Neighbor sum distinguishing total coloring of graphs embedded in surfaces of nonnegative Euler characteristic

A total coloring of a graph $$G$$G is a coloring of its vertices and edges such that adjacent or incident vertices and edges are not colored with the same color. A total $$[k]$$[k]-coloring of a graph $$G$$G is a total coloring of $$G$$G by using the color set $$[k]=\{1,2,\ldots ,k\}$$[k]={1,2,…,k}. Let $$f(v)$$f(v) denote the sum of the colors of a vertex $$v$$v and the colors of all incident edges of $$v$$v. A total $$[k]$$[k]-neighbor sum distinguishing-coloring of $$G$$G is a total $$[k]$$[k]-coloring of $$G$$G such that for each edge $$uv\in E(G)$$uv∈E(G), $$f(u)\ne f(v)$$f(u)≠f(v). Let $$G$$G be a graph which can be embedded in a surface of nonnegative Euler characteristic. In this paper, it is proved that the total neighbor sum distinguishing chromatic number of $$G$$G is $$\Delta (G)+2$$Δ(G)+2 if $$\Delta (G)\ge 14$$Δ(G)≥14, where $$\Delta (G)$$Δ(G) is the maximum degree of $$G$$G.