Long Monotone Paths on Simple 4-polytopes

TheMonotone Upper Bound Problem (Klee, 1965) asks if the numberM(d, n) 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.