New formula for the sum number for the complete bipartite graphs

Given a graph G=(V,E), a labelling is a function f : V→Z+ which has different values on different vertices of G. Graph G is a sum graph if there exists a labelling f : V→Z+ such that for every pair of distinct vertices u,v∈V, there is an edge uv∈E if and only if there exists a vertex w∈V with f(w)=f(u)+f(v). It is clear that every sum graph has at least one isolated vertex. The sum number σ(G) of the graph G is the least number of isolated vertices one must add to G to turn it into a sum graph. It was stated by Hartsfield and Smyth (in: R. Rees (Ed.), Graphs, Matrices and Designs, Marcel Dekker, New York, 1993, pp. 205) that for the complete bipartite graphs Km,n where m⩾n⩾2 the sum number is σ(Km,n)=⌈(3n+m−3)/2⌉. Unfortunately, this formula is wrong when m⩾3n. The new construction given in this paper shows that σ(Km,n) in this case is much smaller. The new formula for σ(Km,n) is proved.