On Layer-Rainbow Latin Cubes Containing Layer-Rainbow Latin Cubes

TL;DR

This paper proves the optimal embedding condition for layer-rainbow latin cubes, showing they can be embedded into larger such cubes if and only if the larger cube's order is at least twice the smaller one.

Contribution

It establishes the first optimal bound for embedding layer-rainbow latin cubes, improving previous bounds significantly.

Findings

Embedding of layer-rainbow latin cubes is possible if and only if the larger cube's order is at least twice the smaller one's.

The result is optimal, matching the lower bound for embedding.

Provides a precise characterization of when such embeddings exist.

Abstract

Despite the fact that latin cubes have been studied since in the 1940's, there are only a few results on embedding partial latin cubes, and all these results are far from being optimal with respect to the size of the containing cube. For example, the bound of the 1970's result of Cruse that a partial latin cube of order $n$ can be embedded into a latin cube of order $16n^4$, was only improved very recently by Potapov to $n^3$. In this note, we prove the first such optimal result by showing that a layer-rainbow latin cube of order $m$ can be embedded into a layer-rainbow latin cube of order $n$ if and only if $n\geq 2m$. A layer-rainbow latin cube $L$ of order $n$ is an $n\times n\times n$ array filled with $n^2$ symbols such that each layer parallel to each face (obtained by fixing one coordinate) contains every symbol exactly once.