Perfect codes in Cayley graphs of abelian groups

TL;DR

This paper investigates the existence and properties of perfect and total perfect codes within Cayley graphs constructed from finite abelian groups.

Contribution

It provides new theoretical results characterizing perfect codes and total perfect codes specifically in Cayley graphs of finite abelian groups.

Findings

Established conditions for perfect codes in these graphs

Characterized total perfect codes in the same setting

Extended previous results to a broader class of 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 total perfect code in $\Gamma$ is a subset $C$ of $V$ such that every vertex of $\Gamma$ is adjacent to exactly one vertex in $C$. In this paper we prove several results on perfect codes and total perfect codes in Cayley graphs of finite abelian groups.