Complete graphs whose topological symmetry groups are polyhedral

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 $m$, the complete graph $K_m$ has an embedding in $S^3$ whose topological symmetry group is isomorphic to one of the polyhedral groups: $A_4$, $A_5$, or $S_4$.