Perfect codes in proper reduced power graphs of finite groups

Abstract

Let Γ be a graph with vertex set A subset C of is a perfect code of Γ if C is an independent set such that every vertex in is adjacent to exactly one vertex in C. A subset T of is a total perfect code of Γ if every vertex of Γ is adjacent to exactly one vertex in T. Let G be a finite group. The proper reduced power graph of G is the undirected graph whose vertex set consists of all nonidentity elements, and two distinct vertices x and y are adjacent if or In this paper, we give a necessary and sufficient condition for a proper reduced power graph to contain a perfect code. In particular, we determine all perfect codes of a proper reduced power graph provided that the proper reduced power graph admits a perfect code. Moreover, we give some necessary conditions for a proper reduced power graph to contain a total perfect code. As applications, we determine the abelian groups and generalized quaternion groups whose proper reduced power graphs admit a total perfect code. We also characterize all finite groups whose proper reduced power graphs admit a total perfect code of size 2.