Conjugate Orthogonal Latin Squares with Equal-sized Holes

Abstract

In this paper, we shall extend the earlier investigations undertaken by J.H. Dinitz, R.C. Mullin, D.R. Stinson, and L. Zhu of MOLS (mutually orthogonal Latin squares) with holes, and SOLS (self-orthogonal Latin squares) with holes. Here we shall consider the existence of (3,2,1)-conjugate orthogonal Latin squares (briefly COLS) with holes, corresponding to missing sub-COLS, which are disjoint and spanning. We denote by (3,2,1)-HCOLS(hn) a (3,2,1)-COLS with n holes of equal size h. It is shown that a (3,2,1)-HCOLS(hn) exists for h ≥ 2 if and only if n ≥ 4, except possibly when n ∈ {8,9,12} and h=2, and when n=6 and h ∈ {5,7,13,61}, which is quite similar to an earlier result for HSOLS (hn) due to Stinson and Zhu. As a consequence, we also have the identical result for (1,3,2)-HCOLS(hn). It is hoped that these results will be of some importance in the construction of other combinatorial designs, such as Mendelsohn designs.