Name
Abstract | ||
|---|---|---|
Let $$\Gamma $$ be a graph with vertex set V. If a subset C of V is independent in $$\Gamma $$ and every vertex in $$V\setminus C$$ is adjacent to exactly one vertex in C, then C is called a perfect code of $$\Gamma $$. Let G be a finite group and let S be a square-free normal subset of G. The Cayley sum graph of G with respect to S is a simple graph with vertex set G and two vertices x and y are adjacent if $$xy\in S$$. A subset C of G is called a perfect code of G if there exists a Cayley sum graph of G which admits C as a perfect code. In particular, if a subgroup of G is a perfect code of G, then the subgroup is called a subgroup perfect code of G. In this paper, we give a necessary and sufficient condition for a non-trivial subgroup of an abelian group with non-trivial Sylow 2-subgroup to be a subgroup perfect code of the group. 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. As an application, we classify the abelian groups whose every non-trivial subgroup is a subgroup perfect code. Moreover, we determine all subgroup perfect codes of a cyclic group, a dihedral group and a generalized quaternion group. |
Year | DOI | Venue |
|---|---|---|
2020 | 10.1007/s10623-020-00758-3 | Designs, Codes and Cryptography |
Keywords | DocType | Volume |
Perfect code, Subgroup perfect code, Cayley sum graph, Finite group, 05C25, 05C69, 94B25 | Journal | 88 |
Issue | ISSN | Citations |
7 | 0925-1022 | 0 |
PageRank | References | Authors |
0.34 | 0 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Xuanlong Ma | 1 | 13 | 3.42 |
| Min Feng | 2 | 4 | 1.96 |
| Kaishun Wang | 3 | 227 | 39.82 |