Perfect codes in Cayley graphs of abelian groups

Abstract

A perfect code in a graph $\Gamma =(V,E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V\setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$ of $V$ such that every vertex of $\Gamma$ is adjacent to exactly one vertex in $C$. In this paper we prove several results on perfect codes and total perfect codes in Cayley graphs of finite abelian groups.

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