An Eberhard-Like Theorem for Pentagons and Heptagons
Eberhard proved that for every sequence (p (k) ), 3a parts per thousand currency signka parts per thousand currency signr, k not equal 6, of nonnegative integers satisfying Euler's formula a (ka parts per thousand yen3)(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 ka parts per thousand yen3, where p (k) =0 if k > r. In this paper we prove a similar statement when nonnegative integers p (k) are given for 3a parts per thousand currency signka parts per thousand currency signr, 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 ka parts per thousand yen3. 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.
eberhard theorem simple polyhedron planar graph face-width