Geodesic
A counterexample to the Hirsch Conjecture
Francisco Santos1
1 Departamento de Matemáticas, Universidad de Cantabria, Av. de los Castros 48, E-39005 Santander, Spain
Annals of mathematics, Tome 176 (2012) no. 1, pp. 383-412.

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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, 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 that violates a certain generalization of the $d$-step conjecture of Klee and Walkup.
DOI : 10.4007/annals.2012.176.1.7

Francisco Santos 1

1 Departamento de Matemáticas, Universidad de Cantabria, Av. de los Castros 48, E-39005 Santander, Spain 
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Francisco Santos. A counterexample to the Hirsch Conjecture. Annals of mathematics, Tome 176 (2012) no. 1, pp. 383-412. doi : 10.4007/annals.2012.176.1.7. http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.176.1.7/

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