Perfect codes in Cayley graphs of Haj\'os groups

TL;DR

This paper classifies all Cayley graphs of Hajós groups that admit perfect codes and determines all such codes, advancing understanding of perfect codes in algebraic graph structures.

Contribution

It provides a complete classification of perfect codes in Cayley graphs of Hajós groups, a significant step in algebraic graph theory and coding theory.

Findings

Classified all Cayley graphs of Hajós groups with perfect codes.

Determined all perfect codes within these Cayley graphs.

Extended understanding of perfect codes in algebraic and directed graph contexts.

Abstract

A perfect code in a graph $\Gamma$ is a subset $C$ of the vertex set of $\Gamma$ such that every vertex of $\Gamma$ outside $C$ has exactly one neighbour in $C$. A perfect code in a directed graph can be defined similarly by requiring that for every vertex $v$ outside $C$ there exists exactly one vertex $u$ in $C$ such that the arc from $u$ to $v$ exists in $\Gamma$. A subset $X$ of an abelian group $G$ is said to be periodic if there exists a non-identity element $g$ of $G$ such that $g + X = X$. A factorization of $G$ is a pair of nonempty subsets $(A, B)$ of $G$ such that every element $g$ of $G$ can be expressed uniquely as $g = a+b$ with $a \in A$ and $b \in B$. If for every factorization $(A, B)$ of an abelian group $G$ at least one of $A$ and $B$ is periodic, then $G$ is said to be a Haj\'os group. In this paper we classify all Cayley graphs (directed or undirected) of Haj\'os groups which admit perfect codes, and moreover we determine all perfect codes in such Cayley graphs.