An Upper Bound for the Representation Number of Graphs with Fixed Order

A graph has a representation modulo n if there exists an injective map f: {V (G)} → {0, 1,...,n − 1} such that vertices u and v are adjacent if and only if |f(u) − f(v)| is relatively prime to n. The representation number is the smallest n such that G has a representation modulo n. We seek the maximum value for the representation number over graphs of a fixed order. Erd˝os and Evans provided an upper bound in their proof that every finite graph can be represented modulo some positive integer. In this note we improve this bound and show that the new bound is best possible.