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Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10. (English) Zbl 1426.05051
Summary: Given a simple graph $$G$$, a proper total-$$k$$-coloring $$\phi : V(G) \cup E(G) \rightarrow \{1, 2, \ldots, k \}$$ is called neighbor sum distinguishing if $$S_{\phi}(u) \neq S_{\phi}(v)$$ for any two adjacent vertices $$u,v \in V(G)$$, where $$S_{\phi}(u)$$ is the sum of the color of $$u$$ and the colors of the edges incident with $$u$$. It has been conjectured by M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] that $$\mathop{\Delta}(G) + 3$$ colors enable the existence of a neighbor sum distinguishing total coloring. The conjecture is confirmed for any graph with maximum degree at most 3 and for planar graph with maximum degree at least 11. We prove that the conjecture holds for any planar graph $$G$$ with $$\mathop{\Delta}(G) = 10$$. Moreover, for any planar graph $$G$$ with $$\mathop{\Delta}(G) \geq 11, \mathop{\Delta}(G) + 2$$ colors guarantee such a total coloring, and the upper bound $$\mathop{\Delta}(G) + 2$$ is tight.

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