An Eberhard-like theorem for pentagons and heptagons

TL;DR

This paper extends Eberhard's theorem by establishing the existence of infinitely many convex polyhedra with prescribed face counts, including pentagons and heptagons, under certain conditions, and proposes a general method for such combinatorial constructions.

Contribution

It proves a new version of Eberhard's theorem allowing for prescribed counts of pentagons and heptagons, and introduces a general method for constructing such polyhedra.

Findings

Existence of infinitely many convex polyhedra with given face counts including pentagons and heptagons.

Extension of Eberhard's theorem to broader face configurations.

A proposed general method for constructing maps with specified face counts.

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 $\sum_{k\ge3} (6-k) p_k = 12$, 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 $k\ge3$, 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 $k\ge3$. %, 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.