On the faces of unigraphic 3-polytopes

A 3-polytope is a 3-connected, planar graph. It is called unigraphic if it does not share its vertex degree sequence with any other 3-polytope, up to graph isomorphism. The classification of unigraphic 3-polytopes appears to be a difficult problem.

In this paper we prove that, apart from pyramids, all unigraphic 3-polytopes have no $n$-gonal faces for $n\ge 10$. Our method involves defining several planar graph transformations on a given 3-polytope containing an $n$-gonal face with $n\ge 10$. The delicate part is to prove that, for every such 3-polytope, at least one of these transformations both preserves 3-connectivity, and is not an isomorphism.