Abstract
An additive coloring of a graph G is an assignment of positive integers \({\{1,2,\ldots ,k\}}\) 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 \({\eta (G)}\) . We prove that \({\eta (G) \, \leqslant \, 468}\) for every planar graph G. This improves a previous bound \({\eta (G) \, \leqslant \, 5544}\) due to Norin. The proof uses Combinatorial Nullstellensatz and the coloring number of planar hypergraphs. We also demonstrate that \({\eta (G) \, \leqslant \, 36}\) for 3-colorable planar graphs, and \({\eta (G) \, \leqslant \, 4}\) for every planar graph of girth at least 13. In a group theoretic version of the problem we show that for each \({r \, \geqslant \, 2}\) there is an r-chromatic graph G r with no additive coloring by elements of any abelian group of order r.
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Sebastian Czerwiński was partially supported by Polish Ministry of Science and Higher Education grant (MNiSW) no. N N201 271335.
Jarosław Grytczuk was partially supported by the Ministry of Science and Higher Education of Poland as grant no. 884/N-ESFEuroGIGA/10/2011/0 within the ESF EuroGIGA project Graph Drawings and Representations and by Polish Ministry of Science and Higher Education grant (MNiSW) no. N N206 257035.
Bartłomiej Bosek and Grzegorz Matecki were partially supported by Polish National Science Center grant (NCN) no. 2011/03/B/ST6/01367.
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Bartnicki, T., Bosek, B., Czerwiński, S. et al. Additive Coloring of Planar Graphs. Graphs and Combinatorics 30, 1087–1098 (2014). https://doi.org/10.1007/s00373-013-1331-y
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DOI: https://doi.org/10.1007/s00373-013-1331-y