# “A Counterexample to the Hirsch Conjecture,” is Now Out

Francisco (Paco) Santos’s paper “A Counterexample to the Hirsch Conjecture” is now out

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 to each other by a path of at most $n-d$ edges. This paper presents the first counter-example 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.