A note on nearly platonic graphs

TL;DR

This paper investigates nearly platonic graphs, focusing on their face degree uniformity, proving the impossibility of having exactly one different face, and proposing conjectures about the structure of graphs with two such faces.

Contribution

It establishes a new impossibility result for finite nearly platonic graphs and introduces conjectures about their structural families.

Findings

Impossible for finite graphs to have exactly one disparate face.

Proposes conjectures on the limited families of graphs with two disparate faces.

Abstract

A nearly platonic graph is a k-regular simple planar graph in which all but a small number of the faces have the same degree. We show that it is impossible for a finite graph to have exactly one disparate face, and offer some conjectures, including the conjecture that graphs with two disparate faces come in a small set of families.