Lucky labelings of graphs

Suppose the vertices of a graph G were labeled arbitrarily by positive integers, and let S(v) denote the sum of labels over all neighbors of vertex v. A labeling is lucky if the function S is a proper coloring of G, that is, if we have S(u) S(v) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set {1,2,...,k} is the lucky number of G, denoted by @h(G). Using algebraic methods we prove that @h(G)=