2-nearly Platonic graphs are unique

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 m1, m2, both different from d. Such a graph is called balanced if m1 = m2. We show that all 2-nearly Platonic graphs are necessarily balanced. This proves a recent conjecture by Keith, Froncek, and Kreher.