Plane Cubic Graphs with Prescribed Face Areas
Carsten Thomassen · Combinatorics Probability Computing · 1992
If G is a plane, cubic graph, then G has a drawing such that each edge is a straight line segment and each bounded face has any prescribed area. The special case where all areas are the same proves a conjecture of G. Ringel, who gave an example of a plane triangulation that cannot be drawn in this way.