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arXiv · 1906.07368

Characterization of subgroup perfect codes in Cayley graphs

Jiyong Chen·Yanpeng Wang·Binzhou Xia
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Abstract

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code in $\Gamma$ if every vertex of $\Gamma$ is at distance no more than $1$ to exactly one vertex of $C$. A subset $C$ of a group $G$ is called a perfect code of $G$ if $C$ is a perfect code in some Cayley graph of $G$. In this paper we give sufficient and necessary conditions for a subgroup $H$ of a finite group $G$ to be a perfect code of $G$. Based on this, we determine the finite groups that have no nontrivial subgroup as a perfect code, which answers a question by Ma, Walls, Wang and Zhou.

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Jiyong Chen, Yanpeng Wang, Binzhou Xia. 2019-06-18. Characterization of subgroup perfect codes in Cayley graphs. https://arxiv.org/abs/1906.07368

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