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

Abstract

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