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Neighbor sum distinguishing total colorings of triangle free planar graphs. (English) Zbl 1317.05065
Summary: A total $$k$$-coloring $$c$$ of a graph $$G$$ is a proper total coloring $$c$$ of $$G$$ using colors of the set $$[k]=\{1,2,\dots ,k\}$$. Let $$f(u)$$ denote the sum of the color on a vertex $$u$$ and colors on all the edges incident to $$u$$. A $$k$$-neighbor sum distinguishing total coloring of $$G$$ is a total $$k$$-coloring of $$G$$ such that for each edge $$uv\in E(G)$$, $$f(u)\neq f(v)$$. By $$\chi''_{\mathrm{nsd}}(G)$$, we denote the smallest value $$k$$ in such a coloring of $$G$$. M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] conjectured that $$\chi''_{\mathrm{nsd}}(G)\leq \Delta (G)+3$$ for any simple graph with maximum degree $$\Delta(G)$$. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that the conjecture holds for any triangle free planar graph with maximum degree at least 7.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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