Subgroup perfect codes in Cayley graphs

Abstract

Let Γ be a graph with vertex set V (Γ). A subset C of V (Γ) is called a perfect code in Γ if C is an independent set of Γ and every vertex in V (Γ) \ C is adjacent to exactly one vertex in C. A subset C of a group G is called a perfect code of G if there exists a Cayley graph of G which admits C as a perfect code. A group G is said to be code-perfect if every proper subgroup of G is a perfect code of G. In this paper we prove that a group is code-perfect if and only if it has no elements of order 4. We also prove that a proper subgroup H of an abelian group G is a perfect code of G if and only if the Sylow 2-subgroup of H is a perfect code of the Sylow 2-subgroup of G. This reduces the probl..