Constrained completion of partial latin squares

In this paper, we combine the notions of completing and avoiding partial latin squares. Let P be a partial latin square of order n and let Q be the set of partial latin squares of order n that avoid P . We say that P is Q -completable if P can be completed to a latin square that avoids Q ∈ Q. We prove that if P has order 4t and contains at most t − 1 entries, then P is Q -completable for each Q ∈ Q when t ≥ 9. © 2011 Elsevier B.V. All rights reserved.