LAUSR

A proper k-total coloring of a graph G is a mapping from $$V(G)\cup E(G)$$V(G)∪E(G) to $$\{1,2,\ldots ,k\}$${1,2,…,k} such that no two adjacent or incident elements in $$V(G)\cup E(G)$$V(G)∪E(G) receive the same color. Let f(v) denote the sum of the colors on the edges incident with v and the color on vertex v. A proper k-total coloring of G is called neighbor sum distinguishing if $$f(u)\ne f(v)$$f(u)≠f(v) for each edge $$uv\in E(G)$$uv∈E(G). Let $$\chi ''_{\Sigma }(G)$$χΣ′′(G) denote the smallest integer k in such a coloring of G. Pilśniak and Woźniak conjectured that for any graph G, $$\chi ''_{\Sigma }(G)\le \Delta (G)+3$$χΣ′′(G)≤Δ(G)+3. In this paper, we show that if G is a 2-degenerate graph, then $$\chi ''_{\Sigma }(G)\le \Delta (G)+3$$χΣ′′(G)≤Δ(G)+3; Moreover, if $$\Delta (G)\ge 5$$Δ(G)≥5 then $$\chi ''_{\Sigma }(G)\le \Delta (G)+2$$χΣ′′(G)≤Δ(G)+2.