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Neighbor sum distinguishing total coloring of planar graphs without 4-cycles. (English) Zbl 1378.05073
Summary: Let $$G=(V,E)$$ be a graph and $$\phi:V\cup E\rightarrow \{1,2,\dots,k\}$$ 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 $$f(u)\neq f(v)$$ 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 $$\chi^{\prime\prime}_\Sigma(G)$$. M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] conjectured that $$\chi^{\prime\prime}_\Sigma(G)\leq \Delta (G)+3$$ for any simple graph. By using the famous Combinatorial Nullstellensatz, we prove that $$\chi^{\prime\prime}_\Sigma(G)\leq\max\{\Delta (G)+2,10\}$$ for planar graph $$G$$ without 4-cycles. The bound $$\Delta (G)+2$$ is sharp if $$\Delta (G)\geq 8$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory
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