Perfect codes in 2-valent Cayley digraphs on abelian groups

For a digraph $\Gamma$, a subset $C$ of $V(\Gamma)$ is a perfect code if $C$ is a dominating set such that every vertex of $\Gamma$ is dominated by exactly one vertex in $C$. In this paper, we classify strongly connected 2-valent Cayley digraphs on abelian groups admitting a perfect code, and determine completely all perfect codes of such digraphs

Characterizing subgroup perfect codes by 2-subgroups

A perfect code in a graph $\Gamma$ is a subset $C$ of $V(\Gamma)$ such that no two vertices in $C$ are adjacent and every vertex in $V(\Gamma)\setminus C$ is adjacent to exactly one vertex in $C$. Let $G$ be a finite group and $C$ a subset of $G$. Then $C$ is said to be a perfect code of $G$ if there exists a Cayley graph of $G$ admiting $C$ as a perfect code. It is proved that a subgroup $H$ of $G$ is a perfect code of $G$ if and only if a Sylow $2$-subgroup of $H$ is a perfect code of $G$. This result provides a way to simplify the study of subgroup perfect codes of general groups to the study of subgroup perfect codes of $2$-groups. As an application, a criterion for determining subgroup perfect codes of projective special linear groups $\mathrm{PSL}(2,q)$ is given

On the subgroup regular set in Cayley graphs

A subset $C$ of the vertex set of a graph $\Gamma$ is said to be $(a,b)$-regular if $C$ induces an $a$-regular subgraph and every vertex outside $C$ is adjacent to exactly $b$ vertices in $C$. In particular, if $C$ is an $(a,b)$-regular set of some Cayley graph on a finite group $G$, then $C$ is called an $(a,b)$-regular set of $G$ and a $(0,1)$-regular set is called a perfect code of $G$. In [Wang, Xia and Zhou, Regular sets in Cayley graphs, J. Algebr. Comb., 2022] it is proved that if $H$ is a normal subgroup of $G$, then $H$ is a perfect code of $G$ if and only if it is an $(a,b)$-regular set of $G$, for each $0\leq a\leq|H|-1$ and $0\leq b\leq|H|$ with $\gcd(2,|H|-1)\mid a$. In this paper, we generalize this result and show that a subgroup $H$ of $G$ is a perfect code of $G$ if and only if it is an $(a,b)$-regular set of $G$, for each $0\leq a\leq|H|-1$ and $0\leq b\leq|H|$ such that $\gcd(2,|H|-1)$ divides $a$

On subgroup perfect codes in Cayley sum graphs

A perfect code $C$ in a graph $\Gamma$ is an independent set of vertices of $\Gamma$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code $C$ in $\Gamma$ is a set of vertices of $\Gamma$ such that every vertex of $\Gamma$ is adjacent to a unique vertex in $C$. Let $G$ be a finite group and $X$ a normal subset of $G$. The Cayley sum graph $\mathrm{CS}(G,X)$ of $G$ with the connection set $X$ is the graph with vertex set $G$ and two vertices $g$ and $h$ being adjacent if and only if $gh\in X$ and $g\neq h$. In this paper, we give some necessary conditions of a subgroup of a given group being a (total) perfect code in a Cayley sum graph of the group. As applications, the Cayley sum graphs of some families of groups which admit a subgroup as a (total) perfect code are classified

Perfect codes in quintic Cayley graphs on abelian groups

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code of $\Gamma$ if every vertex of $\Gamma$ is at distance no more than one to exactly one vertex in $C$. In this paper, we classify all connected quintic Cayley graphs on abelian groups that admit a perfect code, and determine completely all perfect codes of such graphs