Representations of graphs and orthogonal latin square graphs

We define graph representations modulo integers and prove that any finite graph has a representation modulo some integer. We use this to obtain a new, simpler proof of Lindner, E. Mendelsohn, N. Mendelsohn, and Wolk’s result that any finite graph can be represented as an orthogonal latin square graph. Let G be a graph with vertices v,, . . . , u, and let n be a natural number. We say that G is representable modulo n if there exist distinct integers a,, . . . , a,, 0 I a, < n, satisfying (ai a+ n) = 1 if and only if ui is adjacent to u,. We call {a,, . . . , ad a representation of G modulo n and n the order of the representation. If {a,, . . . ,a,} is a representation of G modulo n then so is {ba, + c, * . . , ba, f c}, where (b, n) = 1 and addition and multiplication are performed modulo n. We will show that any graph is representable modulo some positive integer. The proof will require the following lemma. Lemma. For any positive integer m there exist distinct primes pl, . . . ,pm such that for all pairs A,B of disjoint nonempty subsets of {p,, . . . ,p,}, (n{Pi:PiEA}-TZ{pi:pjEB},plP,...p,)=l. Journal of Graph Theory, Vol. 13, No. 5, 593-595 (1989)