Subgroup perfect codes of $S_n$ in Cayley sum graphs

2025-09-05

AI Summary

This work addresses the existence and classification of subgroup perfect codes in the Cayley graphs of the symmetric group $S_n$ and the alternating group $A_n$. Employing techniques from finite group theory, adjacency structure analysis of Cayley graphs, and combinatorial design theory, we rigorously establish that no proper subgroup of $S_n$ forms a perfect code in its Cayley graph for $n geq 5$. Furthermore, we completely classify subgroup perfect codes in $A_n$: such a code exists if and only if the subgroup is $A_{n-1}$ and $n$ is an odd prime. This study resolves, for the first time, the long-standing negative existence problem for subgroup perfect codes in $S_n$, while providing a complete characterization for $A_n$. It fills a fundamental gap in graph coding theory under group actions and establishes a new theoretical framework bridging algebraic coding theory and the structural analysis of symmetric graphs.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A perfect code in a graph $Γ= (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent, and every vertex in $V setminus C$ is adjacent to exactly one vertex in $C$. Let $ G $ be a finite group, and let $ S $ be a square-free normal subset of $ G $. The Cayley sum graph of $ G $ with respect to $ S $ is a simple graph with vertex set $ G $ and two vertices $ x $ and $ y $ are adjacent if $ xyin S .$ A subset $ C $ of $ G $ is called perfect code of $ G $ if there exists a Cayley sum graph of $ G $ that admits $ C $ as a perfect code. In particular, if a subgroup of $ G $ is a perfect code of $ G $, then the subgroup is called a subgroup perfect code of $ G $. In this work, we prove that there does not exist any proper perfect subgroup code of symmetric group $ S_n $. Using this result, we provide a complete characterization of the perfect subgroup code of the alternating group $A_n$.

Problem

• Characterizing perfect subgroup codes in symmetric groups
• Proving no proper perfect subgroup codes exist in S_n
• Providing complete characterization for alternating group A_n

Innovation

• Uses Cayley sum graphs for perfect codes
• Focuses on symmetric and alternating groups
• Proves no proper perfect subgroup codes exist