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arXiv · 1904.01858

Subgroup perfect codes in Cayley graphs

Xuanlong Ma·Gary L. Walls·Kaishun Wang·Sanming Zhou
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Abstract

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 $V(Γ)\setminus C$ 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.

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Xuanlong Ma, Gary L. Walls, Kaishun Wang, Sanming Zhou. 2020-07-16. Subgroup perfect codes in Cayley graphs. https://arxiv.org/abs/1904.01858

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