Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz

Let G = (V E) be a graph and i be a total coloring of G by using the color set {1,2,...,k} Let f(v) denote the sum of the color of the vertex v and the colors of all incident edges of v. We say that phi is neighbor sum distinguishing if for each edge uv is an element of E(G), f(u) not equal f(v). The smallest number k is called the neighbor sum distinguishing total chromatic number, denoted by chi(nsd)''(G). Pilsniak and Wozniak conjectured that for any graph G with at least two vertices, chi(nsd)''(G) <= Delta(G) + 3. In this paper, by using the famous Combinatorial Nullstellensatz, we show that chi(nsd)''(G) <= 2 Delta (G) + col(G) - 1, where col(G) is the coloring number of G. Moreover, we prove this assertion in its list version.