The sum number of a disjoint union of graphs

In this paper we consider the disjoint union of graphs as sum graphs. We provide an upper bound on the sum number of a disjoint union of graphs and provide an application for the exclusive sum number of a graph. We conclude with some open problems

A Family Of Sparse Graphs Of Large Sum Number

Given an integer r 0, let G r = (V r ; E) denote a graph consisting of a simple finite undirected graph G = (V; E) of order n and size m together with r isolated vertices K r . Then jV j = n, jV r j = n + r, and jEj = m. Let L : V r ! Z + denote a labelling of the vertices of G r with distinct positive integers. Then G r is said to be a sum graph if there exists a labelling L such that for every distinct vertex pair u and v of V r , (u; v) 2 E if and only if there exists a vertex w 2 V r whose label L(w) = L(u) +L(v). For a given graph G, the sum number oe = oe(G) is defined to be the least value of r for which G r is a sum graph. Gould and Rodl have shown that there exist infinite classes G of graphs such that, over G 2 G, oe(G) 2 \Theta(n 2 ), but no such classes have been constructed. In fact, for all classes G for which constructions have so far been found, oe(G) 2 o(m). In this paper we describe constructions which show that for wheels Wn of (sufficiently large) order n +..