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2011 Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets
David Bremner, Lars Schewe
Experiment. Math. 20(3): 229-237 (2011).

Abstract

We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the $d$-step conjecture of Klee and Walkup in the case $d = 6$. This implies that for all pairs $(d, n)$ with $n − d ≤ 6$, the diameter of the edge graph of a $d$-polytope with $n$ facets is bounded by 6, which proves the Hirsch conjecture for all $n − d ≤ 6$. We prove this result by establishing this bound for a more general structure, so-called matroid polytopes, by reduction to a small number of satisfiability problems.

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David Bremner. Lars Schewe. "Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets." Experiment. Math. 20 (3) 229 - 237, 2011.

Information

Published: 2011
First available in Project Euclid: 6 October 2011

zbMATH: 1266.52017
MathSciNet: MR2836249

Subjects:
Primary: 52B05 , 52B40

Keywords: diameter , Hirsh conjecture , oriented matroids , polytopes , satisfiability

Rights: Copyright © 2011 A K Peters, Ltd.

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Vol.20 • No. 3 • 2011
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