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Neighbor sum distinguishing total colorings of planar graphs with maximum degree $$\varDelta$$. (English) Zbl 1316.05041
Summary: A (proper) total [k]-coloring of a graph $$G$$ is a mapping $$\phi : V(G) \cup E(G) \to [k] = \{1, 2, \ldots, k \}$$ such that any two adjacent elements in $$V(G) \cup E(G)$$ receive different colors. Let $$f(v)$$ denote the sum of the color of a vertex $$v$$ and the colors of all incident edges of $$v$$. A total $$[k]$$-neighbor sum distinguishing-coloring of $$G$$ is a total $$[k]$$-coloring of $$G$$ such that for each edge $$u v \in E(G)$$, $$f(u) \neq f(v)$$. By $$\chi_{\mathrm{nsd}}^{\prime\prime}(G)$$, we denote the smallest value $$k$$ in such a coloring of $$G$$. In this paper, we show that if $$G$$ is a planar graph with $$\varDelta(G) \geq 14$$, then $$\chi_{\mathrm{nsd}}^{\prime\prime}(G) \leq \varDelta(G) + 2$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory 05C07 Vertex degrees 05C35 Extremal problems in graph theory
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