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) 6= S(v) whenever u and v are adjacent. The least integer k for which a graph G has a lucky labeling from the set f1; 2; : : : ; kg is the lucky number of G, denoted by (G). Using algebraic methods we prove that (G) k + 1 for every bipartite graph G whose edges can be oriented so that the maximum out-degree of a vertex is at most k. In particular, we get that (T ) 2 for every tree T , and (G) 3 for every bipartite planar graph G. By another technique we get a bound for the lucky number in terms of the acyclic chromatic number. This gives in particular that (G) 100 280 245 065 for every planar graph G. Nevertheless we o¤er a provocative conjecture that (G) (G) for every graph G.