On conjectures and problems of Ruzsa concerning difference graphs of S-units
Abstract
Given a finite nonempty set of primes S, we build a graph G with vertex set 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 G, and also a problem of Ruzsa concerning the existence of subgraphs of G which are not induced subgraphs.
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