one latin square. Also, if any element of the critical set is deleted, the remaining set can be embedded in more than one latin square. A critical set is strong if the embedding latin square is particularly easy to find because the remaining squares of the latin square are "forced" one at a time. A semi-strong critical set is a generalization of a strong critical set. It is proved that the size of the smallest strong or semi-strong critical set of a latin square of order n is n2/4 . An example of a critical set that is not strong or semi-strong is also displayed. It is also proved that the smallest critical set of a latin square of order 6 is 9.

• Type: Journal
• Year: 1999
• Where: ARSCOM
• Authors: John A. Bate, G. H. John van Rees
• Keywords: ARSCOM 1999, Critical Set, Latin Square, Smallest Critical Set