On families 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 size d and the remaining two are of sizes d 1 , d 2 , both different from d. Such a graph is called balanced if d 1 = d 2 . We show that all connected 2-nearly Platonic graphs are necessarily balanced. This proves a recent conjecture by Keith, Froncek, and Kreher.