A counterexample to the Hirsch Conjecture
Francisco Santos · Annals of Mathematics · 2012
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.