Complete solutions to the Oberwolfach problem for an infinite set of orders

Let n ⩾ 3 and let F be a 2-regular graph of order n. The Oberwolfach problem OP ( F ) asks for a 2-factorisation of K n if n is odd, or of K n − I if n is even, in which each 2-factor is isomorphic to F. We show that there is an infinite set N of primes congruent to 1 ( mod 16 ) such that OP ( F ) has a solution for any 2-regular graph F of order n ∈ N . We also show that for each of the infinitely many n ≡ 10 ( mod 48 ) with n 2 prime, OP ( F ) has a solution for any 2-regular graph F of order n.