An Eberhard-like theorem for pentagons and heptagons

Eberhard proved that for every sequence (p k ), 3≤k≤r, k≠6, of nonnegative integers satisfying Euler’s formula ∑ k≥3(6−k)p k =12, there are infinitely many values p 6 such that there exists a simple convex polyhedron having precisely p k faces of size k for every k≥3, where p k =0 if k>r. In this paper we prove a similar statement when nonnegative integers p k are given for 3≤k≤r, except for k=5 and k=7 (but including p 6). We prove that there are infinitely many values p 5,p 7 such that there exists a simple convex polyhedron having precisely p k faces of size k for every k≥3. We derive an extension to arbitrary closed surfaces, yielding maps of arbitrarily high face-width. Our proof suggests a general method for obtaining results of this kind