A few remarks on the theory of non-nilpotent graphs

TL;DR

This paper investigates the properties of non-nilpotent graphs associated with symmetric and alternating groups, proving they contain Hamiltonian cycles, satisfy specific conjectures, and lack local properties.

Contribution

It establishes the existence of Hamiltonian cycles in non-nilpotent graphs of symmetric groups and confirms a conjecture for alternating groups, also showing these graphs lack local properties.

Findings

Non-nilpotent graphs of $S_n$ have Hamiltonian cycles.

They satisfy a conjecture of Nongsiang and Saikia.

The class of non-nilpotent graphs has no local properties.

Abstract

We prove a few results about non-nilpotent graphs of symmetric groups $S_n$ -- namely that they have a Hamiltonian cycle and they satisfy a conjecture of Nongsiang and Saikia. The latter is likewise proven for alternating groups $A_n$. We also show that the class of non-nilpotent graphs does not have any ''local'' properties, ie. for every simple graph $X$ there is a group $G$, such that its non-nilpotent graph has $X$ as an induced subgraph.