The sum number and integral sum number of complete bipartite graphs

摘要

Let N denote the set of positive integers. The sum graph G~+(S) of a finite subset S of N is the graph (S, E) with vertex set S and edge set E such that for u, v ∈ S, uv ∈ E if and only if u + v ∈ S. A graph G is called a sum graph if it is isomorphic to the sum graph G~+(S) of some finite subset S of N. The sum number σ(G) of a graph G is defined as the smallest nonnegative integer m for which G ∪ mK_1 is a sum graph. Let Z be the set of all integers. By extending the set N to Z in the above definitions of sum graphs and sum numbers, Harary [3] introduced the corresponding notions of integral sum graphs and integral sum number of a graph. In this paper, we evaluate the value of the sum number and integral sum number of the complete bipartite graph K_(r,s). While the former one corrects the result given in [4], the latter settles completely a problem proposed in [3].