Critical sets in Latin squares given that they are symmetric

Abstract

A uniquely completable (UC) set U is a subset of a Latin square L such that L is the only superset of U which is a Latin square. A critical set C of L is a subset of L such that C is uniquely completable and no subset of C has this property. We show that there is a symmetric Latin square with fixed main diagonal entries for each even number, and obtain a uniquely completable partial symmetric Latin square of order 2n for each n and prove that, it is critical set for n = 3, 4, 5 and 6, and make a problem.