Polyhedral Maps of Cubic Graphs with given Automorphism Groups

2026-07-22

AI Summary

This work addresses the problem of constructing, for any given finite group \( G \), a cubic graph that can be embedded as a polyhedral map such that both the graph and the map have automorphism groups precisely isomorphic to \( G \). Building on Babai’s ideas, the authors develop a novel construction method by carefully modifying Cayley graphs of \( G \), integrating techniques from group theory, graph theory, and topological embeddings. This approach simultaneously controls the automorphism groups of both the cubic graph and its polyhedral embedding, rigorously ensuring that each is isomorphic to \( G \). The result establishes that every finite group can be realized as the exact automorphism group of both a cubic graph and a polyhedral map derived from it.

Abstract

L. Babai introduced a method for constructing a cubic graph whose automorphism group is isomorphic to a given finite group $G$, obtained by modifying a corresponding Cayley graph of $G$. Building on this approach, we construct a cubic graph that admits a polyhedral map whose automorphism group, as well as the automorphism group of the polyhedral map itself, is isomorphic to $G$.

Problem

• cubic graph
• polyhedral map
• automorphism group
• finite group
• graph construction

Innovation

• cubic graph
• automorphism group
• polyhedral map
• Cayley graph
• group representation