The spectrum for large set of disjoint incomplete Latin squares Original Research Article

An incomplete Latin square LS(n+a,a) is a Latin square of order n+a with a missing subsquare of order a. A large set of disjoint LS(n+a,a)s, denoted by LDILS(n+a,a), consists of n disjoint LS(n+a,a)s. About the existence of LDILS, Zhu, Wu, Chen and Ge have already obtained some results (see Wu and Zhu, Bull Inst. Combin. Appl., to appear.). In this paper, we introduce a kind of auxiliary design LSm(n) and, using it, completely solve the existence problem of LDILS. The conclusion is that for any positive integer n and any integer a, 0⩽a⩽n, there exists an LDILS(n+a,a) if and only if (n,a)≠(2,1) and (6,5).