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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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