Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree

A proper [h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [h] = {1, 2, ... ,h}. Let w(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. For each edge uv is an element of E(G), if w(u) not equal w(v), then we say the coloring c distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing [h]-total coloring of G. By tndi(Sigma)(G), we denote the smallest value h in such a coloring of G. In this paper, we obtain that G is a graph with at least two vertices, if mad(G) < 3, then tndi(Sigma)(G) <= k + 2 where = max{Delta(G), 5}. It partially confirms the conjecture proposed by Pilsniak and Wozniak.