Perfect codes in Cayley graphs of Hajós groups

A perfect code in a graph $Γ$ is a subset $C$ of the vertex set of $Γ$ such that every vertex of $Γ$ 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 $Γ$. 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 Hajós group. In this paper we classify all Cayley graphs (directed or undirected) of Hajós groups which admit perfect codes, and moreover we determine all perfect codes in such Cayley graphs.