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.