On the coverings of the d-cube for d≤6

Authors:
M. R. Emamy-K;M. Ziegler
Affiliations:
Department of Mathematics, PO BOX 23355, University of Puerto Rico, Rio Piedras 00931, Puerto Rico;Heinz Nixdorf Institute, University of Paderborn, 33095 Paderborn, Germany
Venue:
Discrete Applied Mathematics
Year:
2008

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Abstract

A cut of the d-cube is any maximal set of edges that is sliced by a hyperplane, that is, intersecting the interior of the d-cube but avoiding its vertices. A set of k distinct cuts that cover all the edges of the d-cube is called a k-covering. The cut number S(d) of the d-cube is the minimum number of hyperplanes that slice all the edges of the d-cube. Here by applying the geometric structures of the cuts, we prove that there are exactly 13 non-isomorphic 3-coverings for the 3-cube. Moreover, an extended algorithmic approach is given that has the potential to find S(7) by means of largely-distributed computing. As a computational result, we also present a complete enumeration of all 4-coverings of the 4-cube as well as a complete enumeration of all 4-coverings of 78 edges of the 5-cube.