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

TL;DR

This paper investigates the structure of difference graphs formed by S-units over rationals, resolving conjectures about cycle sizes and subgraph existence posed by Ruzsa.

Contribution

It proves two conjectures on the sizes of induced cycles and addresses a problem on subgraph existence in the difference graph of S-units.

Findings

Resolved conjectures on cycle sizes in the difference graph.

Established conditions for the existence of non-induced subgraphs.

Enhanced understanding of the combinatorial structure of S-unit difference graphs.

Abstract

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.