Solutions to the ultradiscrete Toda molecule equation expressed as   minimum weight flows of planar graphs
  We define a function by means of the minimum weight flow on a planar graph and prove that this function solves the ultradiscrete Toda molecule equation, its B\"acklund transformation and the two dimensional Toda molecule equation. The method we employ in the proof can be considered as fundamental to the integrability of ultradiscrete soliton equations. 
Yoichi Nakata
nlin.SI