Graphs of transportation polytopes

Authors:
Jesús A. De Loera;Edward D. Kim;Shmuel Onn;Francisco Santos
Affiliations:
University of California, Dept. of Mathematics, One Shields Avenue, Davis, CA, United States;University of California, Dept. of Mathematics, One Shields Avenue, Davis, CA, United States;Technion -- Israel Institute of Technology, 32000 Haifa, Israel;University of Cantabria, E-39005 Santander, Spain
Venue:
Journal of Combinatorial Theory Series A
Year:
2009

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Abstract

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 mxn transportation polytope is a multiple of the greatest common divisor of m and n.