Abstract
Let \(G=(V,E)\) be a graph and \(\phi : V\cup E\rightarrow \{1,2,\ldots ,k\}\) be a proper total coloring of G. Let f(v) denote the sum of the color on a vertex v and the colors on all the edges incident with v. The coloring \(\phi \) is neighbor sum distinguishing if \(f(u)\ne f(v)\) for each edge \(uv\in E(G)\). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number of G, denoted by \(\chi _{\Sigma }''(G)\). Pilśniak and Woźniak conjectured that \(\chi _{\Sigma }''(G)\le \Delta (G)+3\) for any simple graph. By using the famous Combinatorial Nullstellensatz, we prove that \(\chi _{\Sigma }''(G)\le \max \{\Delta (G)+2, 10\}\) for planar graph G without 4-cycles. The bound \(\Delta (G)+2\) is sharp if \(\Delta (G)\ge 8\).
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Acknowledgements
The authors thank the referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11671232), the Natural Science Foundation of Hebei Province (A2015202301) and the University Science and Technology Project of Hebei Province (ZD2015106).
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Song, H., Xu, C. Neighbor sum distinguishing total coloring of planar graphs without 4-cycles. J Comb Optim 34, 1147–1158 (2017). https://doi.org/10.1007/s10878-017-0137-x
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DOI: https://doi.org/10.1007/s10878-017-0137-x