On conjectures and problems of Ruzsa, concerning difference graphs of S-units

Custic, A. and Hajdu, Lajos and Kreso, D. and Tijdeman, R. (2015) On conjectures and problems of Ruzsa, concerning difference graphs of S-units. ACTA MATHEMATICA HUNGARICA, 146 (2). pp. 391-404. ISSN 0236-5294

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Abstract

Given a finite nonempty set of primes S, we build a graph G with vertex set Q by connecting x, y ∈ Q 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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