Kotzig's theorem

An extreme case is the triakis icosahedron, where no edge has smaller total degree.

The result is named after Anton Kotzig, who published it in 1955 in the dual form that every convex polyhedron has two adjacent faces with a total of at most 13 sides.

[4] If all triangular faces of a polyhedron are vertex-disjoint, there exists an edge with smaller total degree, at most eight.

[5] Generalizations of the theorem are also known for graph embeddings onto surfaces with higher genus.

However, for planar graphs with vertices of degree lower than three, variants of the theorem have been proven, showing that either there is an edge of bounded total degree or some other special kind of subgraph.