Perfect codes in Cayley sum graphs (Q2073316)

Summary: 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 \(A\).