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The Hirsch Conjecture Fails for Triangulated 27-Spheres

David W. Walkup
Mathematics of Operations Research
Vol. 3, No. 3 (Aug., 1978), pp. 224-230
Published by: INFORMS
Stable URL: http://www.jstor.org/stable/3689492
Page Count: 7
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The Hirsch Conjecture Fails for Triangulated 27-Spheres
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Abstract

A triangulation Z of a 27-sphere is generated in which every sequence of adjacent 27-simplices joining a specified pair of disjoint 27-simplices revisits at least one vertex previously left behind. This shows that the Hirsch conjecture, which asserts the existence of a nonredundant feasible pivot sequence between any pair of bases of a linear program, or equivalently a nonrevisiting sequence of facets of a simplicial polytope, is false when generalized in the obvious way to triangulated spheres. The complex Z is constructed from a complex with 400 4-simplices which has been shown to be shellable, and hence a 4-ball, by a computer computation.

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