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 nite 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 = (G) is deened to be the least value of r for which G r is a sum graph. Gould and RR odl have shown that there exist innnite classes G of graphs such that, over G 2 G, (G) 2 (n 2), but no such classes have been constructed. In fact, for all classes G for which constructions have so far been found, (G) 2 o(m). In this paper we describe constructions which show that for wheels W n of (suuciently large) order n + 1 and size m = 2n, (W n) = n=2 + 3 if n is even and n (W n) n + 2 if n is odd. Hence for wheels (W n) 2 (m).