Neighbor Sum (Set) Distinguishing Total Choosability Via the Combinatorial Nullstellensatz

Let $$G=(V,E)$$G=(V,E) be a graph and $$\phi :V\cup E\rightarrow \{1,2,\ldots ,k\}$$?:V?E?{1,2,?,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 $$C(u)\ne C(v)$$C(u)?C(v) for each edge $$uv\in E(G)$$uv?E(G). We say that $$\phi $$? is neighbor sum distinguishing if $$f(u)\ne f(v)$$f(u)?f(v) for each edge $$uv\in E(G)$$uv?E(G). In both problems the challenging conjectures presume that such colorings exist for any graph G if $$k\ge \varDelta (G)+3$$k?Δ(G)+3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that in both problems $$k\ge \varDelta (G)+2\mathrm{col}(G)-2$$k?Δ(G)+2col(G)-2 is sufficient, moreover we prove that if G is not a forest and $$\varDelta \ge 4$$Δ?4, then $$k\ge \varDelta (G)+2\mathrm{col}(G)-3$$k?Δ(G)+2col(G)-3 is sufficient, where $$\mathrm{col}(G)$$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 $$\varDelta (G)+C$$Δ(G)+C for some families of graphs, e.g. $$\varDelta +9$$Δ+9 for planar graphs. In particular, we therefore obtain that when $$\varDelta \ge 4$$Δ?4 two conjectures we mentioned above hold for 2-degenerate graphs (with coloring number at most 3) in their list versions.