On families of 2-nearly Platonic graphs

Dalibor Froncek, Mahdi Reza Khorsandi, Seyed Reza Musawi, Jiangyi Qiu


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.


Planar graphs; regular graphs; Platonic graphs

Full Text:


DOI: http://dx.doi.org/10.5614/ejgta.2022.10.2.23


M. Atiyah and P. Sutcliffe, Polyhedra in physics, chemistry and geometry, Milan J. Math., 71 (2003), 33–58.

D.W. Crowe, Nearly regular polyhedra with two exceptional faces, The Many Facets of Graph Theory (Proc. Conf. Western Mich. Univ., Kalamazoo, MI, 1968), 63–76, Springer, Berlin 1969.

M. Deza and M. Dutour Sikirić, Geometry of Chemical Graphs: Polycycles and Two-Faced Maps. Cambridge University Press, 2008.

M. Deza, M. Dutour Sikirić and M. Shtogrin, Fullerene-like spheres with faces of negative curvature, Chapter 13 in “Diamond and Related Nanostructures” M.V. Diudea and C.L. Nagy ed., Springer, 2013.

D. Froncek and J. Qiu, Balanced 3-nearly Platonic graphs, Bull. Inst. Combin. Appl., 86 (2019), 34–63.

D. Froncek, M.R. Khorsadi, S.R. Musawi, and J. Qiu, A note on non-existence of nearly Platonic graphs with connectivity one, Electron. J. Graph Theory Appl. (EJGTA), 9(1) (2021), 195–205.

D. Froncek, M.R. Khorsadi, S.R. Musawi, and J. Qiu, 2-nearly Platonic graphs are unique, https://arxiv.org/abs/1911.05648

B. Grünbaum (Prepared and with a preface by V. Kaibel, V. Klee and G.M. Ziegler) Convex polytopes, Graduate Texts in Mathematics, 221. Springer-Verlag, New York, 2003

M. Horňák, A theorem on nonexistence of a certain type of nearly regular cell-decompositions of the sphere, Časopis Pěst. Mat., 103(4) 1978, 333–338.

M. Horňák and S. Jendroľ, On a conjecture by Plummer and Toft, J. Graph Theory 30(3) 1999, 177–189.

M. Horňák and E. Jucovič, Nearly regular cell-decompositions of orientable 2-manifolds with at most two exceptional cells. Math. Slov. 27(1) 1977, 73–89.

S. Jendroľ, 2-nearly Platonic graphs. Discussiones Mathematicae Graph Theory, preprint. https://doi.org/10.7151/dmgt.2446

S. Jendroľ, On the non-existence of certain nearly regular planar maps with two exceptional faces. Mat. Čas. 25(2) 1975, 159–164.

S. Jendroľ and E. Jucovič, On a conjecture by B. Grünbaum. Disc. Math. 2(1) 1972, 35–49.

W. Keith, D. Froncek, D. Kreher, A note on nearly Platonic graphs, Australas. J. Combin., 70 (2018), 86–103.

W. Keith, D. Froncek, D. Kreher, Corrigendum to: A note on nearly Platonic graphs, Australas. J. Combin., 72 (2018), 163.

D.R. Lloyd, How old are the Platonic solids?, BSHM Bull., 27(3) (2012), 131–140.

J. Malkevitch. Properties of planar graphs with uniform vertex and face structure. Memoirs of the AMS, No. 99, American Mathematical Society, Providence, R.I. 1970. MR0260616 (41 #5240).

D. West, Introduction to Graph Theory, Second Edition, Pearson Education, Inc., 2001.


  • There are currently no refbacks.

ISSN: 2338-2287

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

View EJGTA Stats