Perfect codes in Cayley graphs

TL;DR

This paper investigates the properties and construction methods of perfect and total perfect codes within Cayley graphs, focusing on subgroup conditions and automorphism-based constructions.

Contribution

It provides new theoretical results on when subgroups form perfect codes and introduces methods to generate new codes via automorphisms of the underlying group.

Findings

Subgroups can serve as perfect codes under specific conditions.

Automorphisms can be used to construct new perfect codes from existing ones.

Several theorems characterizing perfect codes in Cayley graphs are proved.

Abstract

Given a graph $\Gamma$, a subset $C$ of $V(\Gamma)$ is called a perfect code in $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$, and a subset $C$ of $V(\Gamma)$ is called a total perfect code in $\Gamma$ if every vertex of $\Gamma$ is adjacent to exactly one vertex in $C$. In this paper we study perfect codes and total perfect codes in Cayley graphs, with a focus on the following themes: when a subgroup of a given group is a (total) perfect code in a Cayley graph of the group; and how to construct new (total) perfect codes in a Cayley graph from known ones using automorphisms of the underlying group. We prove several results around these questions.