A note on nearly Platonic graphs with connectivity one
D. Froncek, M.R. Khorsandi, S.R. Musawi, and J. Qiu
TL;DR
This paper proves that nearly Platonic graphs with connectivity one cannot exist, extending previous results that excluded their existence in 2-connected cases, thus clarifying the structural limitations of such graphs.
Contribution
It establishes the non-existence of nearly Platonic graphs with connectivity one, complementing earlier work on 2-connected cases.
Findings
No nearly Platonic graphs with connectivity one exist.
The result extends the understanding of structural constraints of planar graphs.
Complements previous non-existence results for 2-connected nearly Platonic graphs.
Abstract
A k-regular planar graph G is nearly Platonic when all faces but one are of the same degree while the remaining face is of a different degree. We show that no such graphs with connectivity one can exist. This complements a recent result by Keith, Froncek, and Kreher on non-existence of 2-connected nearly Platonic graphs.
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Taxonomy
TopicsAdvanced Graph Theory Research · Interconnection Networks and Systems · Advanced Materials and Mechanics