Distance Graphs on the Integers

We consider several extremal problems concerning representations of graphs as distance graphs on the integers. Given a graph G = (V, E), we wish to find an injective function φ : V → Z = {1, 2, . . . } and a set D ⊂ Z such that {u, v} ∈ E if and only if |φ(u)− φ(v)| ∈ D. Let s(n) be the smallest N such that any graph G on n vertices admits a representation (φG,DG) such that φG(v) ≤ N for all v ∈ V (G). We show that s(n) = (1 + o(1))n as n → ∞. In fact, if we let sr(n) be the smallest N such that any r-regular graph G on n vertices admits a representation (φG,DG) such that φG(v) ≤ N for all v ∈ V (G), then sr(n) = (1 + o(1))n 2 as n → ∞ for any r = r(n) logn with rn even for all n. Given a graph G = (V, E), let De(G) be the smallest possible cardinality of a set D for which there is some φ : V → Z so that (φ,D) represents G. We show that, for almost all n-vertex graphs G, we have De(G) ≥ 1 2 ( n 2 ) − (1 + o(1))n(logn), whereas for some n-vertex graph G, we have De(G) ≥ ( n 2 ) − n(logn). Further extremal problems of similar nature are considered.