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On the monotone upper bound problem
dc.contributor.author | Pfeifle, Julián |
dc.contributor.author | Ziegler, Günter M. |
dc.contributor.other | Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II |
dc.date.accessioned | 2010-06-18T18:23:45Z |
dc.date.available | 2010-06-18T18:23:45Z |
dc.date.created | 2004 |
dc.date.issued | 2004 |
dc.identifier.citation | Pfeifle, J.; Ziegler, G. M. On the monotone upper bound problem. "Experimental mathematics", 2004, vol. 13, núm. 1, p. 1-11. |
dc.identifier.issn | 1058-6458 |
dc.identifier.uri | http://hdl.handle.net/2117/7737 |
dc.description.abstract | 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. |
dc.format.extent | 11 p. |
dc.language.iso | eng |
dc.rights | Attribution-NonCommercial-NoDerivs 3.0 Spain |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/3.0/es/ |
dc.subject | Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria |
dc.subject.lcsh | Polytopes |
dc.subject.lcsh | Combinatory logic |
dc.subject.lcsh | Graph theory |
dc.title | On the monotone upper bound problem |
dc.type | Article |
dc.subject.lemac | Politops |
dc.subject.lemac | Combinatoria |
dc.subject.lemac | Grafs, Teoria de |
dc.contributor.group | Universitat Politècnica de Catalunya. MD - Matemàtica Discreta |
dc.rights.access | Open Access |
local.identifier.drac | 2510367 |
dc.description.version | Postprint (published version) |
local.citation.author | Pfeifle, J.; Ziegler, G. M. |
local.citation.publicationName | Experimental mathematics |
local.citation.volume | 13 |
local.citation.number | 1 |
local.citation.startingPage | 1 |
local.citation.endingPage | 11 |
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