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arXiv · 1006.2814

A counterexample to the Hirsch conjecture

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.

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BibTeXRIS

Francisco Santos. 2011-11-08. A counterexample to the Hirsch conjecture. https://doi.org/10.4007/annals.2012.176.1.7

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