Perfect Codes in Cayley Sum Graphs

Abstract

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 $C$. Let $A$ be a finite abelian group and $T$ a square-free subset of $A$. The Cayley sum graph of $A$ with respect to the connection set $T$ is a simple graph with $A$ as its vertex set, and two vertices $x$ and $y$ are adjacent whenever $x+y\in T$. A subgroup of $A$ is said to be a subgroup perfect code of $A$ if the subgroup is a perfect code of some Cayley sum graph of $A$. In this paper, we give some necessary and sufficient conditions for a subset of $A$ to be a perfect code of a given Cayley sum graph of $A$. We also characterize all subgroup perfect codes of $A$.

Subset Perfect Codes of Finite Commutative Rings Over Induced Subgraphs of Unit Graphs

“…In 2021, Zhang and Zhou [25] focused on some families of groups namely, 2-groups, metabelian groups, nilpotent groups and generalized dihedral groups associated with Cayley graphs and provided some conditions for their subgroups to be perfect codes. Recently, Ma et al [15] focused on the perfect codes of Cayley sum graphs of finite abelian groups G and established some necessary and sufficient conditions for the Cayley sum graphs admitting a subset of G as a perfect code.…”