A family of sparse graphs of large sum number

Abstract

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 | V | = n, | V r | = n+ r, and | E| = 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) ϵE if and only if there exists a vertex wϵ V r whose label L( w) = L( u) + L( v). For a given graph G, the sum number σ = σ( G) is defined to be the least value of r for which G r is a sum graph. Gould and Rödl have shown that there exist infinite classes G of graphs such that, over Gϵ G , σ( G) ϵΘ( n 2), but no such classes have been constructed. In fact, for all classes G for which constructions have so far been found, σ( G) ϵo( m). In this paper we describe constructions which show that for wheels W n of (sufficiently large) order n + 1 and size m = 2 n, σ( W n ) = n/2 + 3 if n is even and n ⩽ σ ( W n ) ⩽ n + 2 if n is odd. Hence for wheels σ ( W n ) ϵ Θ( m).