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Mathematics > Combinatorics

arXiv:1006.2814 (math)
[Submitted on 14 Jun 2010 (v1), last revised 8 Nov 2011 (this version, v3)]

Title:A counterexample to the Hirsch conjecture

Authors:Francisco Santos
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Abstract:The Hirsch Conjecture (1957) stated that the graph of a $d$-dimensional polytope with $n$ facets cannot have (combinatorial) diameter greater than $n-d$. That is, that any two vertices of the polytope can be connected by a path of at most $n-d$ edges.
This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope with 48 facets which violates a certain generalization of the $d$-step conjecture of Klee and Walkup.
Comments: 28 pages, 10 Figures: Changes from v2: Minor edits suggested by referees. This version has been accepted in the Annals of Mathematics
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Optimization and Control (math.OC)
MSC classes: 52B05, 52B55, 90C05
Cite as: arXiv:1006.2814 [math.CO]
  (or arXiv:1006.2814v3 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.1006.2814
Journal reference: Annals of Math. (2), 176 (July 2012), 383-412
Related DOI: https://doi.org/10.4007/annals.2012.176.1.7

Submission history

From: Francisco Santos [view email]
[v1] Mon, 14 Jun 2010 19:38:50 UTC (297 KB)
[v2] Wed, 4 May 2011 18:41:19 UTC (87 KB)
[v3] Tue, 8 Nov 2011 20:06:06 UTC (90 KB)
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