Hirsch equality conjecture for non-degenerate axial transportation polytopes

Let PP be a non-degenerate l×m×nl\times m\times n axial transportation polytope with l,m,n3l,m,n\geq 3. Its dimension is

d=lmnlmn+2.d=lmn-l-m-n+2.

Let ff be the number of facets of PP, and let G(P)G(P) be its graph. Hirsch equality conjecture. The diameter of G(P)G(P) is equal to fdf-d. The Hirsch conjecture concerns the general upper bound fdf-d for the diameter of a dd-dimensional polytope with ff facets; the supplied source does not state a resolution of this stronger equality claim, so it remains open.

Sources & referencesView supporting material

Primary source

Jesús A. De Loera, Edward D. Kim, Shmuel Onn and Francisco Santos, “Graphs of Transportation Polytopes”, arXiv:0709.2189 (2009).

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