Perfect codes in quintic Cayley graphs on abelian groups

Abstract

A subset $C$ of the vertex set of a graph $\hat{I}“$ is called a perfect code of $\hat{I}“$ if every vertex of $\hat{I}“$ is at distance no more than one to exactly one vertex in $C$. In this paper, we classify all connected quintic Cayley graphs on abelian groups that admit a perfect code, and determine completely all perfect codes of such graphs.