Lucky labelings of graphs

Abstract

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 η ( G ) . Using algebraic methods we prove that η ( 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 η ( T ) ⩽ 2 for every tree T, and η ( 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 η ( G ) ⩽ 100 280 245 065 for every planar graph G. Nevertheless we offer a provocative conjecture that η ( G ) ⩽ χ ( G ) for every graph G.