Critical Sets for a Pair of Mutually Orthogonal Cyclic Latin Squares of Odd Order Greater than $9$

Abstract

To date, investigations on critical sets for a set of mutually orthogonal Latin squares (MOLS) have been carried out only for small orders $\le 9$. In this paper, we deal with a pair of cyclic orthogonal Latin squares of order $n$, $n\ge 11$, $n$ odd. Through the construction of a uniquely completable set, we give an upper bound on the size of the minimal critical set. In particular, for $n=15$, a critical set achieving this bound is obtained.