  We solve a problem in the combinatorics of polyhedra motivated by the network
simplex method. We show that the Hirsch conjecture holds for the diameter of
the graphs of all network-flow polytopes, in particular the diameter of a
network-flow polytope for a network with $n$ nodes and $m$ arcs is never more
than $m+n-1$. A key step to prove this is to show the same result for classical
transportation polytopes.
