On families of 2-nearly Platonic graphs

Dalibor Froncek (Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, Minnesota 55812, U.S.A)
Mahdi Reza Khorsandi (Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 36199-95161, Shahrood, Iran)
Seyed Reza Musawi (Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 36199-95161, Shahrood, Iran)
Jiangyi Qiu (Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003-9305, U.S.A)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 10, No 2 (2022): Electronic Journal of Graph Theory and Applications

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.