Neighbor sum distinguishing total coloring of planar graphs without 4-cycles

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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 \(\Delta (G)\ge 8\).