Refined weight of edges in normal plane maps

By a k(ℓ)-vertex we mean a k-vertex incident with precisely ℓ triangular faces. The purpose of our paper is to prove that every NPM has an edge of one of the following types: (3(3),10), (3(2),9), (3(1),7), (4(4),7), (4(3),6), (5(5),6), or (5,5), where all bounds are best possible. In particular, this implies that the bounds in (3,10), (4,7), and (5,6) can be attained only at NPMs having a simplicial 3-, 4-, or 5-vertex, respectively.