Perfect codes in vertex-transitive graphs

Abstract

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