Perfect codes on t-cayley graphs

Let G be a finite group with the identity e and t ∈ N. In this dissertation, we study perfect codes in a t-Cayley graph of G. Let H be a subgroup of G and k ∈ {1, . . . , t}. We have a necessary and sufficient condition for a collection Ω of subsets of G such that H × {1, . . . , k} is a perfect code in t-Cayley graph Cay(G× {1, . . . , t}, Ω). In addition, we obtain criterions for determining if H × {1, . . . , t} is a perfect code in some t-Cayley graph of G. Moreover, let m ∈ N. We have a necessary and sufficient condition for a subset S of G ∖ {e} such that a subgroup H is a perfect code in m-Cayley hypergraph m- Cay(G, S) and obtain some conditions for a subgroup H that guarantee the existence of a subset S ⊆ G ∖ {e} such that H is a perfect code in m- Cay(G, S).