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Neighbor sum distinguishing total colorings of IC-planar graphs with maximum degree 13. (English) Zbl 1434.05057
Summary: A graph is IC-planar if it admits a drawing on the plane with at most one crossing per edge, such that two pairs of crossing edges share no common end vertex. For a given graph $$G$$, a proper total coloring $$\phi:V(G)\cup E(G)\rightarrow\{1,2,\dots,k\}$$ is called neighbor sum distinguishing if $$f_{\phi }(u)\ne f_{\phi }(v)$$ for each $$uv\in E(G)$$, where $$f_{\phi }(u)$$ is the sum of the color of $$u$$ and the colors of the edges incident with $$u$$. The smallest integer $$k$$ in such a coloring of $$G$$ is the neighbor sum distinguishing total chromatic number, denoted by $$\chi''_{\Sigma}(G)$$. M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] conjectured $$\chi''_{\Sigma}(G)\leq\Delta (G)+3$$ for any simple graph with maximum degree $$\Delta (G)$$. This conjecture was confirmed for IC-planar graph with maximum degree at least 14. In this paper, by using the discharging method, we prove that this conjecture holds for any IC-planar graph $$G$$ with $$\Delta (G)=13$$.
##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C10 Planar graphs; geometric and topological aspects of graph theory
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