Additive Coloring of Planar Graphs

Abstract

An additive coloring of a graph G is an assignment of positive integers {1, 2,..., k} to the vertices of G such that for every two adjacent vertices the sums of numbers assigned to their neighbors are different. The minimum number k for which there exists an additive coloring of G is denoted by η(G). We prove that η(G)≤5544 due to Norin. The proof uses Combinatorial Nullstellensatz and the coloring number of planar hypergraphs. We also demonstrate that η(G)≤36 for 3-colorable planar graphs, and η(G)≤4 for every planar graph of girth at least 13. In a group theoretic version of the problem we show that for each r≥ there is an r-chromatic graph G r with no additive coloring by elements of any abelian group of order r. © 2013 The Author(s).