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.
翻译:尽管自1940年代以来对拉丁立方体进行了研究,但是在嵌入部分拉丁立方体方面只取得了少数一些结果,而且所有这些结果远非与包含的立方体的大小相比是最佳的。例如,1970年克鲁斯产物结果的约束是,一个部分拉丁立方体的美元可以嵌入一个按单16n ⁇ 4美元排列的拉丁立方体中,只是最近Potapov将部分拉丁立方体改进为$n ⁇ 3美元。在本说明中,我们证明第一个最理想的结果是,一个分层的拉丁立方体,即一个按单层的拉丁立方体,如果而且只有当一美元时,它才能嵌入一个按单层的双层拉丁立方方体中。一个分立方体的拉丁立方体美元是一美元,一个按单项的立方体是n ⁇ 2美元,每层的符号都与每个面平行的一层(通过确定一个坐标而实现),每层都包含一次。