Subgroup perfect codes in Cayley sum graphs
Subgroup perfect codes in Cayley sum graphs
复制标题
凯莱和图中的子群完美码
DOI：
10.1007/s10623-020-00758-3
复制
发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Wang Kaishun
中科院分区：
文献类型：
--
作者：
Ma Xuanlong;Feng Min;Wang Kaishun
设为一个顶点集为V的图。如果子集V在中独立，且每个顶点都恰好与C中的一个顶点相邻，则称C是的完美码。设G是有限群，S是G的无平方正规子集. G关于S的凯莱和图是顶点集G的简单图，且两个顶点x和y相邻，如果。一个子集G称为G的完美码，如果存在G的Cayley和图，且该图承认C为完美码。特别地，如果G的一个子群是G的一个完全码，则称该子群为G的一个子群完全码。本文给出了具有非平凡Sylow 2-子群的交换群的非平凡子群是该群的子群完全码的一个充要条件。这减少了确定阿贝尔群的给定子群何时是阿贝尔2-群的完美码的问题。作为应用，我们对每个非平凡子群是子群完全码的阿贝尔群进行了分类。此外，我们还确定了循环群、二面体群和广义四元数群的所有子群完美码。
Letbe a graph with vertex setV. If a subsetCofVis independent inand every vertex inis adjacent to exactly one vertex inC, thenCis called a perfect code of. LetGbe a finite group and letSbe a square-free normal subset ofG. The Cayley sum graph ofGwith respect toSis a simple graph with vertex setGand two verticesxandyare adjacent if. A subsetCofGis called a perfect code ofGif there exists a Cayley sum graph ofGwhich admitsCas a perfect code. In particular, if a subgroup ofGis a perfect code ofG, then the subgroup is called a subgroup perfect code ofG. 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.