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

Abstract

A total coloring of a graph $G$ 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 $[k]$-coloring of a graph $G$ is a total coloring of $G$ by using the color set $[k]=\{1,2,\ldots ,k\}$. Let $f(v)$ denote the sum of the colors of a vertex $v$ and the colors of all incident edges of $v$. A total $[k]$-neighbor sum distinguishing-coloring of $G$ is a total $[k]$-coloring of $G$ such that for each edge $uv\in E(G)$, $f(u)\ne f(v)$. Let $G$ 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 $G$ is $\Delta (G)+2$ if $\Delta (G)\ge 14$, where $\Delta (G)$ is the maximum degree of $G$.