Perfect codes in Cayley sum graphs

TL;DR

This paper investigates perfect codes in Cayley sum graphs over finite abelian groups, providing conditions for subsets to be perfect codes and characterizing subgroup perfect codes.

Contribution

It offers necessary and sufficient conditions for subsets to be perfect codes in Cayley sum graphs and characterizes all subgroup perfect codes in this setting.

Findings

Conditions for subsets to be perfect codes in Cayley sum graphs

Complete characterization of subgroup perfect codes

Extension of perfect code theory to Cayley sum graphs over abelian groups

Abstract

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code of $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$. Let $A$ be a finite abelian group and $T$ a square-free subset of $A$. The Cayley sum graph of $A$ with respect to the connection set $T$ is a simple graph with $A$ as its vertex set, and two vertices $x$ and $y$ are adjacent whenever $x+y\in T$. A subgroup of $A$ is said to be a subgroup perfect code of $A$ if the subgroup is a perfect code of some Cayley sum graph of $A$. In this paper, we give some necessary and sufficient conditions for a subset of $A$ to be a perfect code of a given Cayley sum graph of $A$. We also characterize all subgroup perfect codes of $A$.