Representations of Graphs Modulo n Anthony B. Evans ∗ Garth Isaak † Darren A. Narayan ‡ Abstract 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 their labels are relatively prime to n. The representation number of graph G is the smallest n representing G. We review known results and investigate representation numbers for several new classes. 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. 1 Introduction For a finite graph G, with vertices {v 1 , ..., v r }, a representation of G modulo n is a set {a 1 , .., a r } of distinct, nonnegative integers, 0 ≤ a i <n satisfying gcd(a i − a j ,n)=1 if and only if v i is adjacent to v j . The representation number , Rep(G), is the smallest n such that G has a representation modulo n. It was shown by Erd˝os and Evans [1] that any finite graph can be represented modulo some positive integer, and so the representation number of a finite graph is well defined. We will survey known results and techniques for determining representation num- bers. The representation number of a graph is related to its product dimension as defined by Neˇ setˇ ril and Pultr [8]. Reviewing work on both of these problems we can observe new results in each case. We will also obtain some new results on repre- sentation numbers for certain graphs classes. These include complete multipartite graphs and graphs whose complements are paths, cycles and stars along with isolated * Department of Mathematics and Statistics, Wright State University, Dayton, OH 45434 tevans@math.wright.edu † Department of Mathematics, Lehigh University, Bethlehem, PA 18015 gisaak@lehigh.edu Partially supported by a grant from the Reidler Foundation ‡ Department of Mathematics, Lehigh University, Bethlehem, PA 18015 dan6@lehigh.edu Partially supported by NSF REU program at the University of Dayton 1