{"ID":22955973,"CreatedAt":"2026-09-17T02:12:05.498442134Z","UpdatedAt":"2026-09-17T02:12:05.498442134Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.19114","arxiv_id":"2609.19114","title":"Hamiltonicity in graphs defined by primes and primitive elements","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 $q\u003e18\\,888\\,871$, and for the graph on a full prime field $\\F_p$, the bound improves to $p\u003e61$.","short_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 b...","url_abs":"https://arxiv.org/abs/2609.19114","url_pdf":"https://arxiv.org/pdf/2609.19114v1","authors":"[\"Yue-Feng She\"]","published":"2026-09-16T17:41:23Z","proceeding":"math.CO","tasks":"[\"math.CO\"]","methods":"[]","has_code":false}