An Eberhard-Like Theorem for Pentagons and Heptagons.

Eberhard proved that for every sequence ( p), 3≤ k≤ r, k≠6, of nonnegative integers satisfying Euler's formula ∑(6− k) p=12, there are infinitely many values p such that there exists a simple convex polyhedron having precisely p faces of size k for every k≥3, where p=0 if k> r. In this paper we prove a similar statement when nonnegative integers p are given for 3≤ k≤ r, except for k=5 and k=7 (but including p). We prove that there are infinitely many values p, p such that there exists a simple convex polyhedron having precisely p 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. [ABSTRACT FROM AUTHOR]