The Sum Number of the Cocktail

A graph G is called a sum graph if there exists a labelling of the vertices of G by distinct positive integers such that the vertices labelled u and v are adjacent if and only if there exists a vertex labelled u + v. If G is not a sum graph, adding a nite number of isolated vertices to it will always yield a sum graph, and the sum number (G) of G is the smallest number of isolated vertices that will achieve this result. A labelling that realizes G + K (G) as a sum graph is said to be optimal. In this paper we consider G = H m;n , the complete n-partite graph on n 2 sets of m 2 nonadjacent vertices. We give an optimal labelling to show that (H 2;n) = 4n ? 5, and in the general case we give constructive proofs that (H m;n) 2 (mn) and (H m;n) 2 O(mn 2). We conjecture that (H m;n) is asymptotically greater than mn, the cardinality of the vertex set; if so, then H m;n is the rst known graph with this property. We also provide for the rst time an optimal labelling of the complete bipartite graph K m;n whose smallest label is 1.