| Títol: | On the monotone upper bound problem |
| Autor: | Pfeifle, Julián
Ziegler, Günter M. |
| Altres autors/autores: | Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II |
| Matèries: | Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria
Polytopes
Combinatory logic
Graph theory
Politops
Combinatoria
Grafs, Teoria de |
| Tipus de document: | Article |
| Descripció: | 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. |
| Altres identificadors i accés: | Pfeifle, J.; Ziegler, G. M. On the monotone upper bound problem. "Experimental mathematics", 2004, vol. 13, núm. 1, p. 1-11.
1058-6458
http://hdl.handle.net/2117/7737 |
| Disponible al dipòsit: | E-prints UPC
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