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Neighbor sum distinguishing total coloring of 2-degenerate graphs. (English) Zbl 1394.05037
A proper $$k$$-total coloring of a graph $$G$$ is a mapping from $$V(G)\cup E(G)$$ to $$\{1,2,\dots,k\}$$ such that no two adjacent or incident elements in $$V(G)\cup E(G)$$ receive the same color. Let $$f(v)$$ denote the sum of the colors on the edges incident with $$v$$ and the color on vertex $$v$$. A proper $$k$$-total coloring of $$G$$ is called neighbor sum distinguishing if $$f(u)\neq f(v)$$ for each edge $$uv\in E(G)$$. The smallest number $$k$$ in the neighbor sum distinguishing $$k$$-total coloring of $$G$$ is the neighbor sum distinguishing total chromatic number. M. Pilśniak and M. Woźniak [Graphs Comb. 31, No. 3, 771–782 (2015; Zbl 1312.05054)] conjectured that for any graph G the neighbor sum distinguishing total chromatic number is at most $$\Delta(G)+3$$. In this paper, the authors confirm this conjecture for 2-degenerate graphs. Moreover, they improve this bound for graphs with maximum degree at least 5. They prove that if $$G$$ is 2-degenerate with $$\Delta(G)\geq 5$$ then the neighbor sum distinguishing total chromatic number is at most $$\Delta(G)+2$$. The proof is based on the combinatorial Nullstellensatz. Recently, L. Ding et al. [ibid. 33, No. 4, 885–900 (2017; Zbl 1371.05078)] proved that if $$G$$ is not a forest and $$\Delta(G)\geq 4$$ then the neighbor sum distinguishing total chromatic number of $$G$$ is at most $$\Delta (G)+2\mathrm{col}(G)-1$$, where col$$(G)$$ is the coloring number of $$G$$, in particular, the neighbor sum distinguishing total chromatic number of 2-degenerate graph $$G$$ with $$\Delta(G)\geq 4$$ is at most $$\Delta(G)+3$$.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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