Characterizing finite groups whose order supergraphs satisfy a connectivity condition
Ramesh Prasad Panda, Papi Ray
TL;DR
Problem: characterize finite groups whose order supergraphs $\\mathcal{S}(G)$ are cyclically separable. Approach: analyze element orders, clique structures $[x]$ induced by elements with the same order, and divisibility relations within families such as dihedral $D_{2n}$, dicyclic $Q_{4n}$, EPPO groups, nilpotent groups, and symmetric/alternating groups to determine cyclic cutsets. Contributions: exact necessary and sufficient criteria for $\\mathcal{S}(G)$ to be cyclically separable across the listed families, including explicit dihedral conditions on $n$, a non-power-of-two condition for $Q_{4n}$, EPPO-Sylow criteria, nilpotent-factorizations, and $n\ge 4$ for $S_n$ and $A_n$. Significance: advances understanding of how group order structure governs cyclic connectivity in order-based graphs, with potential implications for graph invariants and group-graph classifications.
Abstract
Let $Γ$ be an undirected and simple graph. A set $ S $ of vertices in $Γ$ is called a {cyclic vertex cutset} of $Γ$ if $Γ- S$ is disconnected and has at least two components each containing a cycle. If $Γ$ has a cyclic vertex cutset, then it is said to be {cyclically separable}. For any finite group $G$, the order supergraph $\mathcal{S}(G)$ is the simple and undirected graph whose vertices are elements of $G$, and two vertices are adjacent if as elements of $G$ the order of one divides the order of the other. In this paper, we characterize the finite nilpotent groups and various non-nilpotent groups, such as the dihedral groups, the dicyclic groups, the EPPO groups, the symmetric groups, and the alternating groups, whose order supergraphs are cyclically separable.