Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz

Let G = (V,E) be a graph and φ be a total coloring of G by using the color set {1, 2, ...,k}. Let f(ν) denote the sum of the color of the vertex ν and the colors of all incident edges of ν. We say that φ is neighbor sum distinguishing if for each edge uν ∈ E(G), f(u) ≠ f(ν). The smallest number k is called the neighbor sum distinguishing total chromatic number, denoted by χ _nsd″(G). Pil?niak and Wo?niak conjectured that for any graph G with at least two vertices, χ _nsd″(G) ? Δ(G) + 3. In this paper, by using the famous Combinatorial Nullstellensatz, we show that χ _nsd″(G) ? 2Δ(G)+col(G)?1, where col(G) is the coloring number of G. Moreover, we prove this assertion in its list version.