Subgroup perfect codes of $S_n$ in Cayley sum graphs

Abstract: A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent, and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. Let $G$ be a finite group, and let $S$ be a square-free normal subset of $G$. The Cayley sum graph of $G$ with respect to $S$ is a simple graph with vertex set $G$ and two vertices $x$ and $y$ are adjacent if $xy\in S .$ A subset $C$ of $G$ is called perfect code of $G$ if there exists a Cayley sum graph of $G$ that admits $C$ as a perfect code. In particular, if a subgroup of $G$ is a perfect code of $G$, then the subgroup is called a subgroup perfect code of $G$. In this work, we prove that there does not exist any proper perfect subgroup code of symmetric group $S_n$. Using this result, we provide a complete characterization of the perfect subgroup code of the alternating group $A_n$.