On non-normal subgroup perfect codes

Abstract

Let $X=(V,E)$ be a graph. A subset $C\subseteq V(X)$ is a \emph{perfect code} of $X$ if $C$ is a coclique of $X$ with the property that any vertex in $V(X)\setminus C$ is adjacent to exactly one vertex in $C$. Given a finite group $G$ with identity element $e$ and $H\le G$, $H$ is a \emph{subgroup perfect code} of $G$ if there exists an inverse-closed subset $S\subseteq G\setminus \{e\}$ such that $H$ is a perfect code of the Cayley graph $\mathrm{Cay}(G,S)$ of $G$ with connection set $S$. In this short note, we give an infinite family of finite groups $G$ admitting a non-normal subgroup perfect code $H$ such that there exists $g\in G$ with $g^2\in H$ but $(gh{)}^2\ne e$, for all $h\in H$; thus, answering a question raised by Wang, Xia, and Zhou in [Perfect sets in Cayley graphs. {\it arXiv preprint} arXiv:2006.05100, 2020].

6 pages

References in corpus (2)

• Perfect codes in circulant graphs
• Corrigendum to "On subgroup perfect codes in Cayley graphs" [European J. Combin. 91 (2021) 103228]