Perfect codes in vertex-transitive graphs

Given a graph Γ, perfect code in Γ is an independent set C of such that every vertex outside adjacent to unique and total vertices C. To study (total) codes vertex-transitive graphs, we generalize the concept subgroup finite group introduced by Huang et al. as follows: G H G, A containing called pair (G,H) if there exists coset Cos(G,H,U) consisting left cosets Cos(G,H,U). We give necessary sufficient condition for be few known results groups. also construct some examples propose problems further research.