(untitled) (Journal of Mathematical Study, 2023)
For a given simple graph G = (V(G),E(G)), a proper total-k-coloring c : V(G)∪E(G)→ {1,2,...,k} is neighbor sum distinguishing if f(u) ≠ f(v) for each edge uv ∈ E(G), where f(v) = _wv∈E(G) c(wv)+c(v). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χ^''_ (G). It has been conjectured that χ ^''_ (G) ≤ ∆(G)+3 for any simple graph G. Let mad (G)=max{ 2|E(H)| |V(H)| :H⊆G} be the maximum average degree of G. In this paper, by using the famous Combinatorial Nullstellensatz, we prove χ^'' _(G) ≤ max{9,∆(G)+2} for any graph G with mad (G)