Neighbor sum distinguishing total coloring of 2-degenerate graphs

A proper k-total coloring of a graph G is a mapping from (V(G)cup E(G)) to ({1,2,ldots ,k}) such that no two adjacent or incident elements in (V(G)cup 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)) for each edge (uvin E(G)). Let (chi ''_{Sigma }(G)) denote the smallest integer k in such a coloring of G. Pil?niak and Wo?niak conjectured that for any graph G, (chi ''_{Sigma }(G)le Delta (G)+3). In this paper, we show that if G is a 2-degenerate graph, then (chi ''_{Sigma }(G)le Delta (G)+3); Moreover, if (Delta (G)ge 5) then (chi ''_{Sigma }(G)le Delta (G)+2).