On the monotone upper bound problem
The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n) ≤ Mubt(d,n) provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(d,n) i...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2004 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/7737 |
| Acceso en línea: | https://hdl.handle.net/2117/7737 |
| Access Level: | acceso abierto |
| Palabra clave: | Polytopes Combinatory logic Graph theory Politops Combinatoria Grafs, Teoria de Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria |
| Sumario: | The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n) ≤ Mubt(d,n) provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(d,n) is the number of vertices of a dual-to-cyclic d-polytope with n facets. It was recently shown that the upper bound M(d,n) ≤ Mubt(d,n) holds with equality for small dimensions (d ≤ 4: Pfeifle, 2003) and for small corank (n ≤ d + 2: Gärtner et al., 2001). Here we prove that it is not tight in general: In dimension d=6 a polytope with n=9 facets can have Mubt(6,9)=30 vertices, but not more than 27 ≤ M(6,9) ≤ 29 vertices can lie on a strictly-increasing edge-path. The proof involves classification results about neighborly polytopes, Kalai’s (1988) concept of abstract objective functions, the Holt-Klee conditions (1998), explicit enumeration, Welzl’s (2001) extended Gale diagrams, randomized generation of instances, as well as non-realizability proofs via a version of the Farkas lemma. |
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