Hamiltonicity in graphs defined by primes and primitive elements

A prime circle of order $2n$ is a circular ordering of $1,\dots ,2n$ 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 ${\text{F}}_q$ is Hamiltonian for every prime power $q>18888871$, and for the graph on a full prime field ${\text{F}}_p$, the bound improves to $p>61$.