2019/04/03 by Xuanlong Ma, Ma, Xuanlong, Gary L. Walls +5 · 5 citations
Engineering · Computer Science · Mathematics · #graph theory and CDMA systems #Cooperative Communication and Network Coding #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.1904.01858
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 problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian 2-groups. Finally, we determine all subgroup perfect codes in any generalized quaternion group.