Which maximal subgroups are perfect codes?

TL;DR

This paper investigates which maximal subgroups of a group can serve as perfect codes in Cayley graphs, providing a systematic characterization based on their local complements.

Contribution

It introduces a systematic approach to identify subgroup perfect codes among maximal subgroups of groups, with a new characterization involving local complements.

Findings

Characterization of subgroup perfect codes via local complements

Identification of conditions for maximal subgroups to be perfect codes

Systematic classification of subgroup perfect codes in Cayley graphs

Abstract

A perfect code in a graph $\Gamma=(V, E)$ 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 $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if it is a perfect code in some Cayley graph of $G$. In this paper, we undertake a systematic study of which maximal subgroups of a group can be perfect codes. Our approach highlights a characterization of subgroup perfect codes in terms of their ``local'' complements.