Latin squares with non-partitioning disjoint subsquares

Abstract

A latin square of order $n$ with pairwise disjoint subsquares of orders $h_1,\dots ,h_k$ such that $h_1+\cdots +h_k=n$ is known as a realization. The existence of realizations is a partially solved problem with a few general results for an arbitrary number of subsquares, $k$. Requiring only that $h_1+\cdots +h_k\le n$ gives a variation of the problem that has few known results. In this paper we prove a general necessary condition for existence and completely determine existence when there are at most three subsquares or the subsquares are all of the same order. Importantly, we prove that if $h_1\ge h_2\ge \cdots \ge h_k$ and $n\ge h_1+{\sum }_{i=1}^k{h}_i$ then such a latin square always exists.