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

Abstract

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.