# Perfect codes in circulant graphs

## TL;DR

This paper characterizes when circulant graphs of certain degrees and orders admit perfect and total perfect codes, providing necessary and sufficient conditions based on prime factorization.

## Contribution

It establishes exact criteria for the existence of perfect and total perfect codes in circulant graphs with degrees related to prime powers.

## Key findings

- Conditions for perfect codes in circulant graphs of degree p-1 where p is an odd prime.
- Criteria for perfect codes in circulant graphs of order n with degree p^l-1.
- Results for total perfect codes in similar circulant graph settings.

## Abstract

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ that is an independent set such that 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 $V$ is adjacent to exactly one vertex in $C$. A perfect code in the Hamming graph $H(n, q)$ agrees with a $q$-ary perfect 1-code of length $n$ in the classical setting. In this paper we give a necessary and sufficient condition for a circulant graph of degree $p-1$ to admit a perfect code, where $p$ is an odd prime. We also obtain a necessary and sufficient condition for a circulant graph of order $n$ and degree $p^l-1$ to have a perfect code, where $p$ is a prime and $p^l$ the largest power of $p$ dividing $n$. Similar results for total perfect codes are also obtained in the paper.