Graphs of Transportation Polytopes

TL;DR

This paper investigates the properties of transportation polytopes' graphs, providing bounds on diameters, cataloging small cases, disproving conjectures, and revealing new structural insights such as vertex count divisibility.

Contribution

It offers a quadratic diameter bound for axial 3-way transportation polytopes and disproofs five longstanding conjectures, advancing understanding of their combinatorial structure.

Findings

Quadratic bound on the diameter of axial 3-way transportation polytopes

Catalogue of small non-degenerate transportation polytopes

Number of vertices divisible by gcd of dimensions

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 $m\times n$ transportation polytope is a multiple of the greatest common divisor of $m$ and $n$.