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 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.

After we (DF and JQ) posted v1, we learned from M.R. Khorsandi and S.R. Musawi that they had obtained independently the same results, using very similar methods. We decided to merge the papers, and this is the first joint draft