Neighbor Sum (Set) Distinguishing Total Choosability of d-Degenerate Graphs

Abstract

Let $G=(V,E)$ be a graph with maximum degree $\varDelta (G)$ and $\phi :V\cup E\rightarrow \{1,2,\ldots ,k\}$ be a proper total coloring of the graph G. Let S(v) denote the set of the color on vertex v and the colors on the edges incident with v. Let f(v) denote the sum of the color on vertex v and the colors on the edges incident with v. The proper total coloring $\phi$ is called neighbor set distinguishing or adjacent vertex distinguishing if $S(u)\ne S(v)$ 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 $k\ge \varDelta (G)+3$. Ding et al. proved in both problems $k\ge \varDelta (G)+2d$ is sufficient for d-degenerate graph G. In this paper, we improve this bound and prove that $k\ge \varDelta (G)+d+1$ is sufficient for d-degenerate graph G with $d\le 8$ and $\varDelta (G)\ge 2d$ or $d\ge 9$ and $\varDelta (G)\ge \frac{5}{2}d-5$. In fact, we prove these results in their list versions. As a consequence, we obtain an upper bound of the form $\varDelta (G)+C$ for some families of graphs, e.g. $\varDelta (G)+6$ for planar graphs with $\varDelta (G)\ge 10$. In particular, we therefore obtain that when $\varDelta (G)\ge 4$ two conjectures we mentioned above hold for 2-degenerate graphs in their list versions.