On prime coprime graphs of certain finite groups

TL;DR

This paper studies the prime coprime graph of certain finite groups, analyzing properties like Hamiltonicity, clique number, and vertex degree for cyclic, dihedral, and dicyclic groups, and establishes specific partitions.

Contribution

It introduces new structural results on prime coprime graphs of finite groups, including Hamiltonicity and partition properties for specific group classes.

Findings

Hamiltonicity and clique number characterized for these graphs.

Established $(k,1)$-partitions for cyclic, dihedral, and dicyclic groups.

Results apply to groups of specified orders.

Abstract

The prime coprime graph $\Theta(G)$ of a finite group $G$ is the graph whose vertex set is $G$ and any two distinct vertices are adjacent if the greatest common divisor of their orders is either $1$ or a prime. In this paper, we investigate Hamiltonicity, clique number, and vertex degree of $\Theta(G)$ for cyclic, dihedral, and dicyclic groups $G$. We establish that $\Theta(G)$ admits a $(k,1)$-partition for cyclic, dihedral, and dicyclic groups $G$ of specified orders.