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On the neighbor sum distinguishing total coloring of planar graphs. (English) Zbl 1331.05084
Summary: Let $$c$$ be a proper total coloring of a graph $$G = (V, E)$$ with integers $$1, 2, \ldots, k$$. For any vertex $$v \in V(G)$$, let $$\sum_c(v)$$ denote the sum of colors of the edges incident with $$v$$ and the color of $$v$$. If for each edge $$uv \in E(G)$$, $$\sum_c(u) \neq \sum_c(v)$$, then such a total coloring is said to be neighbor sum distinguishing. The least $$k$$ for which such a coloring of $$G$$ exists is called the neighbor sum distinguishing total chromatic number and denoted by $$\chi_{\Sigma}^{\prime\prime}(G)$$. M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] conjectured $$\chi_{\Sigma}^{\prime\prime}(G) \leq \Delta(G) + 3$$ for any simple graph with maximum degree $$\Delta(G)$$. It is known that this conjecture holds for any planar graph with $$\Delta(G) \geq 13$$. In this paper, we prove that for any planar graph, $$\chi_{\Sigma}^{\prime\prime}(G) \leq \max \{\Delta(G) + 3, 14 \}$$.

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