Neighbor sum distinguishing total chromatic number of 2-degenerate graphs.

Let G=(V(G),E(G)) be a graph and ϕ be a proper total k-coloring of G by using the color set {1,2,…,k}. For any v∈V(G), let f(v)=∑uv∈E(G)ϕ(uv)+ϕ(v). The coloring ϕ is neighbor sum distinguishing, if f(u)≠f(v) for each edge uv∈E(G). The neighbor sum distinguishing total chromatic number of G, denoted by χΣ′′(G), is the smallest integer k such that G admits a k-neighbor sum distinguishing total coloring. In this paper, by using the famous Combinatorial Nullstellensatz, we determine χΣ′′(G) for any 2-degenerate graph G with Δ(G)≥6.