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arXiv · 2507.15993

On prime coprime graphs of certain finite groups

Ravi Ranjan·Shubh N. Singh
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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.

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Ravi Ranjan, Shubh N. Singh. 2025-07-21. On prime coprime graphs of certain finite groups. https://arxiv.org/abs/2507.15993

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