Representation of finite graphs as difference graphs of S-units, I
Combinatorics
2014-08-26 v1
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.
Keywords
Cite
@article{arxiv.1408.5873,
title = {Representation of finite graphs as difference graphs of S-units, I},
author = {K. Győry and L. Hajdu and R. Tijdeman},
journal= {arXiv preprint arXiv:1408.5873},
year = {2014}
}