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.