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 (G) ( (G)), was de:ned 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 de:nition 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 satis:es 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 de:nition, 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: (Kr;s) = (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 :nd sk for any positive integers s¿r and we point out that the result (Kr;s) = (3r + s − 3)=2 , obtained by Harts:eld and Smyth (Graphs and Matrices, Marcel Dekker, New York, 1992, pp. 205–211), is not true. c © 2001 Elsevier Science B.V. All rights reserved.