Complete graphs whose topological symmetry groups are polyhedral
Erica Flapan, Blake Mellor, Ramin Naimi

TL;DR
This paper classifies which complete graphs can be embedded in three-dimensional space with symmetries matching polyhedral groups like A4, A5, or S4, expanding understanding of graph symmetries in topology.
Contribution
It provides a complete characterization of complete graphs with embeddings in S^3 that have topological symmetry groups isomorphic to specific polyhedral groups.
Findings
Identifies all m for which K_m has the desired embeddings.
Connects graph embeddings with polyhedral symmetry groups.
Advances classification of symmetric graph embeddings in topology.
Abstract
We determine for which , the complete graph has an embedding in whose topological symmetry group is isomorphic to one of the polyhedral groups: , , or .
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