Perfect codes in quintic Cayley graphs on abelian groups

TL;DR

This paper classifies all connected quintic Cayley graphs on abelian groups that admit perfect codes and fully determines the structure of these codes, advancing understanding in algebraic graph theory.

Contribution

It provides a complete classification of perfect codes in a specific class of Cayley graphs on abelian groups, a novel result in the field.

Findings

All connected quintic Cayley graphs on abelian groups admitting perfect codes are classified.

Complete characterization of all perfect codes in these graphs is provided.

The structure of such perfect codes is fully determined.

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 $C$. In this paper, we classify all connected quintic Cayley graphs on abelian groups that admit a perfect code, and determine completely all perfect codes of such graphs.