Neighbor sum distinguishing total coloring of graphs with bounded treewidth

A proper total k-coloring (phi ) of a graph G is a mapping from (V(G)cup E(G)) to ({1,2,dots , k}) such that no adjacent or incident elements in (V(G)cup E(G)) receive the same color. Let (m_{phi }(v)) denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if (m_{phi }(u)not =m_{phi }(v)) for each edge (uvin E(G).) Let (chi _{Sigma }^t(G)) be the neighbor sum distinguishing total chromatic number of a graph G. Pil?niak and Wo?niak conjectured that for any graph G, (chi _{Sigma }^t(G)le Delta (G)+3). In this paper, we show that if G is a graph with treewidth (ell ge 3) and (Delta (G)ge 2ell +3), then (chi _{Sigma }^t(G)le Delta (G)+ell -1). This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when (ell =3) and (Delta ge 9), we show that (Delta (G) + 1le chi _{Sigma }^t(G)le Delta (G)+2) and characterize graphs with equalities.