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arXiv · 2609.19114

Hamiltonicity in graphs defined by primes and primitive elements

Yue-Feng She
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Abstract

A prime circle of order $2n$ is a circular ordering of $1,\ldots,2n$ such that the sum of every two adjacent terms is prime. We prove that a prime circle exists for every sufficiently large $n$. The proof is based on a perfect matching and robust expansion. We also study Hamilton cycles in graphs and digraphs defined by primitive sums and differences over finite fields. In particular, the primitive-sum graph on $\F_q$ is Hamiltonian for every prime power $q>18\,888\,871$, and for the graph on a full prime field $\F_p$, the bound improves to $p>61$.

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Yue-Feng She. 2026-09-16. Hamiltonicity in graphs defined by primes and primitive elements. https://arxiv.org/abs/2609.19114

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