Abstract
The hypercube cut number S(d) is the minimum number of hyperplanes in the d-dimensional Euclidean space that slice all the edges of the d-cube. The determination of S(d) in dimensions 5, 6, and 7, is one of the Victor Klee’s unresolved problems presented in one of his invited talks on problems from discrete geometry. The value of S(d) is unknown for d≥7, but for d≤3 it is trivial to show S(d)=d. In the late nineteen eighties, Emamy-K has shown S(4)=4 via two different proofs. More than a decade later, Sohler and Ziegler obtained a computational proof of S(5)=5 that took about two months of CPU computing time. From S(5)=5 it can be verified that S(6)=5. A short mathematical proof for d=5 remains to be a challenging problem that also leads one to the insight of the still open 7-dimensional problem. In this article we present a vertex coloring approach on the hypercube that gives a simplified proof for d=4 and also helps toward a short mathematical proof for d=5, free of computer computations.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.