A note on nearly platonic graphs
  A nearly platonic graph is a k-regular simple planar graph in which all but a small number of the faces have the same degree. We show that it is impossible for a finite graph to have exactly one disparate face, and offer some conjectures, including the conjecture that graphs with two disparate faces come in a small set of families. 
Dalibor Froncek, William J. Keith, Donald L. Kreher
math.CO