Representation of finite graphs as difference graphs of S-units, I

TL;DR

This paper explores the representation of finite graphs as difference graphs of S-units, establishing conditions under which graphs can be represented for various prime sets, and characterizing cubical graphs through number-theoretic methods.

Contribution

It proves that every finite graph can be represented as an S-graph for infinitely many prime sets and characterizes graphs representable for all S as cubical, using advanced Diophantine techniques.

Findings

Existence of infinitely many S for which a given graph is an S-graph.

Characterization of graphs that are S-graphs for all S as cubical.

Deep Diophantine results applied to S-unit equations in graph representation.

Abstract

Let G be a simple finite graph such that each vertex has an integer value and different vertices have different values. Let S be a finite non-empty set of primes. We call G an S-graph if any two vertices are connected by an edge if and only their values differ by a number which is composed of primes from S. We prove e.g. that for every G there exist infinitely many finite sets S such that G is an S-graph. We deal with cycles and complete bipartite graphs G. We consider the triangles in G for a deeper analysis. Finally we prove that G is an S-graph for all S if and only if G is cubical. Besides combinatorial and numbertheoretical arguments some deep Diophantine results concerning S-unit equations are used in our proofs.