SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1909-1921, January 2020.
Let $\Gamma$ be a graph with vertex set $V(\Gamma)$. A subset $C$ of $V(\Gamma)$ is called a perfect code in $\Gamma$ if $C$ is an independent set of $\Gamma$ and every vertex in $V(\Gamma)\setminus C$ 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.

中文翻译：

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Cayley图中的子组完美代码

SIAM离散数学杂志，第34卷，第3期，第1909-1921页，2020年1月。
假设$ \ Gamma $是顶点设置为$ V（\ Gamma）$的图。如果$ C $是$ \ Gamma $的独立集合，并且$ V（\ Gamma）\ setminus C $中的每个顶点，则$ V（\ Gamma）$的子集$ C $被称为$ \ Gamma $中的完美代码。在$ C $中正好与一个顶点相邻。如果存在$ G $的Cayley图将$ C $作为完美代码，则组$ G $的子集$ C $被称为$ G $完美代码。如果$ G $的每个适当子组都是$ G $的完美代码，则$ G $组被认为是代码完美的。在本文中，我们证明了当且仅当它没有阶数为4时，一个组才是代码完美的。我们还证明，如果一个阿贝尔群$ G $的一个适当子组$ H $是$ G $的完美代码，并且仅当$ H $的Sylow 2子组是$ G $的Sylow 2子组的完美代码时。这减少了确定阿贝尔群的给定子群何时是阿贝尔群2的情况的理想代码的问题。最后，我们确定任何广义四元数组中的所有子组完善​​代码。