BibTeX
@TECHREPORT{Pfeifle04longmonotone,
author = {Julian Pfeifle},
title = {Long Monotone Paths on Simple 4-Polytopes},
institution = {},
year = {2004}
}
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Abstract
... of vertices in a monotone path along edges of a d-dimensional polytope with n facets can be as large as conceivably possible: Is M(d, n) = Mubt(d, n), the maximal number of vertices that a d-polytope with n facets can have according to the Upper Bound Theorem? We show that in dimension d = 4, the answer is “yes”, despite the fact that it is “no” if we restrict ourselves to the dual-to-cyclic polytopes. For each n ≥ 5, we exhibit a realization of a polar-to-neighborly 4-dimensional polytope with n facets and a Hamilton path through its vertices that is monotone with respect to a linear objective function. This constrasts an earlier result, by which no polar-to-neighborly 6-dimensional polytope with 9 facets admits a monotone Hamilton path.
