(Total) Perfect codes in (extended) subgroup sum graphs

TL;DR

This paper characterizes when subgroup sum graphs of finite groups contain perfect codes, classifies all code-perfect Dedekind groups, and explores perfect codes in graphs derived from cyclic, dihedral, dicyclic, and abelian groups.

Contribution

It provides necessary and sufficient conditions for the existence of perfect codes in subgroup sum graphs and classifies all code-perfect Dedekind groups and specific group cases.

Findings

Characterization of when subgroup sum graphs admit perfect codes

Classification of all code-perfect Dedekind groups

Identification of groups with total perfect codes in subgroup sum graphs

Abstract

Given a finite group $G$ with identity $e$ and a normal subgroup $H$ of $G$, the subgroup sum graph $\Gamma_{G,H}$ (resp. extended subgroup sum graph $\Gamma_{G,H}^+$) of $G$ with respect to $H$ is the graph with vertex set $G$, in which distinct vertices $x$ and $y$ are adjacent whenever $xy\in H\setminus \{e\}$ (resp. $xy\in H$). A group $G$ is said to be {\em code-perfect} if for any normal subgroup $H$ of $G$, $\Gamma_{G,H}$ admits a perfect code. In this paper, we give a necessary and sufficient condition for which normal subgroups $H$ of $G$ satisfy that a (extended) subgroup sum graph of $G$ with respect to $H$ admits a (total) perfect code, and classify all code-perfect Dedekind groups. As an application, we classify all normal subgroups such that the subgroup sum graph of a cyclic group, a dihedral group or a dicyclic group with respect to such a normal subgroup admits perfect codes, respectively. We also determine all abelian groups $A$ and subgroups $H$ of $A$ such that $\Gamma_{A,H}$ admits a total perfect code.