论文标题
在包含Layer-Rainbow拉丁立方体的Layer-Rainbow拉丁立方体上
On Layer-Rainbow Latin Cubes Containing Layer-Rainbow Latin Cubes
论文作者
论文摘要
尽管自1940年代以来就已经研究了拉丁立方体，但嵌入部分拉丁立方体的结果只有少数结果，而且所有这些结果对于包含立方体的尺寸而言远非最佳。例如，Cruse的结果是，部分拉丁立方体可以将$ n $的部分拉丁立方体嵌入到拉丁单方$ 16n^4 $的拉丁立方体中，而Potapov最近才改善了$ n^3 $。在本说明中，我们证明了第一个最佳结果，即证明订单$ m $的layer-rain-rainbow拉丁立方体可以嵌入到layer-rain-rainbow的拉丁文中，仅当$ n $ n $时，并且只有$ n \ geq 200万美元。 lay-rainbow拉丁立方体$ l $ of订单$ n $是一个$ n \ times n \ times n $阵列，其中填充了$ n^2 $符号，使每个层平行于每个脸部（通过固定一个坐标获得）完全包含每个符号一次。
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.
