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

A total coloring of a graph 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 -coloring of a graph is a total coloring of by using the color set . Let denote the sum of the colors of a vertex and the colors of all incident edges of . A total -neighbor sum distinguishing-coloring of is a total -coloring of such that for each edge , . Let 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 is if , where is the maximum degree of .