Latin cubes with disjoint subcubes of two orders

TL;DR

This paper investigates the existence of Latin cubes with pairwise disjoint subcubes of two specific orders, extending known results from Latin squares to higher dimensions and nearly completing the classification.

Contribution

It provides a near-complete characterization of when such Latin cubes exist for two orders, advancing the understanding from Latin squares to Latin cubes.

Findings

Determined existence conditions for Latin cubes with two disjoint subcube orders in most cases.

Extended known results from Latin squares to Latin cubes.

Partially solved the problem of disjoint subcubes in higher dimensions.

Abstract

Given a partition $h_1+h_2+\dots+h_k = n$, a latin square of order $n$ with pairwise disjoint subsquares of orders $h_1,\dots ,h_k$ is called a realization. When the values $h_i$ are of at most two sizes, the existence of a realization has been completely determined. However, the existence of a latin cube with pairwise disjoint subcubes of two orders is only partially solved. In this paper, we determine existence for such latin cubes in almost all cases.