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Lucky labelings of graphs. (English) Zbl 1197.05125

Summary: 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)100280245065 for every planar graph G. Nevertheless we offer a provocative conjecture that η(G)χ(G) for every graph G.

MSC:
05C78Graph labelling
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