On subgroup perfect codes in Cayley sum graphs

A perfect code C in a graph Γ is an independent set of vertices of Γ such that every vertex outside C is adjacent to a unique vertex in C, and a total perfect code C in Γ is a set of vertices of Γ such that every vertex of Γ 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 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∈X and g≠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.