Defining sets which intersect each Latin trade at least twice

Abstract

A defining set of a Latin square is a partially filled-in Latin square which completes to no other Latin square of the same order. We introduce the concept of a k -strong defining set, in which if less than k entries are deleted, the property of being a defining set is retained. Equivalently, a k -strong defining set intersects every Latin trade in the Latin square at least k times. In the addition table for integers modulo n , when n is even we determine the minimum size of a k -strong defining set for any k . For odd n we give a construction for a minimally 2-strong defining set. We furthermore give computational results for Latin squares of small orders.