The integral sum number of complete bipartite graphs Kr, s

Abstract

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 ( s 1, s 2,…, s k ) of the positive integer s(⩾ r⩾1) is said to be a proper r-partition if it satisfies the following three conditions: (1) s= s 1+ s 2+⋯+ s k ; (2) s 1⩾1, s i⩾s i−1+r−1 (i=2,3,…,k) ; (3) s k is minimum satisfying conditions (1) and (2). Using the definition, the integral sum number and the sum number of the complete bipartite graphs K r, 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 K r, s ( s⩾ r⩾2) are presented as follows: σ(K r,s)=ζ(K r,s)=s k+r−1, where s k 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 s k for any positive integers s⩾ r and we point out that the result σ( K r, s )=⌈(3 r+ s−3)/2⌉, obtained by Hartsfield and Smyth (Graphs and Matrices, Marcel Dekker, New York, 1992, pp. 205–211), is not true.