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

Let S be a finite non-empty set of primes, ZS the ring of those rationals whose denominators are not divisible by primes outside S, and ZS the multiplicative group of invertible elements (S-units) in ZS . For a non-empty subset A of ZS , denote by GS(A) the graph with vertex set A and with an edge between a and b if and only if a − b ∈ ZS . This type of graphs has been studied by many people. In the present paper we deal with the representability of finite (simple) graphs G as GS(A). If A′ = uA + a for some u ∈ ZS and a ∈ ZS , then A and A′ are called S-equivalent, since GS(A) and GS(A) are isomorphic. We say that a finite graph G is representable / infinitely representable with S if G is isomorphic to GS(A) for some A / for infinitely many non-S-equivalent A. We prove among other things that for any finite graph G there exist infinitely many finite sets S of primes such that G can be represented with S. We deal with the infinite representability of finite graphs, in particular cycles and complete bipartite graphs. Further, we consider the triangles in G for a deeper analysis. Finally, we prove that G is representable with every 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. In Part II, we shall investigate these and similar problems over more general domains. 2010 Mathematics Subject Classification. 05C25, 05C62, 11D61.