DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

Journal Impact Factor (JIF 2022): 0.7

5-year Journal Impact Factor (2022): 0.7

CiteScore (2022): 1.9

SNIP (2022): 0.902

Discussiones Mathematicae Graph Theory

Article in volume

Authors:

S. Jendrol'

Stanislav Jendrol'

Department of Geometry and Algebra P. J. Šafárik UniversityJesenná 5041 54 KošiceSLOVAKIA

email: stanislav.jendrol@upjs.sk

0000-0001-6869-2793

Title:

2-nearly Platonic graphs

PDF

Source:

Discussiones Mathematicae Graph Theory 44(1) (2024) 351-362

Received: 2021-08-06 , Revised: 2022-01-13 , Accepted: 2022-01-13 , Available online: 2022-02-02 , https://doi.org/10.7151/dmgt.2446

Abstract:

A $2$-nearly Platonic graph of type $(k, d)$ is a $k$-regular plane 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 balanced. This proves a recent conjecture by Keith, Froncek, and Kreher. We also show that any $2$-nearly Platonic graph belongs to one of 15 well defined infinite classes. The latter states more precisely the statement of Deza, Dutour Sikirič, and Shtogrin from 2013, and of Froncek, Khorsandi, Musawi, and Qui from 2021 that there are only 14 such classes. Moreover, our short proof provides a complete characterization of all $2$-nearly Platonic graphs.

Keywords:

plane graph, Platonic solid, almost Platonic graph

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