Sum Graphs of Small Sum Number
نویسنده
چکیده
Given an integer r > 0, let G r = (V; E) denote a graph consisting of a simple nite undirected connected nontrivial graph G together with r isolated vertices K r. Let L : V ! 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 , (u; v) 2 E if and only if there exists a vertex w 2 V whose label L(w) = L(u) + L(v). For a given subgraph G, the sum number = (G) is deened to be the least number r for which G r is a sum graph; in particular, if G 1 = G K 1 is a sum graph, then the subgraph G is called a unit graph. In this paper it is shown that there exist graphs of every order n and size m whose sum number is O(n). Further, it is shown that for every integer m satisfying bn 2 =4c < m ? n 2 there exists no unit graph, while for each m such that n ? 1 m bn 2 =4c there exists at least one unit graph. Methods of proof are constructive. 1 INTRODUCTION Sum graphs (deened in the Abstract) were introduced by Harary Har88,Har89]. Hao Hao89] showed that a graph of order n is a sum graph if and only if its size m (
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