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Neighbor distinguishing total choice number of sparse graphs via the combinatorial nullstellensatz. (English) Zbl 1336.05046
Summary: Let $$G=(V,E)$$ be a graph and $$\phi:V\cup E\to\{1,2,\dots, k\}$$ be a total-$$k$$-coloring of $$G$$. Let $$f(v)(S(v))$$ denote the sum(set) of the color of vertex $$v$$ and the colors of the edges incident with $$v$$. The total coloring $$\phi$$ is called neighbor sum distinguishing if $$(f(u)\neq f(v))$$ for each edge $$uv\in E(G)$$. We say that $$\phi$$ is neighbor set distinguishing or adjacent vertex distinguishing if $$S(u)\neq S(v)$$ for each edge $$uv\in E(G)$$. For both problems, we have conjectures that such colorings exist for any graph $$G$$ if $$k\geq \Delta(G)+3$$. The maximum average degree of $$G$$ is the maximum of the average degree of its non-empty subgraphs, which is denoted by $$\mathrm{mad}(G)$$. In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that these two conjectures hold for sparse graphs in their list versions. More precisely, we prove that every graph $$G$$ with maximum degree $$\Delta (G)$$ and maximum average degree $$\mathrm{mad}(G)$$ has $$ch_\Sigma^{\prime\prime}(G)\leq \Delta(G)+3$$ (where $$ch_\Sigma^{\prime\prime}(G)$$ is the neighbor sum distinguishing total choice number of $$G$$) if there exists a pair $$(k,m)\in \{ (6,4),(5,\frac{18}{5}),(4,\frac{16}{5})\}$$ such that $$\Delta(G)\geq k$$ and $$\mathrm{mad}(G)<m$$.

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