A counterexample to the Hirsch conjecture
Combinatorics
2013-04-30 v3 Discrete Mathematics
Optimization and Control
Abstract
The Hirsch Conjecture (1957) stated that the graph of a -dimensional polytope with facets cannot have (combinatorial) diameter greater than . That is, that any two vertices of the polytope can be connected by a path of at most 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 -step conjecture of Klee and Walkup.
Cite
@article{arxiv.1006.2814,
title = {A counterexample to the Hirsch conjecture},
author = {Francisco Santos},
journal= {arXiv preprint arXiv:1006.2814},
year = {2013}
}
Comments
28 pages, 10 Figures: Changes from v2: Minor edits suggested by referees. This version has been accepted in the Annals of Mathematics