This article investigates Hamiltonicity and vertex degree in prime coprime graphs of cyclic, dihedral, and dicyclic groups, highlighting new structural insights.
The prime coprime graph $Θ(G)$ 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 $Θ(G)$ for cyclic, dihedral, and dicyclic groups G. We establish that $Θ(G)$ admits a $(k,1)$-partition for cyclic, dihedral, and dicyclic groups G of specified orders.
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Ranjan et al. (2025) studied this question.
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