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Neighbor sum distinguishing total coloring of a kind of sparse graph. (English) Zbl 1413.05111
Summary: For a given graph $$G=(V,E)$$, by $$f(v)$$, we denote the sum of the colour on the vertex $$v$$ and the colours on the edges incident with $$v$$. A proper $$k$$-total coloring $$\phi$$ of a graph $$G$$ is called a neighbor sum distinguishing $$k$$-total coloring if $$f(u)\neq f(v)$$ for each edge $$uv\in E(G)$$. The smallest number $$k$$ in such a coloring of $$G$$ is the neighbor sum distinguishing total chromatic number, denoted by $$\chi_{\sum}^{\prime\prime}(G)$$. The maximum average degree of $$G$$ is the maximum of the average degree of its non-empty subgraphs, which is denoted by $$\text{mad}(G)$$. In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that if $$G$$ is a graph with $$\Delta(G)\geq 6$$ and $$\text{mad}(G)<\frac{18}{5}$$, then $$\chi_{\sum}^{\prime\prime}(G)\leq\Delta(G)+2$$. This bound is sharp.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C42 Density (toughness, etc.)