Mathematics – Combinatorics
Scientific paper
2007-09-14
J. Combin. Theory Ser. A, 116(8), 2009, 1306-1325
Mathematics
Combinatorics
29 pages, 7 figures. Final version. Improvements to the exposition of several lemmas and the upper bound in Theorem 1.1 is imp
Scientific paper
10.1016/j.jcta.2009.03.010
This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way transportation polytopes and a catalogue of non-degenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et al. (1984). It also allowed us to discover some new results. For example, we prove that the number of vertices of an $m\times n$ transportation polytope is a multiple of the greatest common divisor of $m$ and $n$.
de Loera Jesus A.
Kim Edward D.
Onn Shmuel
Santos Francisco
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