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

Let be a graph with maximum degree and 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 is called neighbor set distinguishing or adjacent vertex distinguishing if for each edge . We say that is neighbor sum distinguishing if for each edge . In both problems the challenging conjectures presume that such colorings exist for any graph G if . Ding et al. proved in both problems is sufficient for d-degenerate graph G. In this paper, we improve this bound and prove that is sufficient for d-degenerate graph G with and or and . In fact, we prove these results in their list versions. As a consequence, we obtain an upper bound of the form for some families of graphs, e.g. for planar graphs with . In particular, we therefore obtain that when two conjectures we mentioned above hold for 2-degenerate graphs in their list versions.