Perfect codes in Cayley graphs of abelian groups

Abstract A perfect code in a graph $$\Gamma = (V, E)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a subset C of V such that no two vertices in C are adjacent and every vertex in $$V \setminus C$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>\</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> is adjacent to exactly one vertex in C . A total perfect code in $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> is a subset C of V such that every vertex of $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> 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.