Representations of graphs modulo n

A graph is said to be representable modulo n if its vertices can be labelled with distinct integers between 0 and n−1 inclusive such that two vertices are adjacent if and only if the difference of their labels is relatively prime to n. The representation number of graph G is the smallest n such that G has a representation modulo n. We relate the representation number to the product dimension of a graph and use this connection to obtain new results for each case. We also obtain directly new results on representation numbers for several classes of graphs. In particular, we relate the representation number of the disjoint union of complete graphs to the existence of complete families of mutually orthogonal Latin squares.