  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 $n$.
