The integral sum number of complete bipartite graphs Kr,s

A graph G=(V,E) is said to be an integral sum graph (sum graph) if its vertices can be given a labeling with distinct integers (positive integers), so that uv∈E if and only if u+v∈V. The integral sum number (sum number) of a given graph G, denoted by ζ(G) (σ(G)), was defined as the smallest number of isolated vertices which when added to G result in an integral sum graph (sum graph). In this paper, we shall introduce a new definition of the proper r-partition of the positive integer s on a positive integer r (s⩾r). A partition (s1,s2,…,sk) of the positive integer s(⩾r⩾1) is said to be a proper r-partition if it satisfies the following three conditions: (1) s=s1+s2+⋯+sk; (2) s1⩾1, si⩾si−1+r−1 (i=2,3,…,k); (3) sk is minimum satisfying conditions (1) and (2). Using the definition, the integral sum number and the sum number of the complete bipartite graphs Kr,s, which is an unsolved problem proposed by Harary are investigated and determined. The results on the integral sum number and sum number of graphs Kr,s (s⩾r⩾2) are presented as follows:σ(Kr,s)=ζ(Kr,s)=sk+r−1,where sk is the last term of the proper r-partition of the integer s. Besides, in this paper, we also obtain an analytical method which is able to find sk for any positive integers s⩾r and we point out that the result σ(Kr,s)=⌈(3r+s−3)/2⌉, obtained by Hartsfield and Smyth (Graphs and Matrices, Marcel Dekker, New York, 1992, pp. 205–211), is not true.