The Diameters of Network-flow Polytopes satisfy the Hirsch Conjecture
  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. 
S. Borgwardt, J. A. De Loera, E. Finhold
math.CO