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

TL;DR

The paper investigates the non-cyclic graph $Γ(G)$ of finite non-cyclic groups, establishing Hamiltonicity for all finite non-cyclic nilpotent groups via a detailed analysis of $\mathrm{Cyc}(G)$ and the order structure of elements. It delivers a precise criterion for when $Γ(G)$ possesses a perfect code—namely, when $G$ has a maximal cyclic subgroup of order $2$—and shows that total perfect codes do not exist in this setting. The work combines group-theoretic decompositions (into products of Sylow subgroups and maximal cyclic components) with graph-theoretic constructions on the $Ω_m(G)$ partitions to build Hamiltonian cycles and to derive coding-theoretic classifications. These results deepen the link between algebraic structure and graph properties, with potential implications for combinatorial coding in group-based networks.

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} $Γ(G)$ 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 $Γ$ is called a perfect code of $Γ$ if every vertex of $V(Γ)\setminus C$ is adjacent to exactly one vertex in $C$. A subset \( T \) of the vertex set a graph \( Γ\) is said to be a \emph{total perfect code} if every vertex of \( Γ\) is adjacent to exactly one vertex in \( T \). In this paper, we prove that the graph $Γ(G)$ 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 $Γ(G)$ does not admit total perfect code.

Figures (1)

• Figure 1: Hamiltonian cycle in $\Gamma(\mathbb{Z}_4 \times \mathbb{Z}_2)$.

Theorems & Definitions (34)

• Lemma 1.1
• Proposition 1.2
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3: a.pgroupberkovib.pgroupisaac2006a.pgroupkulakoffa.pgroupmiller
• Corollary 2.4: a.mishra2021lambda
• Theorem 2.5
• Lemma 2.6
• Lemma 3.1
• proof