2-nearly Platonic graphs are unique

TL;DR

This paper proves that all 2-nearly Platonic graphs are balanced, confirming a recent conjecture and deepening understanding of their structural properties.

Contribution

It establishes that all 2-nearly Platonic graphs must be balanced, resolving a conjecture in graph theory.

Findings

All 2-nearly Platonic graphs are necessarily balanced.

The result confirms a recent conjecture by Keith, Froncek, and Kreher.

Provides structural characterization of 2-nearly Platonic graphs.

Abstract

A 2-nearly Platonic graph of type (k|d) is a k-regular planar graph with f faces, f-2 of which are of degree d and the remaining two are of degrees m_1;m_2, both different from d. Such a graph is called balanced if m_1=m_2. We show that all 2-nearly Platonic graphs are necessarily balanced. This proves a recent conjecture by Keith, Froncek, and Kreher.