We prove that, for n equal to 3, 5, and a power of 2, every minimal partition of the edge set of the n-dimensional cube is perfect. As a consequence, we obtain some description of the classes of all minimal parallel-serial contact schemes (π-schemes) realizing the linear Boolean functions that depend essentially on n variables for the corresponding values of n.

中文翻译：

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n维立方体的边集的最小正则分区的完善性

我们证明，对于n等于3、5和2的幂，n维立方体的边集的每个最小分区都是完美的。结果，我们获得了所有最小并行-串行接触方案（π方案）的类别的描述，这些方案实现了线性布尔函数，这些函数基本上依赖于n个变量来获得n的对应值。