Summary: Suppose the vertices of a graph were labeled arbitrarily by positive integers, and let denote the sum of labels over all neighbors of vertex . A labeling is lucky if the function is a proper coloring of , that is, if we have whenever and are adjacent. The least integer for which a graph has a lucky labeling from the set is the lucky number of , denoted by .
Using algebraic methods we prove that for every bipartite graph whose edges can be oriented so that the maximum out-degree of a vertex is at most . In particular, we get that for every tree , and for every bipartite planar graph . By another technique we get a bound for the lucky number in terms of the acyclic chromatic number. This gives in particular that for every planar graph . Nevertheless we offer a provocative conjecture that for every graph .