Neighborly cubical polytopes
| dc.creator | Joswig, Michael | |
| dc.creator | Ziegler, G"unter M. | |
| dc.date | 1998-12-05 | |
| dc.date | 1999-03-23 | |
| dc.date.accessioned | 2026-07-07T05:27:07Z | |
| dc.date.available | 2026-07-07T05:27:07Z | |
| dc.description | Neighborly cubical polytopes exist: for any $n\ge d\ge 2r+2$, there is a cubical convex d-polytope $C^n_d$ whose $r$-skeleton is combinatorially equivalent to that of the $n$-dimensional cube. This solves a problem of Babson, Billera & Chan. Kalai conjectured that the boundary $\partial C^n_d$ of a neighborly cubical polytope $C^n_d$ maximizes the $f$-vector among all cubical $(d-1)$-spheres with $2^n$ vertices. While we show that this is true for polytopal spheres for $n\le d+1$, we also give a counter-example for $d=4$ and $n=6$. Further, the existence of neighborly cubical polytopes shows that the graph of the $n$-dimensional cube, where $n\ge5$, is ``dimensionally ambiguous'' in the sense of Grünbaum. We also show that the graph of the 5-cube is ``strongly 4-ambiguous''. In the special case $d=4$, neighborly cubical polytopes have $f_3=f_0/4 \log_2 f_0/4$ vertices, so the facet-vertex ratio $f_3/f_0$ is not bounded; this solves a problem of Kalai, Perles and Stanley studied by Jockusch. | |
| dc.description | 20 pages, 3 figures, Latex2e (Revised version, with a new result and a major correction) | |
| dc.identifier | https://arxiv.org/abs/math/9812033 | |
| dc.identifier | http://arxiv.org/abs/math/9812033 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/77805 | |
| dc.subject | Combinatorics | |
| dc.subject | 52B12; 52B11; 52B05 | |
| dc.title | Neighborly cubical polytopes | |
| dc.type | text |