Latin squares with non-partitioning disjoint subsquares

TL;DR

This paper investigates the conditions for the existence of Latin squares with disjoint subsquares, providing new necessary conditions and complete solutions for specific cases, advancing understanding of a partially solved combinatorial problem.

Contribution

It establishes a general necessary condition for the existence of such Latin squares and fully characterizes cases with up to three subsquares or all subsquares of equal size.

Findings

Proves a necessary condition for the existence of Latin squares with disjoint subsquares.

Determines existence for cases with at most three subsquares.

Shows that if the largest subsquare size is at most the total order minus the sum of all subsquares, such Latin squares always exist.

Abstract

A latin square of order $n$ with pairwise disjoint subsquares of orders $h_1,\dots,h_k$ such that $h_1+\dots+h_k = n$ 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 $h_1+\dots+h_k\leq n$ 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 $h_1\geq h_2\geq\dots\geq h_k$ and $n\geq h_1+\sum_{i=1}^kh_i$ then such a latin square always exists.