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 $$uv\in E(G)$$ u v ∈ 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 .