On Hamiltonicity and Perfect Codes in Non-Cyclic Graphs of Finite Groups

Let $G$ be a finite non-cyclic group. Define $\mathrm{Cyc}(G)$ as the set of all elements $a \in G$ such that for any $b\in G$, the subgroup $\langle a, b \rangle$ is cyclic. The \emph{non-cyclic graph} $\Gamma(G)$ of $G$ is a simple undirected graph with vertex set $G \setminus \mathrm{Cyc}(G)$, where two distinct vertices $x$ and $y$ are adjacent if the subgroup $\langle x, y \rangle$ is not cyclic. An independent subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code of $\Gamma$ if every vertex of $V(\Gamma)\setminus C$ is adjacent to exactly one vertex in $C$. A subset $T$ of the vertex set a graph $\Gamma$ is said to be a \emph{total perfect code} if every vertex of $\Gamma$ is adjacent to exactly one vertex in $T$. In this paper, we prove that the graph $\Gamma(G)$ is Hamiltonian for any finite non-cyclic nilpotent group $G$. Also, we characterize all finite groups such that their non-cyclic graphs admit a perfect code. Finally, we prove that for a non-cyclic nilpotent group $G$, the non-cyclic graph $\Gamma(G)$ does not admit total perfect code.