Representation numbers of complete multipartite graphs

A graph GG has a representation modulo rr if there exists an injective map f:V(G)→{0,1,…,r−1}f:V(G)→{0,1,…,r−1} such that vertices uu and vv are adjacent if and only if f(u)−f(v)f(u)−f(v) is relatively prime to rr. The representation number rep(G) is the smallest rr such that GG has a representation modulo rr. Following earlier work on stars, we study representation numbers of complete bipartite graphs and more generally complete multipartite graphs.