Joint generalization of the theorems of Lebesgue and Kotzig on the combinatorics of planar maps

The weight of an edge in a map or polyhedron is the sum of the degrees of its end points. A map is normal if it does not contain vertices or faces incident to fewer than three edges. We prove that every planar normal map contains the following: either a 3-face incident to an edge of weight no greater than 13; or a 4-face incident to an edge of weight no greater than 8; or a 5-face incident to an edge of weight 6. All the bounds – 13, 8 and 6 – are attainable.