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arXiv · 2112.06236

Perfect codes in vertex-transitive graphs

Yuting Wang·Junyang Zhang
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Abstract

Given a graph $Γ$, a perfect code in $Γ$ is an independent set $C$ of vertices of $Γ$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code in $Γ$ is a set $C$ of vertices of $Γ$ such that every vertex of $Γ$ is adjacent to a unique vertex in $C$. To study (total) perfect codes in vertex-transitive graphs, we generalize the concept of subgroup (total) perfect code of a finite group introduced in \cite{HXZ18} as follows: Given a finite group $G$ and a subgroup $H$ of $G$, a subgroup $A$ of $G$ containing $H$ is called a subgroup (total) perfect code of the pair $(G,H)$ if there exists a coset graph $Cos(G,H,U)$ such that the set consisting of left cosets of $H$ in $A$ is a (total) perfect code in $Cos(G,H,U)$. We give a necessary and sufficient condition for a subgroup $A$ of $G$ containing $H$ to be a (total) perfect code of the pair $(G,H)$ and generalize a few known results of subgroup (total) perfect codes of groups. We also construct some examples of subgroup perfect codes of the pair $(G,H)$ and propose a few problems for further research.

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Yuting Wang, Junyang Zhang. 2021-12-12. Perfect codes in vertex-transitive graphs. https://arxiv.org/abs/2112.06236

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