  The study of the diameter of the graph of polyhedra is a classical problem in
the theory of linear programming. While transportation polytopes are at the
core of operations research and statistics it is still open whether the Hirsch
conjecture is true for general $m{\times}n$--transportation polytopes. In
earlier work the first three authors introduced a hierarchy of variations to
the notion of graph diameter in polyhedra. The key reason was that this
hierarchy provides some interesting lower bounds for the usual graph diameter.
  This paper has three contributions: First, we compare the hierarchy of
diameters for the $m{\times}n$--transportation polytopes. We show that the
Hirsch conjecture bound of $m+n-1$ is actually valid in most of these diameter
notions. Second, we prove that for $3{\times}n$--transportation polytopes the
Hirsch conjecture holds in the classical graph diameter. Third, we show for
$2{\times}n$--transportation polytopes that the stronger monotone Hirsch
conjecture holds and improve earlier bounds on the graph diameter.
