An Eberhard-like theorem for pentagons and heptagons

Abstract

Eberhard proved that for every sequence $(p_k),3\le k\le r,k\ne 5,7$ of non-negative integers satisfying Euler's formula ${\sum }_{k\ge 3}(6-k)p_k=12$, there are infinitely many values $p_6$ such that there exists a simple convex polyhedron having precisely $p_k$ faces of length $k$ for every $k\ge 3$, where $p_k=0$ if $k>r$. In this paper we prove a similar statement when non-negative integers $p_k$ are given for $3\le k\le r$, except for $k=5$ and $k=7$. 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 length $k$ for every $k\ge 3$. %, where $p_k=0$ if $k>r$. 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.