On the spectrum of critical sets in latin squares of order 2n

Abstract

AbstractSuppose that L is a latin square of order m and P ⊑ L is a partial latin square. If L is the only latin square of order m which contains P, and no proper subset of P has this property, then P is a critical set of L. The critical set spectrum problem is to determine, for a given m, the set of integers t for which there exists a latin square of order m with a critical set of size t. We outline a partial solution to the critical set spectrum problem for latin squares of order 2n. The back circulant latin square of even order m has a well‐known critical set of size m2/4, and this is the smallest known critical set for a latin square of order m. The abelian 2‐group of order 2n has a critical set of size 4n‐3n, and this is the largest known critical set for a latin square of order 2n. We construct a set of latin squares with associated critical sets which are intermediate between the back circulant latin square of order 2n and the abelian 2‐group of order 2n. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 25–43, 2008