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A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles. (English) Zbl 1343.05066
Summary: Let $$G=(V,E)$$ be a graph and $$\phi$$ be a total $$k$$-coloring of $$G$$ using the color set $$\{1,\dots,k\}$$. Let $$\sum_\phi(u)$$ denote the sum of the color of the vertex $$u$$ and the colors of all incident edges of $$u$$. A $$k$$-neighbor sum distinguishing total coloring of $$G$$ is a total $$k$$-coloring of $$G$$ such that for each edge $$uv\in E(G)$$, $$\sum_\phi(u)\neq\sum_\phi(v)$$. By $$\chi^{\prime\prime}_{\mathrm{nsd}}(G)$$, 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)] first introduced this coloring and conjectured that $$\chi^{\prime\prime}_{\mathrm{nsd}}(G)\leq\Delta (G)+3$$ for any simple graph $$G$$. In this paper, we prove that the conjecture holds for planar graphs without intersecting triangles with $$\Delta (G)\geq 7$$. Moreover, we also show that $$\chi^{\prime\prime}_{\mathrm{nsd}}(G)\leq\Delta (G)+2$$ for planar graphs without intersecting triangles with $$\Delta (G)\geq 9$$. Our approach is based on the Combinatorial Nullstellensatz and the discharging method.

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