  Given a finite nonempty set of primes S, we build a graph $\mathcal{G}$ with
vertex set $\mathbb{Q}$ by connecting x and y if the prime divisors of both the
numerator and denominator of x-y are from S. In this paper we resolve two
conjectures posed by Ruzsa concerning the possible sizes of induced
nondegenerate cycles of $\mathcal{G}$, and also a problem of Ruzsa concerning
the existence of subgraphs of $\mathcal{G}$ which are not induced subgraphs.
