Further results on latin squares with disjoint subsquares using rational outline squares

TL;DR

This paper advances the understanding of Latin squares with disjoint subsquares by establishing new necessary conditions, providing a construction for specific cases, and extending the theory of rational outline squares.

Contribution

It introduces a new necessary condition for the sizes of disjoint subsquares and offers a construction method for cases with all but two subsquares of equal size, extending existing theory.

Findings

New necessary condition for subsquare sizes

Construction method for cases with all but two equal-sized subsquares

Extensions to the theory of rational outline squares

Abstract

In this paper we consider the problem of finding latin squares with sets of pairwise disjoint subsquares. We develop a new necessary condition on the sizes of the subsquares which incorporates and extends the known conditions. We provide a construction for the case where all but two of the subsquares are the same size, and in this case the condition is sufficient. We obtain these results using symmetric rational outline squares, and additionally provide several new results and extensions to this theory.