On families of 2-nearly Platonic graphs - Dalibor Froncek; Mahdi Reza Khorsandi; Seyed Reza Musawi; Jiangyi Qiu (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2022)
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 d1, d2, both different from d. Such a graph is called balanced if d1 = d2. We show that all connected 2-nearly Platonic graphs are necessarily balanced. This proves a recent conjecture by Keith, Froncek, and Kreher.