Representation numbers of complete multipartite graphs

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