A counterexample to the Hirsch conjecture

TL;DR

This paper provides the first known counterexample to the long-standing Hirsch conjecture, demonstrating that the maximum diameter of a polytope's graph can exceed the conjectured bounds.

Contribution

It constructs the first explicit counterexample to the Hirsch conjecture, showing that the conjecture does not hold universally for all polytopes.

Findings

Counterexample polytope has dimension 43 and 86 facets.

The polytope violates the generalized $d$-step conjecture.

Demonstrates that the diameter can exceed the conjectured maximum.

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.