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

Abstract

Let $$G=(V,E)$$G=(V,E) be a graph with maximum degree $$\varDelta (G)$$Δ(G) and $$\phi :V\cup E\rightarrow \{1,2,\ldots ,k\}$$?:V?E?{1,2,?,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 $$S(u)\ne S(v)$$S(u)?S(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. Ding et al. proved in both problems $$k\ge \varDelta (G)+2d$$k?Δ(G)+2d is sufficient for d-degenerate graph G. In this paper, we improve this bound and prove that $$k\ge \varDelta (G)+d+1$$k?Δ(G)+d+1 is sufficient for d-degenerate graph G with $$d\le 8$$d≤8 and $$\varDelta (G)\ge 2d$$Δ(G)?2d or $$d\ge 9$$d?9 and $$\varDelta (G)\ge \frac{5}{2}d-5$$Δ(G)?52d-5. In fact, we prove these results in their list versions. As a consequence, we obtain an upper bound of the form $$\varDelta (G)+C$$Δ(G)+C for some families of graphs, e.g. $$\varDelta (G)+6$$Δ(G)+6 for planar graphs with $$\varDelta (G)\ge 10$$Δ(G)?10. In particular, we therefore obtain that when $$\varDelta (G)\ge 4$$Δ(G)?4 two conjectures we mentioned above hold for 2-degenerate graphs in their list versions.