Subgroup perfect codes in Cayley sum graphs
| Autor: | Min Feng, Kaishun Wang, Xuanlong Ma |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Finite group
Hamming bound Applied Mathematics Sylow theorems Quaternion group 020206 networking & telecommunications Cyclic group 0102 computer and information sciences 02 engineering and technology Dihedral group 01 natural sciences Computer Science Applications Vertex (geometry) Combinatorics Mathematics::Group Theory 010201 computation theory & mathematics 0202 electrical engineering electronic engineering information engineering Abelian group Mathematics |
| Zdroj: | Designs, Codes and Cryptography. 88:1447-1461 |
| ISSN: | 1573-7586 0925-1022 |
| DOI: | 10.1007/s10623-020-00758-3 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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