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Neighbor sum (set) distinguishing total choosability via the combinatorial Nullstellensatz. (English) Zbl 1371.05078
Summary: Let $$G=(V,E)$$ be a graph and $$\phi :V\cup E\rightarrow \{1,2,\dots,k\}$$ be a total coloring of $$G$$. Let $$C(v)$$ denote the set of the color of vertex $$v$$ and the colors of the edges incident with $$v$$. Let $$f(v)$$ denote the sum of the color of vertex $$v$$ and the colors of the edges incident with $$v$$. The total coloring $$\phi$$ is called neighbor set distinguishing or adjacent vertex distinguishing if $$C(u)\neq C(v)$$ for each edge $$uv\in E(G)$$. We say that $$\phi$$ is neighbor sum distinguishing if $$f(u)\neq f(v)$$ for each edge $$uv\in E(G)$$. In both problems the challenging conjectures presume that such colorings exist for any graph $$G$$ if $$k\geq \varDelta (G)+3$$. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that in both problems $$k\geq \varDelta (G)+2\mathrm{col}(G)-2$$ is sufficient, moreover we prove that if $$G$$ is not a forest and $$\varDelta \geq 4$$, then $$k\geq \varDelta (G)+2\mathrm{col}(G)-3$$ is sufficient, where $$\mathrm{col}(G)$$ is the coloring number of $$G$$. In fact we prove these results in their list versions, which improve the previous results. As a consequence, we obtain an upper bound of the form $$\varDelta (G)+C$$ for some families of graphs, e.g. $$\varDelta +9$$ for planar graphs. In particular, we therefore obtain that when $$\varDelta \geq 4$$ two conjectures we mentioned above hold for 2-degenerate graphs (with coloring number at most 3) in their list versions.

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